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0 there is anTOOsuch that m > nig and alH e Z From formulas (2.51) and (2.52), it follows that i = af = 0, for j < 0(Sm6) and j > 2"l~1n. Thus, from (2.56), we easily obtain that
for any c e t°°(1).
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To prove that the MSS {Ae : e 6 {0,1}} converges is straightforward. We pick a t e [0,1], write it in its binary expansion t = -6162 • • •, and define im — em + 2e m _i H h2 m - 1 ei. Then, according to (2.34), we have for alH = 0,1,... ,n-1
where x — ( c _ n + i , . . . , CQ) , while from (2.49) and (2.50) we get
Embodied in Theorem 2.4 are a number of necessary conditions for the convergence of Sa. Namely, the refinable function <j> satisfies (2.36) and the refinement equation (2.43) and the mask satisfies (2.55). Necessary and sufficient conditions follow by applying Theorem 1.2 to the matrices (2.35). Our next theorem, from Micchelli and Prautzsch [MPr], is more than sufficient to cover the convergence of the iteration (2.24)-(2.26). THEOREM 2.5. Suppose (a,j : j e Z) is a mask such that for some integers m,l with m — I > 2, we have a,j > 0, if t < j < TO, and zero otherwise, and
Then there exists a unique continuous refinable function (f> C(R) of compact support such that
and
Moreover, the stationary subdivision scheme
converges to
Note that the subdivision scheme (2.24)-(2.26) whose mask is given by (2.27)(2.28) obviously satisfies the hypothesis of Theorem 2.5 with I = 0 and m = n+l for n > 1. Therefore, this result establishes the convergence of (2.24)-(2.26). Proof. We give two proofs of this theorem. The first follows Micchelli and Prautzch [MPr] and uses Theorem 2.3 and Theorem 1.4. For this proof we first
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77
FIG. 2.11. A refinable function: ao — .8, ai = .5, aa = .2, 03 = .5, zero otherwise.
observe that it suffices to prove Theorem 2.5 when I = 0. To see this we consider the mask bj := a,j+e and observe that bj — a r + ( 2 ,-_ 1 w. r = 1, 2 , . . . . Hence, S,, converges if and only if St, converges and their refinable functions are related by the formula fa = <j>a(- + 1). Moreover, bj > 0 if and only if j — 0 , 1 , . . . . m - i. We now claim that the matrices Ae, e e {0,1}, defined in(2.35) ((. = 0, m = n) have a positive column; in fact the jth column of Ar is positive provided that it is chosen to satisfy n — 1 < f. + 1j < n. This means that (i) of Theorem 1.4 is valid (for m = 1). According to our hypothesis A f , f & {0,1}, is a stochastic matrix. Hence, by the remark following Theorem 1.4 and the fact that it has a positive column, there is a unique u£ R™ (with nonnegative components) such that (£) = 1 and A^uf = ue. To see that AQ\II = A^UQ, we argue as we dide,u in the proof of Theorem 2.3. Specifically, since (-Ao)oj = ao^Oji j' — 0 , 1 , . . . , n — 1 ( ^ = 0, m — n), we see that e = ( 1 , 0 , . . . ,0) T is an eigenvector of Aa with eigenvalue CQ. As AQ has a positive column, one must be a simple eigenvalue with corresponding eigenvector e. Moreover, keeping in mind that n > 2 we conclude that ao ^ 1. But (UQ)O = (A)Uo)o = oo(uo)o, and so it follows that (U O )Q — 0. Now we set (u 0 ) n = 0 and introduce the vector x = ( ( u 0 ) i , . . . , (u n ) n ) 7 . Then (e, x) = 1 and a direct computation shows that A'{x = x. This implies that x = Ui and also for i. = 0 , 1 , . . . , n — 1 we obtain the relations
This computation proves (ii) of Theorem 1.4 and hence Theorem 2.3 can be applied to complete our first proof of Theorem 2.5.
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Our next proof of Theorem 2.5 bypasses Theorem 1.4 and Theorem 2.3. It is self-contained and uses arguments from Cavaretta, Dahmen, and Micchelli [CDM] (which are especially useful in the analysis of multivariate SSS). We introduce the linear operator
As a consequence of (2.57) and the nonnegativity of the mask (a,j : j £ Z), Fa is a bounded linear operator on C(R) with norm two. Let ip be any function of compact support. Then F0^ is also of compact support. In fact, if if)(x) = 0 for x ^ (f-,m) then (Fatl>)(x) = 0 outside of (I, m). as well. Thus, for any Cj, j £ Z, one can easily verify the useful formula
which extends for any r 6 Z+ to the equation
Now. pick '0o to be any B-spline who support is (I,m); for instance
will do nicely. We shall show that the sequence of continuous functions
converge uniformly on R. Central to our analysis of this assertion is the semi-norm
used in Cavaretta, Dahmen, and Micchelli [CDM]. We pick any other mask 6j, j; G Z such that
and
Then for any constant p and any bi-innnite sequence c = (GJ : j £ Z) we have the formula
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By a judicious choice of p, we get the inequality
and a similar argument yields
where
Next we observe that
For the proof of this inequality we need to show that for any integers j, r such that | j - r < m — Ithere is an integer k satisfying
This conclusion would imply the strict inequality
and therefore establish (2.70). For definiteness, we suppose j < r, in other words,
Thus, there is an even integer in the interval [r — ra, j — 1} that we call 2k. This integer k satisfies all the requisite inequalities and consequently we have proved (2.70). We are now ready to establish the convergence of the sequence (2.63). We let b = (bj : j Z) be the coefficients appearing in the refinement equation for iba, namely
The coefficients b j , j Z satisfy (2.65) and (2.66) and so inequalities (2.67) and (2.68) apply. Moreover, since
we get
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from which it follows for all x e R, r 6 Z+ that
We conclude that {hr : r & Z+} is a Cauchy sequence in C(R). Therefore there is a continuous function such that
uniformly on R. In particular, <j>(x) > 0, x £ R and is zero outside of (£,m). Certainly satisfies the refinement equation
Also, (2.59) holds because for any r £ Z+ and x G R we have
There remains only the uniqueness of 0 and the convergence of the SSS. To this end, we let tf> be any other continuous function of compact support that satisfies (2.58) and (2.59). Then, for x e R it follows that
The second term above is bounded by a constant times w(0;2 r ), the modulus of continuity of , while to bound the first term it suffices to show that the SSS converges. For this purpose, we proceed as before and note that since
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we have
and consequently for j G Z
Since 4> is of compact support and c £°°(Z), the function / is uniformly continuous on K and so (2.34) follows. Remark 2.1. For later use it is helpful to note that
for x e R, when ip^ is chosen to be the indicator function of any interval of the form [p,p-\- 1) contained in [1. m]. In this case, the mask for ip$ satisfies 6, = 0, if j $ {2p, 2p + 1} and 1 otherwise. In the remainder of this chapter we develop various properties of the function d> constructed above. Generally speaking, these properties center around two issues: smoothness and shape preservation. Principally, we will be concerned with (/> constructed by the averaging algorithm (2.24)-(2.26). But as we shall see, most of these properties hold with greater generality. The function 0 constructed from the averaging scheme (2.24)-(2.26) has a variation diminishing property. Specifically, for any c — (GJ : j e Z) we let S~ (c) be the number of sign changes in the components of c. Likewise, given a continuous function / defined on R we let
be the number of sign changes of / on M. With this definition in mind, we claim that
To see this, we argue as follows. Suppose there are points x\ < - • • < xp+\ such that f(xj)(-l)> 0that f(xj)(-l)> 0, j = 1,2,... ,p + 1, wherej = 1,2,... ,p + 1, where
According to (2.34), there are integers ji r < • • • < j0+1 r such that
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and in particular, S~(S^c) > p. However, because Sac is obtained from c by successively averaging its components, we have S~(Sac) < S~(c) and generally by induction on r it follows that S~(S£c) < S~(c). Consequently, S~(c) > p, as claimed. Next we point out that 0 is Holder continuous. We base our remark on the following general fact already used in Chapter 1, which we restate here. LEMMA 2.1. Let {/„ : n & Z+} be a sequence of real-valued continuous functions on M. Suppos0 (l,oo), p & (0,1), and M,N are positive numberse p such that for all x, y 6 R, n e Z + ,
W |/n(z) - /n(y)| < M(ppo)n\x -y,and
(iin+1(x)-fn(x)\
uniformly on M, and for n e (0, oo) defined by p — p^ we have
for some constant 6 > 0. COROLLARY 2.1. Let 4> be as described in Theorem 2.5. Then there is a constant R > 0 such that
where p — 2~^ and p is defined in (2.69). Proof. We apply Lemma 2.1 to the sequence {hr(x) : r e Z+} used in the proof of Theorem 2.5. We already noted there that Using the derivative recurrence formula for the B-spline, see (2.4) and (2.5), we also get
and so
Next we recall a fact proved in Micchelli and Pinkus [MPi]. PROPOSITION 2.1. Let (/> be as described in Theorem 2.5. Then <j>(x) > 0 for all x 6 R with strict inequality if and only if x 6 (l,m). Proof. Since (f> is constructed as a limit of nonnegative functions that vanish outside of (l,m), we need only prove
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and so by the refinement equation for <j>
Hence 0(x) > 0 for x G IQ := 2~l(m + £ - l,m + f + 1 ) and the length of I0 is one. Next we will show how to "propagate" the positivity of (j> to all of (£, TO). To this end, suppose that 0(y) > 0 for some y e M. Then for any integer k 6 [f., m\. we have
Consequently, if 0 is positive on an interval (a,b] of length at least one, it is positive on the interval 2~1(a + l, b + m). Starting with /0 this process produces intervals Ir — (ar,br),ar := 2~ 1 (a r _i +£), br := 2~ 1 (6 r _i + TO) on which
Hurwitz polynomials and stationary subdivision.
More can be said about the "shape" of the refmable function 4> for the averaging scheme (2.24)-(2.26). For instance, we can resolve the following interpolation problem. Given integers i\ < • • • < ip, real numbers x\ < • • • < xp and y\,...,yp determine a function / of the form
such that
To study this problem we set
Also we say a Laurent polynomial is a Hurwitz polynomial provided that it has zeros only in the left half-plane. The following theorem is from Goodman and Micchelli [GM]. THEOREM 2.6. Let a = (a,j : j & Z) be a mask that satisfies the hypothesis of Theorem 2.5 such that
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is a Hurwitz polynomial. Then
and
for all real numbers x\ < • • • < xp and integers ii < • • • < ip. Moreover, strict inequality holds in (2.75) if and only if Xj — ij £ (I, m), j = 1,... ,p. Note that the averaging scheme (2.24)-(2.26) surely satisfies the hypothesis of the theorem. In fact, in this case, according to (2.27) and (2.28), a(z) only has negative zeros in C. Using Proposition 2.1, another way of describing Theorem 2.6 is to say that the matrix
is nonsingular if and only if its diagonal elements are nonzero. Also observe that whenever a Laurent polynomial is a Hurwitz polynomial all its coefficients are of the same sign. Therefore the mask (aj : j G Z) of Theorem 2.6 has nonnegative entries. For the proof of Theorem 2.6, we first point out that we already proved (2.74) for the averaging scheme (2.24)-(2.26). The general case covered by Theorem 2.6 can be proved by using the determinantal inequalities (2.75) just as we proved Corollary 1.3. The proof of (2.75) is longer and, as in the proof of Theorem 2.5, we assume without loss of generality that I — 0 and m = n + I with n > 1. We define the bi-infinite matrix
and as usual denote the rth power of D by Dr. The essential step in the proof of (2.75) is the next result of some independent interest. PROPOSITION 2.2. Let a = (aj : j 6 Z) be any mask such that aj — 0 if j £ {0,1,..., n + 1}, aoa n +i > 0, and a(z] is a Hurwitz polynomial. Then
for all ii < • • • < ip,ji < • • • < jp, n = 1 , 2 , . . . , and strict inequality holds if and only if 0 < je - Tie. < (2r - l)(n + 1), £ = ! , . . . ,p. The proof of this result is somewhat involved. We postpone it until after we demonstrate how it leads to the proof of (2.75). Proof of Theorem 2.6. First we prove that the determinants in (2.75) are nonnegative. For this purpose, we first observe that (2.75) holds if <j> were the indicator function of [0,1). Let us confirm this observation. In this case, there are two possibilities: either x\ — ie £ [0,1) for all £ = l , . . . , p , so that the
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determinant (2.75) is zero, or x\ — if e [0,1) for some £, 1 < f. < p. Since i\ < • • • < ip, then Xk — ij $. [0,1) for k > 1 and j < (.. as well as for fc = 1 and j > f . In the first case,
while in the second case
If t > 2 then the determinant is zero. Otherwise, I = 1 and the determinant in (2.75) has the same value as the minor corresponding to the last p — I rows and columns. In this way, we see, inductively on p, that the determinant is either zero or one. Next we consider the sequence of functions
where we choose V'o to be the indicator function of [0,1). In view of Remark 2.1 we have lim hr(x) = (x). x G R. Moreover, by (2.46) we have
and so by the Cauchy-Binet formula (see Lemma 1.6 of Chapter 1) we have
From this formula, and (2.76) of Proposition 2.2 with r — 1, it follows, inductively on k, that
whenever x\ < • • • < xp and i\ < • • • < ip. Letting k —> oo proves the desired inequality (2.75). To resolve the case of equality in (2.75) we use the full value of Proposition 2.2. We begin with the sufficiency of our conditions. Thus we suppose 0 < Xk — ik < n + 1. k = 1.... ,p and choose r large enough so that I1 ik < 1rXk — n — 1 and Vk < 2rife + (2r-l)(n + l), k = 1.... ,p. Now pick any integers j^,. < • • • < jp,,-x such that
Consequently, we get the inequalities
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and
According to the refinement equation (2.58), we have
Hence, by (2.76) of Proposition 2.2, inequality (2.75) (which we already proved), and the Cauchy-Binet formula, we get
The minor of Ar above is nonzero by Proposition 2.2 and (2.78). On the other hand, the minor of <j> has positive diagonal elements by (2.79) and Proposition 2.1, and for r —*• oo it has zero off-diagonal elements, for the same reason. This proves the sufficiency of the theorem. For the necessity, we first suppose x3 - is < 0 for some s, 1 < s < p. Then xr - ik — 0 for k = s,... ,p,r = 1,..., s. Thus the matrix in (2.75) has a zero entry in the ( r , k ) position, r = l , . . . , s , k = s,...,p. Consequently, the last p — s + 1 columns must be linearly dependent and the minor in (2.75) is zero. For a similar reason, it is zero if xs - is > n + 1 for some s, 1 < s < p. We now turn our attention to the proof of Proposition 2.2. The first case to deal with is r = 1. This is a result of Kemperman [Ke], which we now prove. As described earlier, a polynomial
is said to be a Hurwitz polynomial if all its zeros are in the (open) left halfplane. Recall that all the coefficients of a Hurwitz polynomial are nonzero and necessarily of one sign. We assume for definiteness that
We adhere to the convention of writing the coefficients of d(z) in reverse order and, in keeping with our previous notation, always make the identification
Central to our discussion is the bi-infinite Hurwitz matrix
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where
and dj is set equal to zero for j < 0 or j > n + 1. The Roth-Hurwitz criterion states that d(z) is a Hurwitz polynomial if and only if for p = 1, 2 , . . . , n + 1,
cf, Gantmacher [G, p. 194]. Note that for p > n + I the last column of the determinant A p is zero and so Ap = 0, for p — n + 1,... . We need the following factorization procedure for Hurwitz matrices, LEMMA 2.2. Suppose d(z) = £]?_o^n-j£ J > n > 1, is a Hurwitz polynomial. Then there is a positive constant c such that the matrix C = (cij)ij£%, defined by
has the property that
where H' is a Hurwitz matrix corresponding to some Hurwitz polynomial of degree
Equivalently, we have
and also
Thus it follows that
When c is specified, equations (2.81)-(2.82) determine all d'^j G Z. To pin down c, we use the requirement that d'- = 0 for j < 0 and so we must have
CHAPTER 2
that is, c = do/di > 0. With this choice of c, we define d'^ for all j e Z using equations (2.81) and (2.82). It is an easy matter to see that d'^ — 0 for j < 0 and j > n — 1. Therefore, it remains to prove that
is a Roth-Hurwitz polynomial. For this task, we use the Roth-Hurwitz criterion. To this end, we again consider the determinant
Multiply the first, third, ... , rows of A p by c and subtract from the second, fourth, ... , rows to get
The first column of the matrix above has zero elements except for d'0. Hence we get From this formula and the Roth-Hurwitz criterion, we conclude d'(z) is a Hurwitz polynomial. To make use of Lemma 2.2 we use the following lemma. LEMMA 2.3. Let C = (cij) l)je z be a one-banded upper triangular matrix, i.e., Cij = 0, wheneverj
Actually, the first equation above holds for ar, ji,..., jr because Cirjrll ii,..., i is zero, if jr — ir < 0 or jr — ir > 1. Proof. For p = 1, the formula above is true by hypothesis. Suppose p > 2 and ji < i\. Ther, r = I,..., s and so the first column of the determinant isn j\ < i
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zero. If ii +1 < Ji then the first row is zero. Hence in either case the determinant is zero. When i± = j\ then all the elements in the first column are zero except perhaps c^^. Thus
Similarly, if j\ = i\ + I we get
Using these formulas the lemma follows by induction on p. LEMMA 2.4 [Ke]. Let
be a Hurwitz polynomial. Then the corresponding Hurwitz matrix H (d2j-i)tjgz is totally positive, that is,
=
for all integers ii < • • • < ip,ji < • • • < ip and strict inequality holds in (2.83) if and only if
Proof. The proof is by induction on n. The case n = 0 is elementary. Now assume n > I and that the result is true for n — 1. Using Lemma 2.2, Lemma 2.3, and the Cauchy-Binet formula we have
where c.ik > 0, if k = i + 1 or k = i = an even integer and zero otherwise. By induction, H' is totally positive and therefore so too is H. Also, by induction we have strict equality in (2.83) if and only if there exist integers k\,... ,kp such that k\ < • • • < kp, 0 < 2je — kg < n — 1, i = 1,... ,p and for each r = 1 , . . . ,p either kr = ir + 1 or kr = ir = an even integer. From this fact, (2.84) easily follows. Lemma 2.4 proves Proposition 2.2 in the case r = 1. To elaborate on the details, we relate the matrices A and H. Since aj = dn+\-j we get
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and so
with strict inequality if and only if 0 < 2(n + l + it)-(n + l+je) < n +1, which simplifies to 0 < jf -2if
Thus, by induction on p, we see that all determinants of Ar+l are nonnegative and moreover, the determinant of Ar+l on the left-hand side of the above equation is positive if and only if there exist integers j\ p such that< ••• < j
and
Clearly, a necessary condition for the solution of these inequalities is that
The fact that these inequalities are also sufficient for the existence of j\,..., jp satisfying the above inequalities would advance the induction hypothesis and prove Proposition 2.2. This result is proved next. LEMMA 2.5. Let r,p Z+\{0} and n Z. Given integers k\ < • • •
and
if and only if
Proof. It is clear that (2.85) and (2.86) imply (2.87). The converse is proved by induction on p. To this end, we assume that it holds for all positive integers smaller than p. Now choose j\ to be the smallest integer satisfying the lower bounds on ji in both (2.85) and (2.86). Then successively choose je, (, = 2 , . . . ,p, to be the smallest integer, not only satisfying the lower bound (2.85) and upper
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bound (2.86), but also greater than jt-\- We know by induction that there are integers ti < • • - < i p _i that satisfy
and
Hence by our choice of j i , . . . ,jp one obtains je < tg, i = 1 . . . . ,p - 1 and consequently, it suffices to prove that
and
to advance the induction hypothesis. We first observe, without loss of generality, that we can assume that ji, • • • , j p are consecutive integers. Suppose to the contrary that for some m, 1 < m < p— 1, we have jm+\ > jm + 2. Then jm+i is necessarily the minimum integer that satisfies the inequalities
and
Hence we may argue as above and conclude by the induction hypothesis that je, I — m + 1,... ,p, would satisfy (2.85) and (2.86). In particular, inequalities (2.90) and (2.91) would follow and the proof would end. Thus we need only consider the case that,
Let us now check the validity of (2.90). For p > 2, we have
as required. When p — I we consider, in turn, two possibilities: cither j\ -1 < 1i\ or 1r(ji — 1)
Simplifying this inequality gives ji < 1 — 2~ r + 2«i + n + 1. which again proves (2.90).
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There remains the proof of (2.91), which we verify by contradiction. Thus we suppose to the contrary that
Using (2.92) and (2.85), inequality (2.93) gives ji + p - 1 > 2ip > 2ii + 2p - 2 or, in other words, ji - 1 > lii- Thus by our choice of ji, we conclude that
This inequality, with the lower bound on pj in (2.85) and (2.87), gives the inequalities
and so p < n + 2. Using (2.94) and the fact that k\ < kp - p + 1, we obtain
which together with (2.93) implies that p > n + 2. Hence, we conclude that p = n + 2. Next, we choose integers s, t such that
and 0 < t < 2r+1. Using (2.87) again, we have
which gives ii+n+l>s + 2~r~l(t + 1) > s and so ii + n > s. Also, by (2.87) we get 2~r~lkp >ip>ii+p—l=ii+n+l, which implies that
Furthermore, from inequality (2.94) we obtain
or, equivalently,
Therefore (2.95) implies that
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which means that jp — j\ + p — 1 — j\ + n + 1 < 1~rkp. This contradicts (2.93) and proves the result. D We now return to the issue of smoothness for the function constructed in Theorem 2.6. There is a simple criterion available to determine how many continuous derivatives has on K. THEOREM 2.7. Let a — (aj : j £ Z) be a mask that satisfies the hypothesis of Theorem, 2.6. Then 0 e C lfc (M) for some k > 1, if and only if a(z) has a zero of order k + I at z = — 1. We base the proof of Theorem 2.7 on two facts from Cavaretta, Dahmen, and Micchelli [CDM]. PROPOSITION 2.3. Let a = (dj : j Z) be a mask such that \{j : a, ^ 0}| < co. // there exists a nontrivial continuous function f and a vector c = (cj : j G Z) such that
then a(-l) — 0 and a(l) = 2. Proof. We assume without loss of generality that a,j = 0 for j < 0 and j > n. Our hypothesis and Theorem 2.4 imply that the MSS determined by the matrices (2.35) converges. Hence by (1.39) of Theorem 1.2 we have Aee - e, e 6 {0,1}, which is easily seen to be equivalent to the desired conclusion. We also need the following proposition. PROPOSITION 2.4. Let a = (o,0 : j 6 Z) be a mask such that \{j : a,j ^ 0}| < oc. If Sa converges in the sense of (2.34), and has an associated refinable function
likewise converges and has a refinable function ip given by
Proof. Our first observation is the formula
where A is the forward difference operator defined earlier in Theorem 2.2. This equation means that
Equivalently, multiplying both sides of this equation by z\ z C\{0}, and summing over i 6 Z gives
which simplifies to the definition of h(z) given in Proposition 2.4. Using formula (2.96) repeatedly yields the equation
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We also need to recall from (2.52) that there is an integer q > 0 such that for all r = l,2,...
and so
Since 50 converges we conclude that for any e > 0 there is an m such that
for r > m and j Z. For any k 6 Z, replace j by fc + j above and sum the left-hand side of the inequality above over all j < 0 for which it is nonzero. There are at most q"2r such integers, for some constant q', independent of r. Hence we get from (2.97) that
We also observe that
If j gives a nonzero summand above then either \x + j/2 r | < q or x +1\ < q for some t e [(j — l)/2 r , j/2r}. In either case we get
and so independent of x, there are at most 1q + 3 nonzero summands. Consequently, we conclude that
Combining these inequalities we obtain for all k £ Z the inequality
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Clearly the upper bound goes to zero as r —» oo. Moreover, for any c G £°°(Z) with the function g defined as
we have
Again, the number of nonzero summands in the sum above is < 2q + 1 (independent of r) and so the above quantities go to zero as r —> oo. We now have at hand the necessary facts to prove Theorem 2.7. Proof of Theorem 2.7. First, we suppose that a(z] can be factored in the form
where 6(1) = 2 and 6( —1) = 0. We define Laurent polynomials bo(z), b i ( z ) . . . . , bk(z) recursively by setting
and
Then each bf(z) satisfies the hypothesis of Theorem 2.6. Therefore, the corresponding subdivision scheme Sbt converges. Let
Since a(z) = bk(z), we have
where 6 is a Laurent polynomial. Since the zeros of b(z) are certainly in the left half-plane, we conclude from Theorem 2.6 and Proposition 2.4 that S& converges and has a refinable function in Ck~l(R). If k > 1 we apply the same procedure to 6 and inductively arrive at the desired conclusion, by induction on k.
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2.6.
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Wavelet decomposition.
Although it is not central to the theme of this chapter we describe an application of Theorem 2.6 to wavelet decomposition. Unfortunately, we must refrain from describing the background motivation for our remarks because this would take us too far afield. For an excellent account of this important subject we refer to Chui [C] and Daubechies [D]. We call the function , constructed in Theorem 2.6, a ripplet. Ripplets are therefore created from Hurwitz polynomials. We will follow Micchelli [Mi] and show how easily ripplets lead us to wavelets. To this end, we introduce the sequence fi = (fj,j : j £ Z) defined by the equation
where 4> and a = (aj : j Z) are as in Theorem 2.6 and assume, without loss of generality, that £ = 0 and m = n + 1. Since $ is positive on (0,n + 1) and zero otherwise, (ij is positive for j {—In — 1,..., n} and zero elsewhere. The vector // leads us to the function
It is the study of this function ip that will occupy us for the remainder of the chapter. Our main goal is to show that the family of functions
provides an orthogonal decomposition of L 2 (R). Moreover, every / e L 2 (R) can be efficiently represented in terms of these functions. We begin our discussion of the function V by observing that ip(x) is zero for x 0 (—ri, n + 1). To describe additional properties of tp, we need to accumulate some facts about <j>. For this purpose, we introduce for every k £ Z the space
Each Vk is a closed subspace of L 2 (R). For the proof of this fact, we note that since (j>(x),..., (j)(x + n) are linearly independent on [0,1], it follows that there are constants K,J>0 such that
From this inequality it easily follows that Vk is a closed subspace of L 2 (K).
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In view of the refinement equation (2.58) satisfied by 4>, the spaces (2.98) are nested, viz.
Moreover, we claim that
To prove the first part of (2.100), we let g e Vk, for all k e Z. Thus, for every k e Z, there is a d = (dj : j e Z) e £ 2 (Z) such that
Let
and use the Cauchy-Schwarz inequality to get
so that letting fc —> — oo proves g = 0. The second claim in (2.100) follows by noting that since (p is continuous, of compact support, and satisfies (2.59), it follows that whenever / is likewise continuous and of compact support the sequence of functions
are also of compact support and converge uniformly to /. Since continuous functions of compact support are dense in L 2 (M), the result follows. Next we let Wk be the orthogonal complement of Vk in Vj,+i. Two important properties of Wk, k e Z, follow from equations (2.99) and (2.100), namely,
and
For instance, to prove the first fact we observe that if k < k' then
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For the second claim we need only show that there is no nontrivial function / e L 2 (R) orthogonal to all Wk, k 6 Z. To this end, let / L 2 (R) be orthogonal to all Wki k e Z. Pick e > 0, and choose a Vk0 Vk0 such that
where || • || is the standard norm on L 2 (R),
Hence we conclude that ||/|| 2 —2(/, Vk0) < £2. Next, we write Vk0 = Vk0-i+Wk0-i, where vko-i £ Vfe 0 -i and wko-i 6 W fco -i- Then
and
In this way for every integer k < &o there is a i>fc G Vk such that
and
Using equation (2.100), and the bound on Vk above, we see that lim^oo Vk — 0, weakly in L 2 (E). This implies, by the inequality above, that ||/|| < e. Since e is arbitrary, it follows that / = 0. For later use, we need to record some facts about the bi-infinite sequences fj, defined earlier. For this purpose, we use the following notation. Given any Laurent polynomial
we split it into its "even"
and "odd"
parts, so that
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99
We also need the sequence g — (g,j : j G Z) denned by
Using the refinement equation for <j> we obtain the formula
Also, referring back to the definition of ju and again using the refinement equation for 4> likewise provide the companion formula
PROPOSITION 2.5. Let a = (a0 : j Z) satisfy the hypothesis of Theorem 2.6. Then
and g(z) has only negative zeros on C\{0}. Furthermore, HQ(Z) and fii(z) have no common zeros for z C\{0}. Proof. First we note by the Cauchy-Binet formula and Theorem 2.6 that the bi-infinite matrix (gi-J)lijGz is totally positive. Therefore, by a result of Aissen, Schoenberg, and Whitney [ASW], we conclude that g(z) only has negative zeros. To show that the inequality (2.102) holds, it suffices to demonstrate that g(ei9) ^ 0 for 9 E R, since g(ei0) is real for all Q e R and also positive for 9 = 0. Suppose to the contrary that g(el'e) = 0 for some 0 & R. Then for all k <E Z
Multiplying both sides of this equation by e the formula
lk6
and summing over fc e Z gives
Therefore for x e [0,1],
which contradicts the linear independence of the functions 4 > ( x ) , . . . . (p(x + n) on [0,1]. Finally we show that /J,Q(Z), ^i(z) have no common zeros. Suppose to the contrary that Ho(z) = ^i(z) — 0 for some z e C\{0}. According to (2.101), we have
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and
We let r 2 := z and observe that the determinant of this linear system (in O,Q(Z~I) and a\(z~1)) is
Since g(z) has only negative zeros and the vector g has only nonnegative components, it follows that A ^ 0. Therefore, we conclude that ao(^~ 1 ) = a\(z~l) = 0 or, equivalently, a(z~~l) = a(-z~l) = 0. This is a contradiction, since a(z) is a Hurwitz polynomial. We now arrive at the main property of t/j that we wish to describe here. THEOREM 2.8. Let a = (a, : j 6 Z) satisfy Theorem 2.6. Then
and there exist two sequences b = (bj : j £ Z), c = (cj : j £ Z) that decay exponentially such that
for all n 6 Z and a; R. Moreover, ip is the function of minimal support in V\ that is orthogonal to V0. We remark that the orthogonality of the spaces Wk, k e Z imply that for any j, j, k, k' Z, with k ^ k'
Thus i/} is what is commonly called a pre-wavelet. Proof. First, let us show that tp J_ VQ. To this end, we compute the inner product
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101
Next, using the refinement equation for 0 and the definition of ifr, we see that to prove (2.103), it suffices to show
where we have set
The solution of equation (2.104) is facilitated by the following lemma. LEMMA 2.6. Let a,b,c,d e £ 1 (Z) (absolutely summable sequences) satisfy (2.104). Then
and
Conversely, if (2.105) holds for a,d ll(Tj then b,c given by (2.106), (2.107) are likewise in ^(Z) and satisfy (2.104). Proof. Let z e D, multiply both sides of (2.104) by zn, and sum over n e Z. This gives us the equivalent system of equations
Let e {0,1}, and write £ in the form I — 2r + e. Then (2.108) simplifies to the matrix equation
Thus we conclude that A(z) ^ 0 and also by Cramer's rule we obtain
This formula proves (2.106) and (2.107). The converse statement of the lemma is obtained by reversing the above steps, using (2.105) and Wiener's lemma, cf. Rudin [R. p. 266], to ensure that 6, c given by (2.106), (2.107) are in £l(Z.).
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Returning to the proof of Theorem 2.8, we verify that for the choice of d given above that A 7^ 0. In fact, do(z) = zni(z), d\(z) = — ^o(z) and thus
Therefore, in view of Proposition 2.5, we conclude that indeed A(z) 7^ 0 for z e D. To finish the proof, we shall establish that ip is the function of minimal support in Vi which is orthogonal to VQ. By this we mean the following. If ft e Vj. is orthogonal to Vb and vanishes outside of (—n,m) where m is some number less than n + 1, then h = 0. Using the linear independence of the integer translates of <j>, we may as well assume m is an integer and that for some sequence d = (dj : j e Z) with dj — 0, whenever j g {—In — 1 , . . . , m — 1},
Now, since ft _L VQ, we conclude that
Proposition 2.5 states that /io(z),A*i(z) have no common zeros in C\{0}. Therefore there exist Laurent polynomials / and r such that
cf. Walker, [W, Thm. 9.6, p. 25]. Define q := d0f - d±r and observe that
or, equivalently, d(z) = -zp,(—z}q(z2). We wish to show that q(z) = 0, for all z e C. Suppose to the contrary that q ^ 0. Then, for some integers, (. < s. and constants < & , . . . , < ? « we have
with qiqs / 0. This implies that
with d-2n-i+2edn+2s+i ¥^ 0- However, dj = 0 whenever j ^ { — 2 n — 1 , . . . ,m — 1}, which leads • us to the contradiction that (. > 0 and s < -1.
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A disadvantage of the representation (2.103) is that both sequences b, c are infinitely supported, albeit decaying exponentially, even though both 4> and ip are of compact support. The next theorem provides a nonorthogonal decomposition of V\ through VQ that remedies this difficulty. THEOREM 2.9. Let a = (cij : j 6 Z) satisfy Theorem 2.6. There, exist a function 0 e Vi, supp # = (0.2n + 1) and sequences of finite support b = (bj : j <E Z), c = (c.j : j <E Z) suc/i £/ia£
/or all n <E Z and x R. Proof. As we already pointed out in the proof of Proposition 2.5, the polynomials aa(z),ai(z) have no common zeros for z e C. Therefore there exist polynomials d $ ( z ) , d\(z) such that
Consequently, the function
satisfies the requirements of the theorem. An advantage of the decomposition in Theorem 2.6 is that every function of finite support in Vj is a unique linear combination of functions of finite support from V(4>) and V(6). References [ASW]
[C] [CDM] [Ch] [Dj [deR] [G] [GM]
[Ke]
A. AISSEN, I. SCHOENBERG, AND A. WHITNEY. On the generating functions of totally positive sequences. I. J. d'Anal. Math.. 2 (1952), 93-103. C. K. CHUI. An Introduction to Wavelets, Academic Press, Boston, 1992. A. S. CAVARETTA, W. DAHMEN AND C. A. MICCHELLI, Stationary subdivision, Mem. Arner. Math. Soc., 93, #453, 1991. G. M. CHAIKIN, An algorithm for high speed curve generation. Computer Graphics and Image Processing. 3 (1974), 346-349. T. DAUBECHIES, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. G. DE R.HAM. Sur une courbe plane, J. Math. Pures Appl.. 39 (1956), 25-42. F. R. GANTMACHER. The Theory of Matrices, Vol. 2, Chelsea Publishing Company, New York, 1960. T. N. T. GOODMAN AND C. A. MICCHELLI, On refinement equations determined by Polya frequency sequences. SIAM J. Math. Anal.. 23 (1992), 766-784.
J. H. B. KEMPERMAN. A Hurwitz matrix is totally positive, SIAM J. Math. Anal., 13 (1982), 331-341.
104 [LR]
[Mi] [MPi] [MPr]
[R] [Ri] [Sc] [Sch] [W]
CHAPTER 2 J. M. LANE AND R. F. RIESENFELD, A theoretical development for the computer generation of piecewise polynomial surfaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2 (1980), 34-46. C. A. MICCHELLI, Using the refinement equation for the construction of prewavelets, Numerical Algorithms, 1 (1991), 75-116. C. A. MICCHELLI AND A. PINKUS, Descartes systems from corner cutting, Constr. Approx., 7 (1991), 161-194. C. A. MICCHELLI AND H. PRAUTZSCH, Refinement and subdivision for spaces of integer translates of a compactly supported function, in Numerical Analysis, eds., D. F. Griffiths and G, A. Watson, eds., Pitman Research Notes in Mathematical Series, Longman Scientific and Technical, Harlow, UK, 1987, 192-222. W. RUDIN, Functional Analysis, McGraw-Hill Book Company, New York, 1993. R. F. RIESENFELD, On Chaikin's algorithm, Computer Graphics and Image Processing, 4 (1975), 304-310. I. J. SCHOENBERG, Cardinal Spline Interpolation, SIAM, Philadelphia, 1973. L. L. SCHUMAKER, Spline Functions: Basic Theory, John Wiley and Sons, New York, 1981. R. J. WALKER, Algebraic Curves, Princeton University Press, Princeton, N.J., 1950.
CHAPTER 3
Piecewise Polynomial Curves
3.0. Introduction.
Basic concepts from the differential geometry of curves are reviewed. These notions give us a means to develop various methods to join together the segments of piecewise polynomial curves that have geometric significance for the composite curve. We demonstrate that these methods can be conveniently described in terms of (connection) matrices with easily identifiable properties. To create curves with these desirable properties we study spaces of piecewise polynomials specified by connection matrices and knots. We show that totally positive connection matrices lead to piecewise polynomial spaces with B-splinc bases that arc variation diminishing. Several algebraic properties of these Bspline bases are obtained, including polynomial identities, dual functioiials, arid knot insertion formulas. 3.1.
Geometric continuity: Reparameterization matrices.
The two previous chapters contain methods useful for the construction of curves by iterative methods. In contrast, in this chapter we focus on modeling curves by a finite number of distinct polynomial curve segments. The distinction between smoothness of a curve in a given parameterization and its visual or geometric smoothness motivates the point of view taken here. It was pointed out early by those interested in geometric modeling, first by Sabiri [S], arid then by Manning [M], that when modeling a continuous planar curve x(i) = (x(t).y(t)) by cubic splines, each component of x ( t ) need not be twice continuously differentiable in its parameterization to obtain a visually pleasing curve. For instance, at a knot of x(£), say, t — a, for x to have a continuous unit tangent requires only that there be a positive constant m\ such that Here, as elsewhere in this chapter, we assume that x(t) is always a regular curve so that the derivatives (right and left) at any point in its parameter domain are nonzero. With this proviso, the unit tangent is given by 105
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and so by the above equation it follows that t(a + ) = t(a~). Conversely, if t is continuous at a then indeed x'(a + ) = mix'(a~) for some positive constant mi > 0. Geometric continuity of order two requires, by definition, not only a continuous unit tangent but also a continuous curvature vector (the second derivative of the curve relative to arc length parameterization). For a planar curve, recall that the curvature vector is given by the expression
Therefore, if there is another constant m-2 such that both equations
and
hold, then indeed both t and k are continuous at a, and conversely. These equations were derived in [M]. It was Barsky [B\] who was the first to introduce a local basis for cubic piecewise polynomials that satisfy at each knot a the conditions
where the same value of mi and m? is used, independent of the knot a. Barsky called mi bias and m^ tension, and his basis f3-splines. In [#2], Barsky examined practical implications of modeling both curves and surfaces by /^-splines. He argues that the introduction of bias and tension into the design paradigm gives added flexibility that has geometric significance and usefulness. Thus a curve given in the form (1.2) of Chapter 1,
where f\,...,nf are /^-splines, has the advantage that the control points of x can be kept fixed while the bias and tension can be adjusted to alter the shape of the curve. This was done in Fig. 3.1. First an ordinary cubic spline was used on the control polygon. Then an adjustment of the m2 value was made (either increased or decreased and indicated by a + or — value, respectively) to finally yield the desired shape.
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FIG. 3.1.
107
Modeling with 0-splines.
The first mathematical treatment of /9-splines for arbitrary degree was given by Goodman [Go]. At each knot, his connection relations were
where he allows m i , . . . , mn to change from knot to knot. He obtained a local basis for his /3-spline and studied interpolation by linear combinations of such piecewise polynomials. In general terms, the principal issue we are concerned with in this chapter is the sense in which different polynomial segments of a curve are joined together and how that is reflected in geometric properties of the curve. Initially we are interested in the behavior of a curve x denned on [0,1] at one point a within
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its domain of definition. All our curves are required to be continuous and have continuous derivatives of all orders, up to and including n, to the left and right of a G (0,1). It is convenient to think of the curve
as composed of two "smooth" segments x- and x+, which denote x restricted to [0,a) and (a, 1], respectively. We form the vector of derivatives
For the value of Dn[x](t) at t — a, we assume that as t —> a" or t —•> a+ the first n derivatives of x converge to a limit and denote them by £) n [x_](a) and -D n [x + ](a), respectively. Now, fix an (n + 1) x (n + 1) nonsingular matrix M. We tie together the two segments x_ and x+ by requiring that the constraint
is satisfied. To explain the geometric motivation for such a requirement we review some basic terminology about the differential geometry of curves. First, it should be understood that, as mentioned above, we always suppose that both x_,x + are parameterized as regular curves on [0,a],[a, 1], respectively. This means that
and
If the matrix M is the identity matrix the curve x is given by an n-times continuously differentiable regular parameterization throughout [0,1]. Even if this were not the case x may be visually smooth, that is, lack of smoothness is only a consequence of a specific parameterization, not an intrinsic property of the curve itself. Let us see what happens if we reparameterize the left segment of x. An admissible parameterization of x_ is an n-times continuously differentiable function r defined on some neighborhood of a. That is. r : [b,c] —> [b, c], for some b < c, r(a) = a, and r'(t) > 0 for t e [b, c] where a £ (b, c). The curve segment
formed by composing x with r, joins together with x+ to form an n-times continuously regular parameterization of x if and only if
To interpret these equations in the form (3.3) we need to compute the derivatives of y by the chain rule. For this purpose, we introduce the (n +1) x (n+l) matrix
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109
R = R(r) defined by the property that for all / having n continuous derivatives in a neighborhood of a
Here for denotes the composite function
We say R is a reparameterization matrix whenever there exists a function r which is n-times continuously differentiable in some neighborhood of a such that r(a) = a, r'(a) / 0, satisfying (3.6). Alternatively, R is determined by any real numbers r 1 ; . . . , rn with r\ > 0 where we choose
in (3.6) and i>, c sufficiently close to a. A formula for the elements of R is known, the Faa di Bruno formula, cf. Goodman [Go] or Knuth [Kn, p. 50]. Alternatively, they can be computed inductively in the following way. The matrix R in (3.6) is a lower triangular matrix. When we increment n by one to n + I , the new matrix is obtained by appending a new row and column to the previous matrix. Thus it is convenient to think of R in (3.6) as the first n + 1 rows and column of an infinite lower triangular matrix. For notational simplicity we also use R to denote this matrix. Therefore, by differentiating both sides of (3.6) with respect to a, it follows that the rows of R can be computed successively by setting Rij ~ Rij(a)\r(a)=a where R0j(a) = j(r(a) - a) J , j - 0,1,..., and
In particular, we have for n = 2
while for n — 3, R is given by
In general, the first column of R is given by R^ = 6io, i = 0,1,..., n, its second column by RH = r.M i = 1 , . . . , n, and its diagonal elements by
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In particular, R is nonsingular. We will say more about this matrix later. At the moment the following geometric characterization of the condition (3.3) for M = R is of interest. PROPOSITION 3.1. Let x : [0,1] —> Mrf be composed of two curve segments
where each segment is given by a regular n-tirnes continuously differentiable parameterization. Then there exists an admissible reparameterization of x_ so that x becomes a regular n-times continuously differentiable curve in this parameterization if and only if there exist constants r j , . . . ,rn, r\ > 0 such that
Proof. We have essentially proved this simple result. The discussion above establishes the necessity. For the sufficiency we choose
and note that r is an admissible parameterization in any sufficiently small neighborhood of a, when ri > 0. The prevailing terminology in the literature is that a curve satisfying (3.11) is called geometrically continuous of order n, denoted by GC n . We formally record this definition below. DEFINITION 3.1. Let x : [0,1] —> Rrf be a curve composed of two segments
for some a & (0,1) where each segment is given by a regular n-times continuously differentiable parameterization. Then x is said to be geometrically continuous of order n if there is an admissible reparameterization r : [b, a] —> [0. a] for some b 6 (0,a) such that the curve y ( t ) = x_(r(t)), t e [b,a], joins with x+ with n continuous derivatives at t = a, that is,
3.2.
Curvatures and Frenet equation: Frenet matrices.
Let us go back to the reparameterization matrix R. We summarize some of the most important properties of R:
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and
These conditions define a larger class of nonsingular matrices which was given a geometric interpretation by Dyn and Micchelli [DM]. To describe this observation we recall some results about the geometry of curves discussed in Spivak [Sp] and also in Gregory [Gr]. We begin with a continuous regular curve x : [0,1] —> Rd having d continuous derivatives. Note that the number of continuous derivatives of x and the dimension of its range are the same. Also, we assume that for each t [0,1] the set of vectors are linearly independent. The Frenet frame of the curve x at the point t is a right-handed orthonormal system of vectors { f i , . . . , f^} formed from (3.12) by a Gram-Schmidt process. These vectors can be denned recursively. Specifically, for i = 1 , 2 , . . . ,d- 1 set where ( , ) denotes the standard inner product on Rd and define
(When i = 1, the sum above is assumed to be zero.) The vector fd(t) is chosen so that
For each i = l,...,d - 1 the Gram-Schmidt process guarantees that the vectors f j . ( £ ) , . . . ,fj(i) span the same linear subspace spanned by the vectors x ^ ) ( i ) , . . . , xW(t). We can express this fact as a matrix equation
where L(t) := (£i.j(t))i,j=i.,..,d is some d x d lower triangular matrix. According to (3.13) and (3.14), we have while in view of (3.15), the sign of tdd(t) is given by the sign of det(xW(i),... ,x^(t)). In fact, by (3.16) we get, quite explicitly,
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The Frenet frame {fi(i),..., fd(t)} flows according to a system of first-order differential equations. These Frenet equations are obtained in the following way. According to (3.13) and (3.14), for each i = 1 , . . . , d — 1, fi(t) is a linear combination of the vectors x^ 1 ^),..., x^(i) and the coefficients that appear in this linear combination are continuously differentiable. Hence, differentiating this equation, we conclude that f t '(£) can be expressed as a linear combination of the vectors x^(t)....,x^z+1^(i) and hence, also by (3.16), as a linear combination of f i ( t ) , . . . , fi+1(f). Since x^(t) exists for t 6 [0,1], so too does f'd(t). Also, it is certainly a linear combination of the vectors f] ( t ) , . . . , fd(t). We summarize these facts by the matrix equation,
where K(t) — (fcij(£))ij=i,...,d ig some lower Hessenberg matrix. Using the fact that (fj(t), fj(t)) = 6ij, i, j = 1,..., d we get from (3.19) that
Moreover, for i,j = 1 , . . . , d we have the equations
In other words,
which means that K is a skew-symmetric matrix. It follows that K is necessarily tri-diagonal. The elements of K on the (upper) secondary diagonal determine the curvatures of x and are defined by the formula
where we set s(t) := ((x^ (t), x^ ( t ) ) ) 1 / 2 . In terms of these curvatures of x, the Frenet equations become
where we set KO(£) = Kd(t) — 0. For later use, we need to relate the curvatures K I ( £ ) , . . . , Ka-i(i) of x to the elements of the lower triangular matrix in (3.16). According to formula (3.16) we have, in particular, that
Now. differentiate both sides of the equation
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and use the orthonormality of the Frenet frame to get
Using the Frenet equations (3.24), this relation simplifies to
and thus we obtain the desired formula
One additional formula is useful. This formula relates the curvatures directly to the derivatives of the curve at t. Specifically, from (3.16) we get
where
is the Gram matrix of the vectors x ^ ^ ( t ) , . . . ,x.^(t). In other words, L(t) is the Cholesky factorization of the positive definite symmetric matrix Gd(t). Computing the determinant of both sides of (3.26) it follows that
for s — I,..., d and so
where Go(t) = 1 and Gt(t) is the determinant of the Gram matrix of the vectors x(1)(t),...,xW(t),i = l,...,d. We now have more than enough formulas. What is needed next are the following definitions. DEFINITION 3.2. Let x : [0,1] —> Kd be a continuous curve composed of two segments
a (0,1) where each curve segment is given by a regular parameterization with d continuous derivatives such that the first d — I derivatives are linearly independent. We say x is Frenet frame continuous provided the Frenet frame and curvatures of x are continuous on [0,1]. Also, for matrices we use the following terminology.
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DEFINITION 3.3. An (n + 1) x (n+ 1) matrix F is called a Frenet matrix provided that it is lower triangular,
and
The following result from Dyn and Micchelli [DM] gives a geometric characterization of Frenet matrices. THEOREM 3.1. Let x : [0,1] —> Md be a continuous curve composed of two segments
a (0,1) where each curve segment is given by a regular parameterization with d continuous derivatives such that the first d—l derivatives are linearly independent. Then x is Frenet frame continuous if and only if there is a (d+1) x (d+1) Frenet F matrix such that
Proof. First suppose (3.29) holds for some Frenet matrix. Since F is, in particular, lower triangular, it follows that for each i,i — l,...,d, the span of the set of vectors {x_ (a),... , x_ (a)} and the span of {x+'(a),..., x+ (a)} are the same. Hence, the Gram-Schmidt process applied to each set yields the same vectors. This means that the Frenet frames of x_ and x+ are the same at a. Therefore using formula (3.16), for each curve segment at t = a, we get the matrix equation
where H is the d x d matrix formed from the last d rows and columns of the Frenet matrix F. In particular, (3.30) gives us the equations
Since HH = Fa = (FnY, i — l , . . . , d , we can use formula (3.25) and equation (3.31) to show that the curvatures of x+ and x_ agree at t — a, that is, («i)(a + ) = («j)(a - ), i = l , . . . , d — 1. Therefore, we have established the sufficiency of condition (3.29). For the necessity, suppose x is Frenet frame continuous at t = a. Then (3.16) implies that the matrix H denned by (3.30) satisfies
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In particular, H is lower triangular, and equations (3.31) and (3.32) yield the formula
that is, HU = (Hu)\i — 1,2, . . . , d , with Hn > 0. Finally, to identify the matrix F that satisfies (3.29) we set
In summary, we have given two instances where equation (3.3) has geometric significance, either M is a reparameterization matrix or a Frenet matrix F. For 3 x 3 matrices these classes are identical, since they both can be put in the form (3.8). Hence, a planar curve is GC2 if and only if it is Frenet frame continuous. The distinction between these concepts begins to occur for curves in three dimensions. In fact, there are Frenet frame continuous space curves that cannot be reparameterized to make them GC3. 3.3.
Projection and lifting of curves.
To gain further insight into the relationship between these two notions of visual smoothness of curves, we explore how they are affected by basic algebraic operations encountered in practical curve design. These computations typically are performed when a curve is specified in terms of control points (see Chapter 1). Thus we begin with vectors c i , . . . , Cjv in K d+1 and scalar valued functions Bi(t),..., BN(t),t [0,1], and consider the curve
Typically, these "blending" functions will be chosen to be piccewise polynomials with a finite number of breakpoints in (0,1). For the moment, we focus on one such breakpoint, say t = a and write each blending function as two segments, viz.
Now, to ensure that our basic connection equation (3.3) holds for any choice of control points, the blending functions must satisfy the connection equations
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The prevailing point of view is that the matrix M takes the role of shape parameters. Thus, a designer specifies a control polygon formed by the control points C i , . . . ,Cjv The curve x(£) above provides the designer with a "smooth" version of the control polygon. However, it may not be exactly what is required and so the curve may be altered by changing M but not the control points. To enrich the design paradigm, piecewise rational curves are formed from x(t) by projection. To this end, we write the control points c t ,z = 1,... ,N in the form Cj = (wtdi, wi), i = 1 , . . . , N, for some scalars w\,..., WN and form the projected curve
To explore the form of the shape parameters of v given that M is either a reparameterization matrix or a Frenet matrix, we formalize the notion of projection. DEFINITION 3.4. Given a curve x(t) = (zi(£),. - - ,Zd+i(i)) T ,x : [0,1] —> d+l R such that Xd+i(t) =£ 0, t e [0,1], its projection is defined as
Some further terminology will be helpful. A nonsingular (n + 1) x (n + 1) matrix M = (Mij)ij—o,i,....n such that
is called a connection matrix. We define C(a, M) to be the class of all realvalued functions / on [0,1] that are n-times continuously differentiable on [0. a] and [a, 1] and satisfy the connection equation
If M is a connection matrix then C(a, M) is a linear space of continuous functions on [0,1] that contains constants. We also say a curve x is in C(a, M) provided that each component of x is in C(a,M). Also, whenever / and g are n-times continuously differentiable on the intervals [0, a] and [a, I], for h := f / g to be in C(a, M) means that g(d) ^ 0 and
This avoids the cumbersome (and for our purpose here, irrelevant) problem that g may be zero somewhere in [0,1] other than at a. With this bit of terminology, we shall prove some facts from Goldman and Micchelli [GM] that show the inherent algebraic flexibility of Frenet frame continuity. As a means of contrast, the first result we describe concerns reparameterization matrices. THEOREM 3.2. Let M be a connection matrix. Then M is a reparameterization matrix if and only if whenever a curve x : [0,1] —> JRd+1 with Xd+\(a) ^ 0 is in C(a,M), it follows that -P(x) is also in C(a,M).
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This result says that the projection of a, curve has the same shape parameters as the original curve if and only if the shape parameters are given by a reparameterization matrix. For Frenet frame continuous curves we will prove the following result. THEOREM 3.3. Let M be an (n+1) x (n+1) Frenet matrix with an associated Frenet frame continuous curve x(t) = ( x i ( t ) , . . . , Zd+i(£)) such that x,i+\(a) ^ 0. Then there exists a Frenet matrix N, depending only on M and x,i+\, such that P(x) is in C(a,N). We begin the proofs of these results by first establishing some properties of the set C(a,M). PROPOSITION 3.2. Let M be. a connection matrix. Then f 6 C(a, M) implies p o / 6 C(a, M) for all polynomials p if and only if M is a reparameterization matrix. Proof. First, suppose M is a reparameterization matrix corresponding to an admissible reparameterization r : [b,c] —> [b,c] with a 6 (b,c). Let / be n-times continuously differcntiable on [O.oj and [a, 1]. Then
and / £ C(a, M) if and only if
Using this equation, and the chain rule twice, we get
In other words, indeed
where in the last equation we used (3.34) on the function p o /_. For the converse, we define the function
Obviously, r is an admissible reparameterization. Moreover, we have
and so r C(a, M). Therefore, by our hypothesis, we have for all polynomials p that
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Let R be the reparameterization matrix corresponding to the function r. Then (3.35) implies that for all polynomials p
from which it follows that R = M. This result requires us to check that whenever / s C(a, M) it follows that pofe C(a, M) for every polynomial p in order to conclude that M is a reparameterization matrix. However, the proof of Proposition 3.2 shows that it suffices to check this implication for the polynomials of degree < n. Next we shall show that it is only necessary to check it for any fixed nonlinear polynomial. PROPOSITION 3.3. Let M be a connection matrix and suppose p is a polynomial of degree at least two. Then f C(a, M) implies p o f e C(a, M) if and only if M is a reparameterization matrix. Proof. Suppose that / C*(a, M) implies that p o / 6 C(a, M) for some polynomial p of degree at least two. We define S to be the set of all polynomials q such that whenever / C(a, M) it follows that q o / e C(a,M). Let us list some properties of the set S. (i) Linear functions are in S. (ii) S is a linear space. (iii) If <7i,92 5 then q\ o g2 S. (iv) p S. We claim that these four conditions imply 5 is the set of all polynomials. First, notice that 5 contains polynomials of arbitrarily large degree. This fact follows from (iii) and (iv), which together imply that p composed with itself any number of times is in S. Next we show that if qm £ 5 and qm has degree m, then all polynomials of degree < m are also in 5. This would prove the result. Without loss of generality, we suppose that qm is a monic polynomial. Then the polynomial is also a monic polynomial and its degree is m — 1. Appropriately specializing properties (i), (ii), and (iii) we see that qm-i £ S. Continuing in this fashion we construct monic polynomials qo, < ? i , . . . , qm S such that the degree of qj is j,j = 0,1,..., m. These polynomials certainly span all polynomials of degree < m. Hence, by property (ii), all polynomial of degree < m are in S. The next result represents the most general form of Proposition 3.3 that is available. THEOREM 3.4. Let M be an (n + 1) x (n + 1) connection matrix and g a nonlinear function with at least n + 2 continuous derivatives on R. Then f C(a, M) implies g o f e C(a, M) if and only if M is a reparameterization matrix. Proof. Let g be as above and choose any / C(a, M). Since g is nonlinear there must be some constant c R such that g"(c) ^ 0. For every real value of x we define the one parameter family of functions
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Because 1 and / are in C(a, M) and C(a, M) is a linear space, f ( x , •) e C(a, M) for all x e R. Consequently, 5 o (/(>,•)) e ^(a, M) for all x e R. Using the fact that g has n + 2 continuous derivatives, we conclude by differentiating both sides of the connection equation
with respect to z, that
for all x e R. However, it is easily verified that
Therefore, g"(c)f2 e C(a,M). Now let p(t) := p"(c)t2 in Proposition 3.3 to conclude that M is a rcparameterization matrix. COROLLARY 3,1. Let M be a connection matrix. Then /,
Specifically, if M = R(r) then /, g 6 C(a, M) if and only if
and
This implies, by Leibniz's rule, that
and so (3.37) gives
In other words, fg C(a, M).
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COROLLARY 3.2. Let M be a connection matrix. Then f , g e C(a,M), g(a) ^ 0 implies that f / g G C(a, M) if and only if M is a reparameterization matrix. Proof. Suppose that M is a connection matrix such that whenever /, g e C(a, M), it follows that fg G C(a, M). Choose any / e C(a, M) with /(a) ^ 0. Since 1 e C(a,M) we conclude that I// G C(a,M) and hence /2 = //(I//) e C(a, M). When /(a) = 0 we set fr£(t) = f(t)+e where e 7^ 0. Then he(a) = e^0 and /i£ 6 C(a, M). Hence by what was just proved, h% G C(a, M). Letting e —> 0 we conclude even in this case that /2 G C(o, M). Thus we can apply Proposition 3.3, again with p(t) = t2, to conclude that M is a reparameterization matrix. The converse statement follows in a manner that parallels the converse of the proof of Corollary 3.1. Here we use the identity
Proof of Theorem 3.2. We are now ready to prove Theorem 3.2. In fact, its proof is now an immediate consequence of Corollary 3.2 (applied to the component functions Xi and xd+i of the curve x). Let us now turn our attention to the proof of Theorem 3.3. For this purpose, we require the following notation. Fix two functions / and g that are n-timcs continuously differentiable in a neighborhood of point a. By Leibniz's rule, there is a matrix L = L(g) such that
Note that L(g) is a lower triangular matrix whose elements depend only on g(a),g(l\a),... ,g^n\a) and, in particular, LH = g ( a ) , i = 0.1,... ,n. Moreover, when g(a) ^ 0 it follows from (3.40), by choosing / = 1/g, that
PROPOSITION 3.4. Let M be a connection matrix. Suppose w : [0,1] —> M is a continuous function that is n times continuously differentiable on [0,a] and [a.1] with w(a) ^ 0 for some a e (0,1). Then f G C(a,M) if and only if f / w 6 C(a,T) where T = T(w) := L(w^)ML(w-). Proof. Choose / C(a,M). Then it follows that
Also, we have the equation
Substituting this equation in the one above gives us the formula
which means that f/w £ C(a,T).
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Conversely, if f / w G C(a, T) then by the previous observation we conclude that / S C(a,L(w+)TL(w~1)). However, by (3.41) we obtain the relations
That is, / e (a, M) COROLLARY 3.3. Let M be a FrK.net matrix and w any function in C(a, M) with w(a) / 0. Then there exists a Frenet matrix T such that whenever f C(a,M) it follows that f / w £ C(a,T). Proof. From Proposition 3.4, the matrix
has the desired property provided we show that it is a Frenet matrix. To this end. we recall that lower triangular matrices form a group under matrix multiplication and hence T is lower triangular. Also, for i = 0 . 1 , . . . , n.
Therefore, it remains to show that the first column of T is ( 1 , 0 , 0 , . . . , 0)T. First, note that since w 6 C(a, M) we conclude that
and, by Leibniz's rule, the first column of L(w_) is Dn[w+](a). Hence, by (3.42) the first column of ML(w-) is Dn[w+](a). Consequently, the first column of T isL(w~l)D n[w+}(a)=Dn[(w-1w)+}(a) = Dn[l}(a) = (l,0,Q,...,0)T. Proof of Theorem 3.3. Corollary 3.3 immediately implies Theorem 3.3. We merely apply it to the component functions Xi and w — x,i+i of the curve x. D Before changing the subject to the construction of blending functions that yield Frenet frame continuous curves, we make one final comment concerning algebraic manipulation of curves and visual smoothness. This observation has to do with the notion of lifting, which is also a typical algebraic operation on curves. Given a curve in the form
we lift it by multiplying each of its components by some fixed functions w, viz.
We ask the question, do Theorems 3.2 and Theorems 3.3 hold for lifting? The answer is yes for Theorem 3.2 but surprisingly, it is no for Theorem 3.3. THEOREM 3.5. Let M be. a Frenet matrix and w e C(a,M) with w(a) ^ 0. // there, exists a Frenet matrix T such that whenever f e C(a, M) it follows that wf C(a,T). then M = T and M is a reparameterization matrix.
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The proof is based on the following fact. LEMMA 3.1. Let R and T be any (n+ 1) x (n+ 1) matrices. If C(a,R) D C(a, T) then R = T. Proof. For every / G C(a,T), we have both equations
and
from which it follows that
For each x = (x0,..., xn)T £ M n+1 , define the function
Then
and so h C(a,T). Hence, choosing / = h in (3.43) gives Tx = Rx for all
x e Mn+1
Proof of Theorem 3.5. Let M and w be as in the hypothesis of Theorem 3.5. Choose an / G C(a, M}. Since 1 is also in C(a, M), we conclude that both w and wf lie in C(a,T). But T is a Frenet matrix and so by Corollary 3.3 there is a Frenet matrix R (depending only on w and T) such that whenever g 6 C(a,T), it follows that g/w e C(a,R). Choosing g = wf, we conclude that / = wf /w C(a, R). Thus we have shown C(a, M) C C(a, R). We can now use Lemma 3.1 to conclude that M = R. But 1 G C(a, T) and so by the definition of R, 1/w e C(a, R) — C(a, M). In summary, we have shown for any w e C(a, M) with w(a) 7^ 0 that 1/w 6 C(a,M). We claim that this implies that M is a reparameterization matrix. To see this, choose any / C(a, M) and e / 0 such that /(a) 7^ ±e. Then / ±e C(a, M) and so by the above observation (/4-e)" 1 and (/ — e)"1 are both in C(a, M). Hence, their difference
is in C(a,M), as well. The function on the right-hand side of this equation is also nonzero at t = a and so its reciprocal is also in C(a, M). In conclusion, we have proved that whenever / e C(a, M) it follows that p o / e C(a, M), where p(t) := (t2 — e 2 )/2e. We now apply Proposition 3.3 to conclude that M is a reparameterization matrix.
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Connection matrices and piecewise polynomial curves.
This finishes our discussion of various ideas centering on the basic connection equation (3.3). Our next goal in this chapter is the construction of linear spaces of piecewise polynomials determined by connection matrices. This motivates us to introduce the following linear spaces. DEFINITION 3.5. Let irn be the space of all polynomials of degree < n. Given points t o , . . . . tk, with —oc < to < t\ < • • • < tk < oo, we set IQ = (—00, to), It = (ti--i,ti),i = l,...,fc and Ik+i = (tfc,oo). Also let A°,Al,...,Akbenxn nonsingular matrices. We introduce the space of functions on R
Observe that when each A1 is the identity matrix
which is the space of spline functions of degree < n with knots at to,... ,tk. We begin with an elementary observation about 5 n .fc. LEMMA 3.2.
Proof. We construct a basis for SHtk in the following way. The space Sn^ does not generally contain even a single polynomial. However, there arc analogues of polynomials in Sn^. We obtain them by a process of extension. Start with any polynomial p in 7rn on the interval 70. We extend it as a polynomial to I\ so that the extension satisfies the connection equation at to and the leading coefficients of the two polynomials are the same. This is done across each knot until we determine a unique function / G S n^ such that
When each A?, j = 0 , 1 . . . . , k, is the identity matrix / — p. In the general case, we let /o, / i , . . . , fn be the extensions into iSn^ of the monomials l,t,..., t". We need fc + 1 more functions. These functions play the roles of the truncated power functions for spline functions. For each knot t j , j = 0,l,...,k let /j+ n +i be the unique function in Sn_k defined by the properties
arid
When each A^j — 0 , 1 , . . . . k , is the identity matrix f:i+n+i(t) = ^(t - £,)" where V\_ := (max(0,t)) n . We claim that these functions are linearly independent on M and span Sn,kTo prove this, it suffices to notice that on /o, /?, j' = 0 . 1 , . . . , n,, are monomials
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and fj+n+ijj = 0,1,...,k, arc zero on I 0. Also, in a neighborhood of any knot t j , j — 0 , 1 , . . . , k, all the functions f(+n+\,l = 0,1,... ,fe, £ / j have a continuous nth derivative while the jump in the nth derivative of /•,•+„+! at tj is one. The fact that they span SHtk now follows by induction on k. We wish to construct a basis of locally supported functions for Sn,k that arc variation diminishing. When the matrices A°,Al,... ,Ak were nonsingular and totally positive, this was done in Dyn and Micchelli [DM]. We shall describe this result next. In view of the importance of Frenet matrices we will always assume here that A°, A1,..., Ak are lower triangular matrices. Nevertheless, the results of Dyn and Micchelli [DM] that we are about to present do not require this hypothesis. Let An denote the class of all totally positive n x n nonsingular lower triangular matrices. First we wish to bound the number of zeros of a function / in Sntk when {j4l}f=o — *4«- As we will see, this information will allow us to construct locally supported basis functions for Sn^ that are variation diminishing. Obviously a function in Sntk may vanish on intervals, and so we must be quite explicit as to how we "count" zeros. Here, as in Dyn and Micchelli [DM], we use the zero counting convention of Goodman [Go]. Thus we say /(a) + = 1,0, — 1, if for a sufficiently small positive number e, / is positive, zero, or negative on the interval (a, a + e}. Also, we set f(a)~ = fo(0)+, where h(t) := /(a - t),t 6 K. Therefore, when /(a) + /(a)~ ^ 0, / is nonzero in a neighborhood of a and there are nonnegative integers I , r < n, such that
Set q = max(^, r). We say that / has a point zero of multiplicity m where
In other words, we always have /(a) /(a) + = (—I)" 1 . As an example, suppose that f ( a ~ ) = f(a+) = 0 while f'(a~)f'(a+) ^ 0. That is, (. — r = 1. There are two cases that can occur depending on the sign of /(o)~"/(o) + . Typical instances of this are depicted in Fig. 3.2.
FlG. 3.2. Zero counting convention.
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Since q — 1, in this case the multiplicity of the zero of / at a is one in the first case, while it is two in the second. If i = 1 and r = 2 then q = 2; the possibilities are shown in Fig. 3.3. Since q = 2. in the first case depicted above m — 3 while in the second case m = 2.
FlG. 3.3. Zero counting convention.
Generally, we count a zero interval of / in the following way. The first case we consider is that a < b, f(a)~/(b)+ ^ 0, and f(x) = 0 for a < x < b. Obviously in this case, a and b correspond to knots of / and we say that / has a zero interval of multiplicity n + 2 + k where k equals the number of knots in (a, 6). When f(x) ~ 0, for all x
and
for values of t distinct from a knot. At a knot there arc corresponding left and right values of these quantities. Our main result about zeros of / G Sn^ is covered by our next result, taken from Dyn and Micchelli [DM]. THEOREM 3.6 Let {A'}jL0 c An. Then for every f <E 5n:fc\{0}
The proof of this result is lengthy. We describe it, piece by piece, with several auxiliary facts, the first of which gives an upper bound on the multiplicity of a zero of / at an interior knot. LEMMA 3.3. Suppose that {A'}f=0 C An. Let f e SHik be of exact polynomial degree p,q in ( t , _ i , ^ ) and (titi+i), respectively, but not identically zero in either
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interval. Let mi be the multiplicity of the zero of f at ti. Then
Proof. Suppose f ( t ~ ) = • • • = /^(tr) = 0 ? /«(*r) and /(*+) = • • • = /^r 1H*i") = 0 T^ /^(O f°r some nonnegative integers g,r
that is,
Next we observe that
since
Finally, we need that
which comes from the inequalities
Now, combining (3.46)-(3.48), we get
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The second case, when m* > n, is much easier to prove. In fact in this case S-(i,+,/) = () and S + ( t t - , / ) = n . For the next result we need the Budaii-Fourier theorem for polynomials. LEMMA 3.4. Let f be a polynomial of exact degree n. Then
We can even allow a — — oc or b = oo above with the convention that S~ (a+, /) = n for a = — oo and S^(b~, /) = 0 for b = oo. Lemma 3.4 is a very useful result and is central to the proof of Theorem 3.6. For this reason, it is worth a brief digression to provide its proof. Proof. For any constants XQ,. .. ,xn we have
Thus, Lemma 3.4 has the equivalent form
It is this version that we will prove, by induction on n. The linear case n — I is straightforward. Assume that the result is valid for all polynomials of degree < n. Since / has exact degree n, we can find a 60 > 0, sufficiently small, such that for all f 6 (CU0), f ( l ] ( b - f ) f ( i ] ( a + e) ^ 0,i = 0 , 1 , . . . , n,
and
Now choose an interval (c,d) — (a + e, b — e) so that the above equations hold and, in addition, all the zeros of / on (a, b) are in ( c , d ) . By induction
and, by Rolle's theorem,
Combining these inequalities proves the result.
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It is worth mentioning that Rolle's theorem only guarantees a zero of fW between two consecutive zeros of /. However, if S ~ ( f ( c ) , -f^(c)) = I then we have an additional zero between c and the first zero of /^ in (c, d). It is this simple improvement of Rolle's theorem that is embodied in the Budan-Fourier lemma (see Fig. 3.4).
FIG. 3.4. Extra zero for the derivative. S~(f(0), -/ (1) (0)) = 1.
The next result extends Lemma 3.4 to functions in Sn^k that do not vanish on intervals. LEMMA 3.5. Suppose {A*}i=0 C An and let f Sn,k have no zero intervals in (a, b). Then
where u(a,b) := #{i : a < ti < b} and pa,Pb are the exact degree of f just to the right of a and left of b, respectively. Proof. When k = — 1, that is, (a, b) contains no knot in its interior, this result is just Lemma 3.4. Otherwise, we suppose that for some i, j with 0 < i < j < k that
Let pm denote the exact degree of / in the interval (tm, tm+i) for m = — 1 , 0 , . . . , k with the convention that t-\ — -oo and tk+i = oo. Then Lemma 3.4, appropriately specialized to the cases at hand, gives
and
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Also, we need the fact that, by Lemma 3.2, the multiplicity mr of the zero of / at tr satisfies for r — i , . . . , j. Since
we conclude, from inequalities (3.51)-(3.54), that
As a preparation for the proof of Theorem 3.6 we list several special cases of this lemma in the next corollary. COROLLARY 3.4. Suppose {Al}^0 C An. Let f £ Sn,k and i,j integers with 0
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Proof of Theorem 3.6. If / has no zero intervals in R the result is given by Corollary 3.4, part (iv). Next, suppose / has no zero intervals in ( t g , t k ) but vanishes identically in either (-00, t0) or (£ fc ,oo). Three possibilities occur depending on whether / vanishes on (—00, t0), (tk, oo), or both. When / vanishes identically on (—00, to), but has no zero interval in (to, oo), we can use Corollary 3.4, part (iii), to get
When / vanishes identically on (tk, oo), but has no zero interval in (—00, tk), we can use Corollary 3.4, part (ii), to get once again
If / vanishes identically outside of (to,ifc), but has no zero intervals in (ioi^fc)i then we use Corollary 3.4, part (i), to get
There remains the case when / does indeed have a zero intervals in (to,tk). Let the zero intervals be (a,Q,bo), •.., (av, bv) with v > 0, to < CQ, 60 < tk, bf < ae+i, I = 0,1,..., i/ - 1, and at < be, (. = 0,1,... ,v. Each at,be, t = 0,1,..., i/ corresponds to some knot of /. First suppose that / is not identically zero outside either (—00,to) or (i/-,oo). Then we have
and
These inequalities follow from Corollary 3.4, parts (ii), (i), and (iii), respectively. Our zero counting convention leads us to the formula
Therefore, by adding up the bounds above we obtain the inequality
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Next, suppose / vanishes on( —oo,£Q) while /(to) + /(^fc) + ^ 0. All the previous bounds remain valid except for Z(f\(—oo,ao)). This quantity is bounded by using the fact that and the inequality
obtained from Corollary 3.4, part (i). Consequently, taking account of the knot to, we still have, even in this case, the inequality
A similar argument takes care of the case when / vanishes identically on (tfc,oc) and f(tk)~f(to)~ / 0 and the case when / identically vanishes outside of(« 0 ,*fe). 3.5.
B-spline basis.
Our next proposition provides the existence of a basis of locally supported functions for the space Sn
Proof. Let / n + i , - • • ,/2n+i bg the functions in <SnjTi+i (constructed in the proof of Lemma 3.2) defined so that for j — 0 , 1 , . . . , n
and
for i = 0 , 1 , . . . , n+1. Obviously there are nonzero constants GO, . . . , cn such that the function
satisfies /^(C+i) = 0 , f. = 0 , 1 , . . . , n . This function is necessarily zero on (-co, to] and (t, J+ i,oc). Therefore, according to Theorem 3.6 with (fc = n + 1) it follows that
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However, we also have by the construction of / that Z(/|R) > 2n + 2, since / vanishes on (—OO,£Q] and (tn+i,oo). Hence we conclude / has no further zeros in (to,tn+i). This means that /^(*o")/^nH*n+i) ^ 0 anc^tnat / nas no zero m (toi*n+i)- Therefore, for some constant c, S := cf satisfies all our requirements. To prove the uniqueness of S we suppose that there are two functions Si and 62 that satisfy (i)-(iv). Then, in view of (iv), the difference g := Si - S2 has at least one zero in (to,t n +i)- However, by (ii) it also has 2n + 2 zeros, as it vanishes on (—00,to) and (tn+i,oo). This gives at least In + 3. But (i) and Theorem 3.6 (with k = n + l) allow only In + 2. Note that when each A\ i = 0 , 1 , . . . , n+1, is the identity matrix the function S in Lemma 3.6 is the B-spline of Curry and Schoenberg [CS]. Moreover, if [to,..., tn+\]f is the n + 1st divided difference of / at to, - •., tn+i then
and
is the function described in Lemma 3.6, since properties (i)-(iv) can be checked directly from this formula. Lemma 3.6 establishes the existence of geometrically continuous B-splines corresponding to any family of totally positive lower triangular matrices. Our intention in the remainder of this chapter is to develop properties of the B-splines that parallel the special case when all the connection matrices are the identity matrix. It is worthwhile observing at this juncture that B-splines exist even if the connection matrices are not totally positive. In fact, when the knots are at the integers and the same connection matrix is used at each knot, the corresponding B-splines are studied in [DEM]. In this generality the B-splines are not necessarily positive and therefore have unexpected shapes. For instance, the functions in Fig. 3.5 are all B-splines of degree three. We now consider piecewise polynomials on R corresponding to an infinite partition of R into bounded intervals. Thus we are given an infinite sequence {ti}icz with limj^±00 U = ±00 and ^ < ti+i,i Z, a corresponding family of matrices {Al}ie% C An and the space
For each i 6 Z, we let 5j(t) = S(t\ti,...,ti+n+i),t 6 R, be the function constructed in Lemma 3.6 corresponding to the knots £ $ , . . . ,£; +n+ i and matrices \i
Ai+n+1
A , . . . ,/i
PROPOSITION 3.5. Suppose {A1}^^ C An. Then / Sn if and only if it can be expressed as a unique linear combination of the functions {5j}iez , that is,
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FlG. 3.5. Geometrically continuous B-splines.
Proof. First we prove that for each i, the functions Si-n,..., Si are linearly independent on the interval (ti,ti+\). To this end, we consider a typical linear combination of these functions, viz.
Then g £ Sn<2n+i(ti-n, • • • j^i+n+i) and therefore, by Theorem 3.6, when g is not identically zero, we obtain the inequality
Now, g vanishes to the left of tl^n and to the right of ti+n+\. This gives us at least 2n + 2 zeros for g. If it vanishes identically on (ti,ti+i) we would get n + 2 more zeros for a total of at least 3n + 4 zeros. This number exceeds the bound above unless g is identically zero on R. Thus we have established the linear independence of the functions Si_ n ,.... Si on (£ z , i i+1 ). On the interval (ti, ti+\) these n + 1 functions are polynomials of degree < n and hence must span all such polynomials. In particular, for / <S there are unique constants Cj _ „ , . . . , Ci such that
Let us now consider the function
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which has been arranged to be zero on (ti,ti+i). In other words.
from which it follows that g\ (t^+l) = 0, I = 0 , 1 , . . . . n - 1. In view of Lemma 3.6, part (iii), we can choose a unique constant c l+ i such that the function
satisfies the equations 5,-+1(<^fl) = 0,1 = 0 , 1 , . . . , n. This means that g vanishes identically on the next interval (^0-1,^4-2)- In other words, we have
Proceeding to the right of ij+2 in this way past each knot gives us unique constants Cj_ n ,Cj_ n +i) • . • such that
Similarly, we proceed to the left of ij, one knot at a time to finally obtain the desired (unique) representation of / We remark that the functions {Sijiez can be used to generate a basis for the space <$„,&. Clearly, every / £ Sn,k is in <5n and hence can be written as
If we restrict this representation to some interval (a, 6), with t-\ < a < t$ < tk < b < £fc+i, we have the formula
Thus the n + k + 2 functions S _ n _ i , . . . , Sk give a locally supported basis for <S n ,fcTo proceed further, we introduce the subclass Bn of all matrices A in An with the property that AJQ = AQJ = SQJ, j = 0 , 1 , . . . , n — 1, that is, A is a connection matrix. Whenever {Aj}j^z C Bn, we can be sure that all functions in Sn are continuous on R and that constants belong to Sn. Thus, from Proposition 3.5 when {Al}i£i C Bn, there are unique constants {oijjgz such that
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We want to show that a^ > 0 for all i G Z. For this purpose, we restrict the above equation to the interval (tl+n,ti+n+i). Then the function
has the property that F(t) = I for t G (£ i+n . t,;+r;+1). Hence, it follows that FW(tt+n)=Q,e=l,...,n-l. Set
For each j G Z. let A* be the (n — 1) x (n — 1) submatrix of A* obtained by deleting its first row and column. Also, denote by Sn-i the space of piecewise polynomials corresponding to the family A :— {AJ}j z on the same partition {tj}j z. Clearly, A C An-i because A C Bn and therefore, by Theorem 3.6, Z(h\K) < 2n (here k — n and n is replaced by n — 1). However, as h vanishes to the left of ti and the right of ti+n it has, by our zero counting convention, at least In zeros. Thus, F' doesn't vanish in (tj,ij+ n ) and so F must be strictly positive there. However, since F^0+) = a,;S|n)(^) and 5^ n) (i+) > 0 we get by (iii) of Lemma 3.6 that a^ > 0, i Z. We now introduce the functions Mi := azSi,i G Z, which are everywhere positive on (tt,ti+n+i), zero otherwise, and satisfy the equation
3.6.
Dual functionals.
In Barry, Dyn, Goldman, and Micchelli [BDGM], a family of linear functionals |Z/i}!6z were constructed that are "dual" to the set of functions {Mj}jez m the sense that
The basic ingredient in this construction is a certain finite dimensional linear space of piecewise polynomials. To explain this idea we consider a family of (n + 1) x (n + 1) matrices {C"}jez C An+\ and define
Note that at each knot we impose n+l connection conditions on the derivatives of a function T1. In contrast, for the space Sn only n arc imposed. When each C1 is the identity matrix, "P becomes the space vn. Generally, since each C1 is nonsingular we can begin on any interval between consecutive knots with some polynomial p and extend it uniquely to all of M as an element of P. In particular, dim "P = n + 1 and moreover any nontrivial element of T cannot vanish on an interval. The analogue of Theorem 3.6 for P is given next.
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THEOREM 3.7. Suppose that {C*}i z C -4n+i and {£i}jgz is a partition of L Then for any f 6 P\{0} we have Proof. The proof of this result parallels the proof of Theorem 3.6, but it is much simpler. First, keep in mind that a function / £ P\{0} only has point zeros, which are counted as before. Let m^ be the multiplicity of the zero of / at <j. With the same notation used in Lemma 3.2, we see that (3.46) and (3.48) become
(The only difference here is n — 1 is replaced by n). Therefore, (3.47) becomes the inequality This means, in the present circumstance, that Hence, if we apply the Budan-Fourier Lemma 3.4 to / on intervals between successive knots in a given interval (a, b), as in Lemma 3.5, we simply get This implies that / has at most a finite number of zeros on any interval (a, b), and hence a finite number of zeros on E. Now, choose (a, b) large enough to include all the zeros of / in its interior. Then S~a (a+, f) = n and S^ (b~, /) = 0 and so (3.60) gives the desired bound (3.56). We use this result in the following way. We return to our space Sn determined by a partition {ti}iez and n x n matrices {A''}iez C An. Define (n+1) x (n + 1) matrices
so that Pc C Sn and every element in Pc has the same leading coefficient on every interval (ti,ti+i). We also introduce the (n + 1) x (n + 1) matrices
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and
We claim that each Ei is in An+i- In fact, since Ci is in An+i-, this observation is a consequence of the following standard fact. LEMMA 3.7. Let C be an (n+1) x (n+1) nonsingular totally positive matrix. Then the matrix B defined by
is also totally positive Proof. For every 0 < ii < • • • < ip < n and 0 < j\ < • • • < jp < n we have
where {i{,..., i'n+l_p}, { j [ , . . . , j'n+i_p} are the complementary indices to {n — z ' p , . . . , n — ii}, {n — j p , . . . , n — ji} relative to ( 0 , 1 , . . . , n}, respectively. In the last equation, we used the formula relating the minors of an inverse of a matrix to the minors of the matrix itself, cf. Karlin [K, p. 5]. Note that (Ei)nj = 8nj,j = 0,1,... ,n and so any piecewise polynomial in P£ has the same leading coefficient on every knot interval (^,i i + 1 ). We specify a family of functions {A^}^ by the following conditions: (i)N,£P£, £:={£,}, ez , (ii) Ni(tj) — 0 , j = i + l , . . . , z + n, and (iii) N(tn\t) = n!(-l)n, teR. The existence of nontrivial functions satisfying (i) and (ii) is clear since dimP£ = n. Theorem 3.7 implies that each Ni has exactly n zeros given by (ii) and no more. Therefore, 7V7; is of exact degree n and can be normalized to satisfy (iii). Note that when each A1 is the identity matrix, Ni is given explicitly
by For the general case, we follow Barry, Dyn, Goldman, and Micchelli [BDGM] and define for every x G R, i G Z and function /, which is n times continuously differentiate in some neighborhood of x, the linear functional
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First we shall show that for any / in <Sn, Li(x)(f) is a continuous function of x on the interval (£j,£i+ n +i)- Clearly Lj(x)f is defined and continuous except at the knots {ti+\,... ,^+n}. Now, choose x — tj for some j {i + 1 , . . . , « + n} and observe that
Since Ni(tj) = 0 this equation becomes
However, (3.62) gives us the formula
Moreover, by the definition of R we see that
and so Cj'REi = R. Using this equation above confirms that L i ( x + ) ( f ) = Li(x~)(f). Thus Li(x)f is indeed a continuous piecewise polynomial on (ti, ti+n+l)-
We claim that it is, in fact, a constant there. To see this, we pick any x G (tj, £; + n + i)\{£j + i,..., ti+n} and compute the derivative of Li(x)f with respect to x, viz.
Therefore I/,;(x)/ is indeed a constant, independent of x, on (tj,£j + n + i). Our principal observation about these linear functions comes next.
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THEOREM 3.8. Suppose that {A1}^-^ c Bn and (£ z }igz is a partition of R. Then Li(x)(f) is constant independent of x in (ti,ti+n+i) and
Proof. To prove equation (3.65), we fix an i G Z. If j < i - n - 1 or j > i + n + 1 then L i(x)(Mj) = 0, because M3 vanishes on (t,:,ii+n+1). For i — n < j < i, choose any x G (ti+n,tl+n+i). Again, Ll(x)(Mj) = 0, since Mj vanishes on (tl+n. t i+n+1 ). Finally, for i < j < i + n, we choose an x G (£;,i i+1 ) and conclude, as before, that Li(x)(Mj) = 0. Since Li(x)(f) is independent of x G (£j, £i+ n +i) when / G <Sn we have proved (3.65), as long as j ^ i. It remains to demonstrate that L i ( x ) ( M i ) = I . Here we use the fact that Mi,i G Z, were chosen to form a partition of unity, that is,
Since L i ( x ) ( l ) = I for x G (ti,ti+n+i), we get, by applying Ll(x) to both sides of (3.66) and using what we have already proved, that
Our next result, from Barry, Dyii, Goldman, and Micchclli [BDGM], ties the families of functions {A^}i6Z and {Mj}ieZ together by a binomial identity. To give meaning to this statement we need the correct interpretation of the function (x — y)n relative to our set of matrices {Al}i&zRecall the fact that the functions {Ni}i % were constructed in P£ relative to the family £ = {Ei}ieZ given by (3.62). In the next result, the family C = {Ci}iez of matrices denned by (3.61) and the corresponding space Pc will also appear. Recall the fact, mentioned earlier, that Pc C <Sn. PROPOSITION 3.6. Suppose that {Al}iez c An. There exists a unique function g(x,y),x,y 6 M such that g ( - . y ) G Pc for each y G K. Moreover, whenever for some i G Z, both x,y £ (ti,£t+i) then g ( x , y ) = (x — y)n. For any k G Z and i = 0 , 1 , . . . , n, let pi(-, £fc), & ( • , £ & ) be the unique functions such thatpi(-,tk) Pc,ql(-,tk) G P£ and
i — 0 , 1 , . . . , n. Then the function g can be expressed in the form
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Note that the left-hand side of (3.67) is independent of tk- Also, in the special case that each A*,i e Z, is an identity matrix, we have
and so (3.67) is just the binomial theorem. Proof. The functions Pi(-,ife),9i(-,tfe), i = 0 , 1 , . . . , n , are well-defined because of the previously mentioned fact that every polynomial of degree < n has a unique extension from any knot interval to all of R in either Pc or P£. The existence of g ( x , y) is not clear since we want it to reduce to (x — y)n whenever x, y are in the same knot interval. We establish its existence in the following way. To this end, we first note that there is at most one function g ( x , y ) , x, y 6 R, with the desired properties. To see this, suppose that there is another such function, say g(x.y). Now fix any knot interval (t i ( tj + 1 ),i 6 Z, and a y e (ij,t i+1 ). For x (ij,ti+i), g ( x , y ) and g ( x , y ) agree. Since they are both in Pc as functions of x, they must agree everywhere on R. But i Z was arbitrarily chosen and so g and g are the same. Next for any integer fc, the right-hand side of (3.67) is in Pc as a function of x for any fixed y. When x, y are in (tk, ijt+i), it reduces to (x — y)n by the binomial theorem. The essence of the proof is to show that the right-hand side of (3.67) agrees with (x — y]n whenever x, y are in any knot interval (ti, ti+i) i &%Then, by what we said above, it follows that this function is independent of k and has all the properties expected of g. Call the right-hand side of (3.67) H(x, y). The proof of our claim is made by induction. We suppose for some r 6 Z that H(x,y) reduces to (x — y)n when x, y 6 (tr-i,tr). We will show that H(x,y) = (x — y)n when x, y (t r ,t r+1 ). This would mean, by induction, that for any x,y 6 (tr,tr+i) with r > k it would follow that H(x,y) = (x — y)n. Similarly, moving to the left we will also show under the same condition above that H(x, y) — (x — y)n, when x, y e (t r _2, t r - l ) -
We begin the proof of these two assertions with a forward induction step. For i Z, let
and
We also need the monomials
and thw vector
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Finally, define the matrices
arid With this notation it follows that for all t e Z
and
since Pi 6 T>c and QJ P£, i = 0 , 1 , . . . , n. Now, suppose for x, y & (t r _i, t,-)
By Taylor's theorem we also have the equations
and
Substituting these equations into the previous one gives us the formulas
Since the binomial theorem says that
we conclude that
if and only if We could just as easily expand P r and Q, on (tr-i,tr) in a monomial basis about t r _i (we will need to do this for our backward induction step). If we proceed in this way. we get also that
as a condition ensuring that (3.70) holds.
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Call the matrix on the left-hand side of equation (3.71) Fr. Suppose that x, y (tr,tr+i), then Taylor's theorem, (3.68), and (3.69) give
and similarly,
Hence, using equations (3.71)-(3.74) we conclude that
where in the last equation we again used the binomial theorem. This computation advances the induction hypothesis in the forward direction. To step backwards from the interval (tT-\,tr) we now suppose x,y (i r _2)^r-i) and now proceed to compute H(x,y), knowing that (3.72) holds. Specifically, we have the equations
According to (3.72), (3.68)-(3.69), and (3.62) we obtain
which is an equivalent way of saying that
Hence, the above expression for H simplifies to
THEOREM 3.9. Suppose that {Al}i^i C B n and {£i}iez is a partition of K. Then
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Proof. As we already pointed out, Pc is a subspace of Sn. Hence, by Proposition 3.5 we can express g ( x , y ) in the form
where necessarily each Cj £ P£. This can easily be seen from formula (3.67) of Proposition 3.6. Now, pick any knot interval (t r ,t r +i). By Proposition 3.6, (3.75) specializes to the formula
Apply the linear functional Lr(y) to both sides of this equation, as a function of x. Using definition (3.64) of LJ(J/), we get on the left Ni(y) and on the right, by Theorem 3.8, we get cr(y}. Thus cr and Nr agree on the interval (tr,tr+i). Since both are in P£ they agree everywhere. D When each A% is the identity matrix, Theorem 3.9 reduces to a well-known identity of Marsden, cf. Schumaker [Sch]
3.7.
Knot insertion and variation diminishing property of the B-spline basis.
We shall now show that our basis functions {Afijiez are variation diminishing (see Chapters 1 and 2 for a definition of this concept). We approach the proof of this claim by using a knot insertion formula for the space Sn, taken from Barry, Goldman, and Micchelli [BGM]. The technique of knot insertion has already been mentioned in Chapter 2. In that chapter, we studied cardinal splines and inserted knots at all half integers. Here we begin with the space Sn and insert one knot at a time. Specifically, we pick any point i 0 {
We have to decide what matrix to associate with the new knot i. Since we want the new space S on the new partition to contain S we add the n x n identity matrix to our family {Al}i z- Specifically, set
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which ensures that S c S. Also, with this definition we lei
and so P£ = P?. Let us now relate the new functions {/Vj}iez, {Mi}ie%, and dual functionals {£j}jgz corresponding to the space S to the original families of functions {Njjiez, {A/i}i z and dual functionals {Li}ie2. For this purpose, we require the following lemma. LEMMA 3.7. For any x $ {tj+i,... ,
Proof. Set Y(t) = N^t) - Ni_i(i). Suppose to the contrary that Y(x) = 0 for some x £ {ti+i,...,ii+n_i}. Then Y has n zeros, since it obviously vanishes at ti+i,... ,ti+n-i. Also, Y is a polynomial of degree n — 1 on each of the knot intervals (tk, tk+i), k 6 Z. Let Ei denote the nxn submatrix of Ei obtained by deleting its last row and column. Then
where £ := {Ei}iez and Theorem 3.7 implies that This contradiction proves the result. PROPOSITION 3.7. Let t 6 (tq,tq+i) and suppose that {A*}iez C Bn. Then
Proof. Only the middle formula needs to be proved. Denote the right-hand side of the above equation by U(t). Then, clearly
since both Nj-i and Nj satisfy these conditions. Moreover, U vanishes at t and tj,..., tj+n-\. But these points are precisely the zeros of Nj. THEOREM 3.10. Suppose that {A*}iez C Bn. Then for x e (tj,tj+n) we have
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Proof. The proof is an immediate consequence of Proposition 3.7 and definition (3.64) of the linear functionals Li(x). The next result is the knot insertion formula for the pair Sn and SnTHEOREM 3.11. Suppose that {A*}jez c Bn and {£j}igz is a partition of K. Then, for any f G Sn given as
we have
where
Proof. To prove the result we merely apply the dual functionals { L , j } j e i t o both sides of the formula
and use Theorem 3.10. Theorem 3.11 allows us to prove the variation diminishing property of the basis (M,,} THEOREM 3.12. Suppose that {Al}ie% c Bn and {^}j6z is a partition of M. Then
Proof. First we point out that the knot insertion formula (3.78) implies that each Cj is a convex combination of GJ and Cj-\. To prove this claim we observe that
which follows from the fact that Ni vanishes only at ti+i,... ,ti+n and lim^^^oc Ni(t) = oo. Hence, for any integer j with q — n + 1 < j < q, we have sgn Nj(i ) sgn Nj-i(i ) = -1. Using this fact, it follows that
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Now we add knots successively and form sequences of partitions {^}jez,fc = 1,2,... , remembering to add at each new knot the identity matrix to our family of matrices. The knots are added in each interval (t(, tm) so that their successive differences go to zero as k —> oo. Thus we have
and from (3.79) we get Let r = S (/) and suppose that y\ < • • • < yr+l are the points at which / alternates in sign. For definiteness, we may as well assume that sgn f(yi) = (—1)*, t — 1.... ,r + 1. At each point y^, there are most n + 1 integers i such that Mf(yi) ^ 0. Call this set JJf and choose k sufficiently large so that Ji < • • • < Jr+i- Since the M*, i e Z, are nonnegative and sum to one, there must be an ikf 6 Jf, t = 1,..., r + 1 such that
Hence, S~({c$}jz) > r and from (3.80) we get S~(f} < S-({cj}ji). References B. A. BARSKY, The Beta-Spline: A Local Representation Based on Shape Parameters and Fundamental Geometric Measures, Ph.D. thesis, Department of Computer Sciences, University of Utah, December 1981. B. A. BARSKY, Computer Graphics and Geometric Modeling Using [B2] Beta-Splines, Springer-Verlag, Berlin, 1988. [BDGM] P. J. BARRY, N. DYN, R. N. GOLDMAN, AND C. A. MICCHELLI, Identities for piecewise polynomial spaces determined by connection matrices, Aequationes Math., 42(1991), 123-136. P. J. BARRY, R. N. GOLDMAN, AND C. A. MICCHELLI, Knot in[BGM] sertion algorithms for piecewise polynomial spaces determined by connection matrices, Adv. Comp. Math., 1(1993), 139-171. [CS] H. B. CURRY AND I. J. SCHOENBERG, On Polya frequency functions IV: The fundamental spline functions and their limits, J. d'Anal. Math., 17(1966), 71-107. N. DYN, A. EDELMAN, AND C. A. MICCHELLI, On locally sup[DEM] ported basis functions for the representation of geometrically continuous curves, Analysis, 7(1987), 313-341. [DM] N. DYN AND C. A. MICCHELLI, Piecewise polynomial spaces and geometric continuity of curves, Numer. Math., 54(1988), 319-337. [GM] R. N. GOLDMAN AND C. A. MICCHELLI, Algebraic aspects of geometric continuity, in Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L.L. Schumaker, eds., Academic Press, Boston, 1989, 353-371. Pi]
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[Go] [Gr]
[K]
[Kn]
[M] [S]
[Sch]
[Sp]
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T. N. T. GOODMAN. Properties of beta-splines, J. Approx. Theory, 44(1985), 132-153. J. A. GREGORY, Geometric continuity, in Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. L. Schumaker, eds., Academic Press, Boston, 1989, 313-332. S. KARLIN, Total Positivity, Stanford University Press, Stanford, CA, 1968. D. E. KNUTH, The Art of Computer Programming, Vol. 1, Fundamental Algorithms, Second ed., Addison-Wesley Publishing Company, Reading, MA, 1973. J. R. MANNING, Continuity conditions for spline curves. Comput. J., 17(1974), 181 186. M. A. SABIN, Spline curves, report VTO/M S/154, British Aircraft Corporation, Weybridge, England, 1969. L. L. SCHUMAKER, Spline Functions: Basic Theory, John Wiley and Sons, New York, 1981. M. SPIVAK, Differential Geometry, Publish or Perish, Inc., Boston, 1975.
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CHAPTER 4
Geometric Methods for Piecewise Polynomial Surfaces 4.0.
Introduction.
This chapter explores the powerful idea of generating multivariate smooth piecewise polynomials as the volume of "slices" of polyhedra. The traditional finite element approach to the problem of constructing smooth piecewise polynomials puts down a grid on the domain of interest, specifies a given smoothness, and then finds, by ad hoc methods, a desirable (compactly supported) B-splinc. Here we are guided by the geometric principle that polyhedral B-splincs can be constructed as volumes. To make this a viable alternate approach to constructing smooth piecewise polynomials, we must develop analytic and computational tools to study these volume functions. This is done via recurrence formulas that enable us to delineate their smoothness properties, describe their knot regions, and provide efficient methods to compute with them. Three specific B-splines are studied: the multivariate B-spline, the truncated power, and the cube spline associated respectively with the simplex, positive orthant, and the cube. First, various formulas and properties of the multivariate B-spline are derived. Then using this function as a starting point we develop analogous facts about the truncated power and cube spline. We do not discuss how one builds spline spaces from multivariate B-spliues. The interested reader can consult the papers [DM1], [DM3], [H] as well as [DMS], [G], [GL], [S], and reference [Se] from Chapter 5 for information on this important topic. At the end of the chapter we include; some correspondence by I.J. Schoenberg and H.B. Curry that describes some of the historical development of B-splincs. 4.1.
B-splines as volumes.
Many years ago Alfred Cavaretta mentioned to me in passing that I.J. Schoenberg had a way to interpret the univariate B-spline geometrically as the volume of the intersection of a simplex and a hyperplane. At the time, this appeared to be no more than a curiosity and I made no effort to look into Curry and Schoenberg [CS], the original source, for a precise description of the result. Only years later, in the spring of 1978 to be precise, when I was visiting the Mathematic Research Center at the University of Wisconsin by invitation of Carl de Boor, did I appreciate the value of their observation. 149
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It was my first day in Madison. I had just driven from Yorktown, New York and arrived at the WARF building on the University of Wisconsin campus. I went to Carl's office, which was filled with visitors. Besides Carl himself, there were B. Sendov from Sofia, Bulgaria and Dick Askey from the Mathematics Department. They were engaged in a lively conversation about some new multivariate interpolation scheme discovered by P. Kergin who was just then completing his Ph.D. in Mathematics at the University of Toronto. My first reaction to Kergin 's result was a sense of confusion. But I was going to be around Madison until the end of the summer and so there was plenty of time to clarify the matter. Things did become clear, mostly because of Curry and Schoenberg's geometric interpretation of the B-spline. What we present in this chapter are the details that have since followed. They took much longer than a summer to discover and represent the collective efforts of many people to understand this elegant way of studying multivariate splines. I think the appropriate way to start this chapter on multivariate piecewise polynomials is to describe Schoenberg's geometric construction of the B-spline. Let us recall the definition of the univariate B-spline of degree n — 1 with knots at t0,...,tn, viz.
Here [to,...,tn]g denotes the divided difference of a function g at tQ,...,tn, which is given explicitly by
when to,..., tn are distinct. We mentioned in Chapter 3 that S is the kernel for the integral representation of the divided difference operator. Thus we have for any g e Cn(R) the formula
There is another useful formula for the divided difference. It says that
where (v, t) = /^?=o i>jtj is the standard inner product on Rn+1 of the vectors v = (VQ, ..., vn)T, t = (t0,..., tn)T,
is the standard n-simplex, and dm(t) = dt\ • • • dtn is Lebesgue measure on A n .
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One way to prove this well-known formula when t^,..., tn are distinct (the general case would follow by continuity) is to use the divided difference recurrence formula
and show, by integrating relative to dtn in (4.3), that the integral satisfies the same recurrence formula. Now, we combine formulas (4.2) and (4.3) to eliminate the divided difference and at the same time replace # (n ' by g. Thus for any g 6 C(R) we obtain the equation
This formula is the crucial observation. To interpret it geometrically, we choose any n + I vectors v°, v 1 , . . . , v™ in R™ such that the first component of v* is k,i = 0 , 1 , . . . , n (see Fig. 4.1). We express this symbolically by saying that
FlG. 4.1. Lifting knots.
(Generally we use v\^d for the vector in Rd composed of the first d components of the vector v.) We also require that the vectors v°, v 1 , . . . , vn are chosen so that the simplex
where V = {v°,...,v n }, has nonzero n-dimensional volume. There are many choices of v ° , . . . , vn with this property, as long as the points t 0 , . . . , tn are not
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all the same. For instance, if we have ta < t\ < - • • < tn with to ^ tn then the vectors
will do, and in this case
We recall that the volume of [V] in (4.7) is given by
where
We consider the linear mapping T : M"+1 —> Kn given by
where u = (UQ, ..., un}T• T maps A™ bijectively onto [V] and the first componenta of T(u) is (u,t). Thus, by a change of variables in (4.5), we get
and we obtain the formula
at all common points of continuity of these functions. This formula, from Curry and Schoenberg [CS], expresses the B-spline as the volume of a "slice" of a simplex (see Fig. 4.2).
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FlG. 4.2. The value oj a univanate qu&dralic B-sp/me by slicing.
In (4.11), we use the convention that for n = 1 the numerator in (4.11) is the characteristic function of the interval [to,tn]. Also we want to point out the remarkable fact that (4.11) holds for any vectors v ° , . . . . v " such that vz|B = tl,i = 0 , 1 , . . . , n , and vo\n[V] > 0. We mention two applications of formula (4.11). The first provides a lower bound for S. To this end, we go back to the special choice of
vectors (4.8) and notice that a typical y — Xl?=o ujvJ' e M w^h ui - ®j = 0,1,..., n, X)"_ 0 Uj = 1, has the property that y^ — t if and only
and
Therefore
where
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To estimate the volume of P we introduce the quantities and
Notice that the simplex
is contained in P. Therefore, we get the lower bound
and, in particular, S is positive on (to,tn). Curry and Schoenberg [CS] had a nice application of formula (4.11). The following result comes from their paper. PROPOSITION 4.1. The function is concave on the open interval (to,tn). Proof. The proof uses the Brunn-Minkowski inequality, which states that for any two nonempty convex sets A\ and A? in R""1
cf. Milman and Schecktman [MS, p. 134] for a proof of the result. We apply this result to the family of convex subsets of R n-1 defined by and use the fact that for every 0:1,2:2 (to,tn) and t 6 [0,1], A(txi + (l — t)x2) = tA(xi) + (1 — t)A(x-z) to conclude that the function
is indeed concave. We are not aware of any other application of formula (4.11). However, Schoenberg clearly had something else in mind. In a letter written to P. Davis of Brown University in 1965, Schoenberg describes a method to construct a bivariate B-spline (see the end of the chapter for a copy of this letter and other historical notes pertaining to the multivariate B-splines). His sketch of the quadratic case (below) clearly shows that his surface has cross sections, near a vertex, that are univariate B-splines.
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FlG. 4.3. Schoenberg's sketch of the bivariate quadratic B-spline.
4.2.
Multivariate B-splines: Smoothness and recursions.
In his letter to Davis. Schoenberg still seemed to be fascinated by formula (4.3), albeit for analytic functions g and complex points t^...., tn and therefore he was restricted to two variables. It was in de Boor [deB] that the multivariat B-spline was first defined. The definition came at the end of his survey paper on univariate B-spline and it follows next. Given a set of vectors X — {x°,..., xn} in Kd with n > d, let [X] denote its convex hull and suppose that
This condition can be also expressed by saying that the (n+ 1) x (d+ 1) matrix
has rank d + 1. In other words, the vectors (l, xn) ,..., (1, x") span I
m
When n = d and M has rank d + 1 we sometimes say that the points x ° . . . . , x" are affinely independent. Choose any v ° , . . . , v" e R™ such that
and the simplex [V], where V = ( v ° , . . . , v"}, has positive volume. The multivariate B-sp/me is denned by the formula
We use the convention that when n = d the numerator above is the characteristic function of the simplex [X]. Note that the multivariate B-spline is unchanged by any reordering of the elements in X.
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FIG. 4.4. Computer generated bivariate quadratic ^-splines.
At first sight, it seems very difficult to make use of definition (4.13). For instance, the following result from Micchelli [Mic2] on the smoothness and piecewise polynomial nature of the multivariate B-spline seems to be inaccessible by means of (4.13). To formulate the result, we require some terminology. We say that the set X = {x°,..., xn} is in general position provided that for every subset Y C X of cardinality d+ 1 it follow that voldjV] > 0. Another way of describing this concept is to say that X is in general position whenever every subset of d + 1 vectors in X is affinely independent. Recall that the dimension of a convex set is defined to be one less than the maximum number of affinely independent vectors it contains. We use (Y) to denote the affine space generated by affine linear combinations of the vectors in Y. that is, if Y = |v 1 ,...,v m } then
Also, we call (Y) an X-plane whenever Y is a subset of X with dim (Y) = d— 1. When X is in general position every Y C X with #Y = d determines an Jf-plane that contains V, namely ( Y ) . We use the following standard multivariate notation for derivatives of functions,
where x = ( x j , . . . ,Xd)T, k = ( f c j , . . . , kd)T Z+, |k|i = fci + • • • + kd, and Finally, we use II n (R d ) for the set of all polynomials of total degree < n. A typical clement p in n n (]R d ) has the form
for some real scalars c^, |k|i < n, and the dimensi
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FlG. 4.5. Knot regions for a bivariate quadratic ^-spline,.
THEOREM 4.1. Let X — {x°,...,x n } c Kd be in general position. Then S(-\X) is in Cn~d~]-(Rd), is positive in the interior of [X], and on any region m E.d\ U {(y) : Y C X, #Y = d}, is a polynomial of total degree < n - d. Before we prove this result we note the following consequence of Theorem 4.1. Let X be any set in Kd that is in general position with. $-X — n + I . Then every line x + ty, t e M must intersect at least n — d + 2 X-planes. The reason for this is that, as a function of t, S(x + ty\X) is a compactly supported spline of degree n — d with knots at points t where x + ty intersects an Jf-plane. Thus it must have at least n - d + 2 knots. The first step in the proof of Theorem 4.1 is to revisit Schoenberg's proof of (4.11). He began with (4.5) and then derived (4.11). Reversing this argument, we conclude that the multivariate B-spline defined by (4.13) has the property that for all / 6 C(R)
where Xt := Sj=o ^jxJ5 t = (Aj> • • • ,tn)T• Here we use "double duty" notation and let X signify both the set of vectors {x°,... , x™} and the d x (n + I) matrix X whose columns are x ° . . . . , x n . By the way, equation (4.14) demonstrates that equation (4.13) is independent of the choice of vectors v ° , . . . , v r ' e K" that satisfy (4.12). Equation (4.14) gives us a way to extend the meaning of the multivariate B-spline. Specifically, we can think of the multivariate B-spline as the linear functional
where X = {x°,... , x™}. One advantage of this point of view is that no hypothesis is required on the set {x°,... ,x™}. If, in fact, voldfJC] > 0 then the distribution (4.15) corresponds to an integrable function that is zero outside of
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[X]. Below we will derive several equations involving the multivariate B-spline and related distributions. They are to be understood as equalities among distributions, unless otherwise noted. We begin to build the proof of Theorem 4.1. For this purpose, we require the next lemma. LEMMA 4.1. Suppose that vold[X] > 0, X C K d , #X < oo. Given any x e [X] there is a Y C X with #Y = d + 1 and vold[Y] > 0 such that x 6 [Y]. Proof. Let
Clearly C is a nonempty compact convex set and hence has an extreme point u = (KO, - - - , un)T (that is, is not a convex combination of two distinct points of C). If the set 7 := {j : Uj > 0,0 < j < n} has more than d + 1 elements we can find a c = (c 0 ,..., cn)T ^ 0 such that
and Cj = 0 , if j $ I. Let u(e) = u + ec and note that there exists an eo > 0 such that for all e, |e| < e0, u(e) C. This contradicts the fact that u is an extreme point of C. Thus we conclude that #7 < d + 1. Now, define m to be the least positive integer such that x 6 [V] for some V C X with #V = m. So far we have shown that 1 < m < d + I Let us reorder the vectors in X so that V = {x°,... , x7™"1}. The argument used above also shows that the vectors (l,x°)T )...,(l,xm~1)T are linearly independent. Since the vectors (l,x°) r ,..., (l,x n ) T span Rd+l, we may extend the vectors (l,x°)T,..., (l,xm~l)T to a basis m 1 r il T (l 1 x 0 ) r ,...,(l,x - ) ) i(l,x ) ,.--,(l,x < d + 1 - m ) T of Rd+1. Hence Y := m 1 il d 1 {x°,...,x ~ ,x ,...,x + -'"} is the set we seek (see Fig. 4.6)Q PROPOSITION 4.2 Gwew any finite set X = {x°,...,x"} C Rd with void [A"] > 0 then
and logS(x\X) is concave on [^T]°.
FIG. 4.6. Proper triangle.
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Proof. The proof of (4.16) is similar to the univariatc case. Choose x G [X] and any set Y C X such that #Y = d + 1, x G [Y], and vo\d[Y] > 0. By reordering the vectors of X, we may as well assume that Y = {x°,.... xd}. Our choice of vectors in (4.13) are
For this choice, it is easy to sec that n!vol n [v°,..., v11] = d!vold[V]. Let AO(X), . . . . Arf(x) be the baryccntric components of x relative to the simplex [Y] and consider following the convex subset of Rn~d
A typical element y = £j=o ^'v^ m Iv°' • • • - V™1 satisfies y^ = x if and only if
and so
Therefore, for all x in the interior of some [y], Y C X,#Y = d + 1 we have S(x\X) > 0 because the numbers a :— min{Aj(x) : j — 0 , 1 , . . . , d} and b := niax{AJ (x1) : j = 0 , 1 , . . . , d. i = d + 1 , . . . . n} are positive and the set
is contained in y(x|y). The points that lie in the interior of some [Y] are dense in [X]° (since the only points excluded must lie on an X-plane). Call the numerator in (4.13) H(x\X). Then, as in the proof of Proposition 4.1, the Brunn-Minkowski inequality implies #i/(«-d)(£u+(l-t)v) > tH1/(-n-V(u) + (l-t)H1/(n-d)(v), for all u,vG [X] and t G [0,1], from which it follows that logS1 is concave on [X]°. Thus for any x G [X]° we choose u, v G [X]° such that x = |(u + v) and both u and v are in the interior of some [Y], Y C X with #Y = d + 1. Then
which proves S is positive on [X]° (see Fig. 4.7).
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FlG. 4.7. Exceptional point.
LEMMA 4.2. Suppose that X = {x°,...,x n } c Kd and vo\d[X\{^\}] > 0, j = 0 , 1 , . . . , n, then S(-\X) is continuous on K d . Proof. By Proposition 4.2 it suffices to show that 5(-|X) is continuous at any boundary point of [X], since a concave (or convex) function is continuous in the interior of its domain, cf. Roberts and Varberg [RV]. To this end, suppose y 6 9[X] := boundary of [X]. We claim that there exists a Y C X with #Y = d + 1 and vold[Y] > 0 such that y lies on some common supporting hyperplane of [Y] and [X]. The proof of this fact is a consequence of Lemma 4.1. To describe it in detail we recall some standard terminology, cf. Nemhauser and Wolsey [NW] or Schrycr [S]. A convex polyhedron P is defined as the intersection of finitely many affine half-spaces. A convex polyhedron P in Rd is bounded if and only if it is of the form [X] for some X C R d , #X < oo, [Sch, Cor. 7.1c, p. 89] and in this case P is called a polytope. The hyperplane H — {(v,x) — b : x K d } supports a convex polyhedron P, if max{(v,x) : x P} — b. A face F is the intersection of the supporting hyperplane H and the polyhedron P and, in this case, we say H represents F. A polyhedron P is of dimension k if the maximum number of affincly independent points in P is k + 1 and it is called full dimensional provided that dim P = d. A face F of P is called a facet of P, if dimF = dimP - 1. According to [NW, Thm. 3.5, p. 91], facets of a polyhedron P determine it in the following sense. For each facet, there is a supporting hyperplane (v%x) = &*, x G Md representing it and P is represented by its facets
Since [X] is a full-dimensional convex polytope, there is a facet F of [X] containing y. Let H be the hyperplane that represents F. Clearly, F = [X'\ where X' := X n H and, since [X] is full dimensional, #X' <#X. We identify y and F with a vector space of dimension d — 1. Hence by Lemma 4.1, there is a set Y' C X' with #r' = d - 1 such that y 6 [Y'} and vold-i[Y'] > 0. Since [X] is full dimensional, there is an x 6 X\H. Now, set Y := {x} U Y' and observe
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FIG. 4.8. Common supporting hyperplane.
that not only is vol^y > 0 but also H is a common supporting hyperplane of [Y] and [X]. By reordering the elements of X we suppose that Y — {x°,...,x d } and also let A 0 ( x ) . . . . , Arf(x) be the barycentric components of x relative to [Y]. Then, for some m 6 {0,1,... ,d}, A m (x) = 0 is the equation of the hyperplane H. Thus A m (x) > 0 for all x [X] and in particular Moreover, for some k £ {d + 1 , . . . , n}. A m (x f c ) > 0, otherwise we would contradict our hypothesis says that voLji^AJx^}] > 0, j = 0,1,... ,n. Now, according to (4.17), if t = (td+i, • • • ,tn)T e V(x.\Y) for some x G [X], we have J^"=d+i tj < 1 and ifc m(xfc) < Am(x). Hence, for all x e [X] we obtain the inequality This means, for x = y, that S'(yjX) = 0 and also, for any sequence y, [X], j = 0 , 1 , 2 , . . . , such that limj^oo y.,- = y, it follows that l\mj-.00S(y:j\X) = S(y\X) = 0 . Since S(x\X) = 0, for x <£ [X]°, the result follows. The next proposition gives a formula for the derivative of the multivariate B-spline. To describe it, we recall the notation for a directional derivative of /
where y = ( j / i , . . . ,yd)T
and x = (x1,,..,xd)^
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PROPOSITION 4.3. Suppose that X := {x°,..., x™} c Ed. Given any y g Rd swc/i £/ia£ /or some constants CQ, . . . , cn
we have
where
Proof. By the definition (4.14) of the multivariate B-spline, equation (4.20) means that
for all / 6 C 1 ([Jf]). It suffices to prove (4.21) for a class of functions that are dense in C 1 ([X]). For example, the set of functions {/u : u e Md} where /u(x) = 0((u,x)),x e M d , and #(£) = el,t K will suffice for this purpose. We compute the left side of (4.21) when / = /u
which, by (4.3), equals
Similarly, the right-hand side of (4.21) for / = /u can be evaluated as
Both sides of this equation are continuous functions of u. Thus, it suffices to prove they are equal when (u, x ° ) , . . . , (u, x n ) are distinct. To this end, we use
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the recurrence formula (4.4) (with tj replaced by (u,x- 7 )) and (4.19) to obtain
We have now accumulated enough facts for the proof of Theorem 4.1. In fact, we will prove the following result, which subsumes Theorem 4.1. THEOREM 4.2. Let X = {x°,... ; xn} C K d , #X = n + 1. and vo\d[X] > 0. Then S ( - \ X ) is positive in \X}°, and on any region in
it is a polynomial of total degree < n — d. Moreover, if for some I £ { 0 , 1 , . . . , n — d - 1}, vold[X\Y] > 0 for all Y C X with #Y < I + 1 then S ( - \ X ) e Proof. The positivity of iS(-|X) in [X]fl has been proved in Proposition 4.2. Let us now prove it is a piecewise polynomial. We prove the result by induction on n. The case n — d is valid by our definition (4.13). Suppose it is true for all X with #X < n and let X be a set with #X — n. Choose any region R in [X]°, not intersected by X-planes (Y) where #Y = d and Y C X. Note that for each of the sets Xj = ^T\{xJ}, j = 0 , 1 , . . . .n, we have #X7- = n - 1 and any Xj-plane is an X-plane. Hence, R is not cut by .Xj-planes for any j { 0 , 1 , . . . , n}. Moreover, since voLj[X] > 0. the vectors ( l . x ° ) T , . . . , (l,x n ) r span M d+1 . Hence, every y e Rd can be written in the form (4.19) for some constants C.Q. . .. ,cn. Thus, for every y £ Mrf we conclude, from (4.20), that on the region R, DyS(-\X) is a polynomial of degree n. — d ~ 1. From this fact it easily follows that S ( - \ X ) is a polynomial of degree < n — d. The last claim is also proved by induction, this time on i. The case t = 0 is covered by Lemma 4.2. Now, suppose that it is true for all 0 < f.' < (. and that vo\d[X\Y] > 0 for all Y C X, #Y < i + 1. Then for any j = 0 , 1 , . . . , n vold[X,-\y] = vo\d[X\({xJ} U Y)} > 0 whenever Y C X and #Y < t. We conclude, by induction, that the right-hand side of (4.20) is continuous and hence so too is DyS(-\X) for all y e Kd. Our next result from Micchelli [Mic2] is a knot insertion formula for the multivariate B-spline. THEOREM 4.3. If X = {xu.... ,x"} and
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then
We give two proofs of this result. The first proof is "geometric" and confirms Theorem 4.3 when voldpf] > 0, Cj > 0, j = 0 , 1 , . . . , n and n > d + 1. However, it proves that (4.23) holds pointwise on M d . This proof depends on the following lemma. LEMMA 4.3. Given any set of vectors V = {v°,..., vn} c R" with voln[V] > 0 and any v [V], the simplices satisfy
and Proof. Suppose that v = ][^?=o7jVJ', with 7, > 0, j = 0 , 1 , . . . , n , and Y^J=Q 7j — 1- Let A = {j : 7j > 0,j = 0,1,...,n} and suppose that x = ]Cj=o ^j vJ '> -\? - °' j = 0,1,... ,n, 2™=0 Aj = 1, is a typical element in [V]. Then, there is a k 6 {0,1,..., n} with Hence, otj :— 7fc 1 (jk^j — Jj^k) > 0, for j = 0 , 1 , . . . , n, ak = 0, and moreover,
This equation proves (4.25). For (4.26), we suppose that x 6 Vg fl Vjt for t ^ k. We wish to show that x 6 [v U (V\{ve, v fc })]. To this end, we represent x as
for some nonnegative constants (J.j,6j, j = 0,1,..., n that necessarily satisfy the equations /Xj = Aj - AWi7j, *>j = Aj - <5n+i7j, 3 = 0,1, - • •,n + 1, m = 6 and
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Hence, it follows that 6j — /ij = (fJ,n+i — #n+i)7j- In particular, if 7^ = 0 then Ht = Hk = 0 and we have x e [vU(y\{v f , v fc })], as desired. The same conclusion holds when 7^ = 0. Thus, we only need to consider the case 7^7^ > 0. But then the equations -/zfe = (/z n+1 - 6 n+ i)7 fe and 6e = (fJ.n+i ~ 8n+i)ie imply that 6n+i = fj,n+i. Therefore, fj,j = Sj, j = 0 , 1 , . . . , n, and. in particular, we get again Vt = Mfe = ^ = <5fc = 0 (see Fig. 4.9).
FIG. 4.9. Knot insertion.
First proof of Theorem 4.3. Assume that n > d + 1. We choose v ° , . . . , v™ £
Mn such that v-7]^ — x>, j = 0,1...., n, and set v = Z]J=o cjvj wncre cj > 0, j = 0 , 1 , . . . , n. Then, by (4.25) and (4.26)
where Vj is given by (4.24). Thus,
and, by formula (4.10) for the volume of a simplex, we have volT1Vj = CjVol n [V], j = 0,1, ...,n The second proof depends only on simple properties of divided differences. Second proof of Theorem 4.3. Equation (4.23) means that
for all / C([Jf]). As in the proof of Proposition 4.3, it suffices to prove (4.28) for all / of the form / = /u and u e R d . In this case, (4.28) becomes
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For any g e Cn(]R), both sides of this equation are continuous functions of u and so to prove that they are equal we may assume, without loss of generality, that (u, y), (u, x ° ) , . . . , (u, x n ) are all distinct. The divided difference on the left side of equation (4.29) is characterized as the unique linear functional that vanishes on all polynomials of degree < n — 1, is one for the function tn,t R, and is a linear combination of the value of g at (u, x°),..., (u, x"). In view of the second equation in (4.22), the right-hand side of equation (4.29) has the first two of these properties. To verify the third property we need to show that the coefficient of g((u, y)) is zero. For this purpose, we use the formula for the divided difference on distinct points for each factor in the right-hand side of (4.29) to identify the coefficient of g((u,y)) as
Our next result from Micchelli [Mic2] concerns a recurrence formula for the multivariate B-spline, in the cardinality of X, which is useful for computational purposes. (It is this recurrence formula that was used to generate the surfaces in Fig. 4,4.) THEOREM 4.4. Suppose that X = {x°,... ,x™} c Rd and
If
then
We base the proof of this formula on the following auxiliary result. PROPOSITION 4.4. Suppose that X C R d , #X = n and
Then
Note that a proof of Theorem 4.4 can be constructed from Proposition 4.4 by first choosing y = x in (4.22), and then using (4.31) to simplify the resulting expression. Proof. We shall verify (4.31) directly from the definition (4.13) of S(-\X) as a ratio of volumes. But first we need a simple fact about volumes.
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Let C be any set in R"^1 and y some vector in E™^ 1 . The "cone" in Rn with base C and vertex (y, I) 2 is denned to be the set
The mapping T : C x [0,1] -> K[C,y] denned by T((x,t) T ) = (z,t)T where z = (1 — t)x + ty has Jacobian (1 — t)n~l and hence
In particular, we see that the volume of K[C,y] is independent of y (see Fig. 4.10).
FlO. 4.10. Equal volumes.
Now suppose that X — {x°,... ,x™~ 1 } and v 0 ,...^* 1 " 1 are vectors in R""1 such that v'L^ = x\z = 0, l,...,n - 1, with vol n _i(T > 0, where a = [v 0 ,...^™- 1 ]. Let v be any vector in K""1 such that vL^ = x. Then the cone in K" with base (
where vl = (v l ,0) T ,z = 0 , 1 , . . . ,n - l,v n = (v,l) r . Thus (4.32) gives the formula
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(This formula also follows by appropriately specializing equations (4.9) and (4.10).) Next let ax := {u a : v\^d = x} and CTX := {u a : u]^ = x}. Then ax — K[0x,v] and hence, by (4.32) (with n replaced by n - d), it follows that.
Using (4.33), (4.34), and also (4.13) twice, we compute
The next result gives a recurrence relation for the derivatives of 5, see Micchelli [Micl], [Mic2]. Remarkably, all derivatives of a given order satisfy the same recurrence relation. We formulate the result below only for the first derivatives. The statement for the higher derivatives is similar. THEOREM 4.5. Let y e Rd and X = {x°,... ,x n } c Rd where vol[Xj] > 0, j = 0 , l , . . . , r a . //
then
Proof. We use the equations in (4.35) and eliminate CQ from (4.30). Then, differentiate both sides of (4.30) with respect to GJ, for each j = l , . . . , n , to obtain the equation
Since volj [X] > 0 we can find constants ai,..., an such that
Next, multiply both sides of (4.36) by a,j and sum from j = 1,..., n to obtain the equation
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The last summand above can be simplified by the derivative recurrence given in Proposition 4.3 to obtain the desired formula
4.3.
Multivariate B-splines: Multiple points and explicit formula.
There are two additional identities for the multivariate B-spliiie that we wish to describe. The first concerns the relationship of the multivariate B-spline to the Bernstein-Bezier polynomials. The second is a degree-raising formula. Both are accessible by seeing what happens to the multivariate B-spline when we repeat vectors in the set X. We use the following notation. For every k =(k X = {x°,...,x"} cM r f we let
and (£) := gj, k! = fc 0 ! • • • & „ ! . For the special case k,- = ( 0 , . . . , 0 , 1 , 0 , . . . , 0)T where 1 occurs in the jth component, we use X^ for the set X •'. Thus we have quite explicitly.
Also, when #X = d + 1 and k e Zl+1, we set
where AQ(X), . . . . A^(x) are the barycentric components of x relative to X. THEOREM 4.6. Lei X c Rd,#X = d+ 1, and vo\d[X] > 0. Then, for k 6 Z++1 we. have
When X = {e°. e 1 , . . . , ed}, where e° = 0 and (e*)j = 6^, i, j = 1 , . . . , d, then [X] = Ad and, according to Theorem 4.6, n\S(yi X^) - ^(x), the BernsteinBczier polynomials for Ad (sec Chapter 1, text preceding Lemma 1.5). The next result represents a multivariate B-splinc. corresponding to a set with n + I vectors, in terms of multivariate B-splines, corresponding to sets of n + 2 vectors. In this sense, it is a "degree-raising" formula. THEOREM 4.7. Let X C Md and #X = n+l. Then
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Proof of Theorems 4.6 and 4.7. We begin the proof of these results by setting X = {x°,...,x™} and considering the distribution S(-\Xn). For any / C(Rd), we have the equation
The mapping G : An x [0,1] -> A"+1 denned by G(j/ 0) .... yn+i) = (j/o, - - - , 2/ n _i, j/ n (l - j/n+i). 2/n2/n+i) T has Jacobian yn. Using this mapping to change the variables of integration in the integral above, yields the equation
Appealing to the symmetry of the multivariate B-spline, under reordering of the elements of X, and repeating all vectors in X one at a time, the above computation gives, for k e Z™ +1 , the formula
Thus, for any polynomial h given by
we have the relation
For example, the choice h(tQ,..., i n ) = to + • • • + tn gives us the formula
because h\^n — 1. This observation proves Theorem 4.7. Actually, the choice
in (4.39) gives the equation
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which generalizes (4.38). Another useful choice for h is to set /i(t) = p ( X t ) , where p is a polynomial on Rd. Then (4.40) gives us the formula
where we set Dyp — Ilygy ®yP f°r an\i se^ ^ *- ^ d - For example, when #X = d +1 and p(x) = AJ(x)/j!, j e Z^.+1, we have p(Xt) = tJ/j! for t e A d+1 , and so (4.41) yields the equation
But according to definition (4.13), we have
Combining these two equations proves Theorem 4.6. Our next formula is a curious identity from Micchelli [Mic2], which represents the square of the multivariate B-spline as a sum of products of B-splines. THEOREM 4.8. Suppose that X c E.d,#X = n + 1, and vold[^] > 0, j = 0 , 1 , . . . , n . Then
Proof. From Theorem 4.4 we have almost everywhere, t = (to,..., tn) An,
Choose any / e C*(K d ), multiply both sides of the above equation by f ( X t ) , and integrate over A". We obtain the eouation
Specializing equation (4.39) to k = kj, j = 0 , 1 , . . . ,n, the standard basis in K n+1 , we get the desired formula
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Our final identity is an explicit formula for S(-\X) whose statistical background is discussed in Dahmen and Micchelli [DM3]. To obtain this formula we make use of two lemmas. LEMMA 4.4. Let f £ Lo°(Md) (bounded functions of compact support) and suppose that for all y Kd
Then /(x) = 0, a.e., x e Md. Proof. Let 5(2) = (1 + z)^"1 where 2 C, |z < 1. Then / u (y) := 0(i(y,u)),y e Ed, has derivatives (£>J/u)(y) = (iu)^g^^(Q). Since p (fc) (0) ^ 0, k 6 Z + . we have from our hypothesis that
Therefore, since polynomials are dense in L°° on compact sets, we get that /(x) = 0 a.e., x e Ed. The next lemma is a multivariate partial fraction decomposition. To motivate what we have in mind, we begin by reviewing the univariate case. Given distinct nonzero real numbers XQ, ..., xn, we consider the rational function
We let IQ, ... ,ln be the Lagrange polynomials of degree n such that
where yj = —1/Xj. They are given explicitly by the formula
where o>(x) := 07=0 (^ + xxj}- Consequently, we get
and this leads us to the partial fraction decomposition
which is valid for all z, z — iy, y R. We need the same type of formula in the multivariate case. To this end, we consider the rational function
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where z = iy, y e R d . We approach the partial fraction decomposition of this function in the same way as in the univariatc case, by considering a certain multivariate interpolation scheme. For this purpose, we begin with vectors X = {x°,..., x™} such that {0, x°,...,x n } arc in general position. For every Y C X with #y = d there is a v(= v y ) such that
Thus, and also 1 + (v. x) ^ 0, for all x £ X\Y, because {0} U X is in general position. (It is to be understood here that the ordering of the vectors in Y used in the numerator and denominator of the right-hand side of equation (4.43) is the same.) Consequently, if Y, Y' are two distinct subsets of X with cardinality d there is a, w <E (X\Y) n Y'. Thus, 1 + (v y , w) + 0, while 1 + (v y ',w) = 0. This means that v y ^ v y/ whenever Y ^ Y'. Define a polynomial ly of total degree < n — d + 1, by the formula
Reasoning as above, we choose any two distinct sets V, Y' C X such that #Y = #Y' = d, then pick a w e (X\Y) n Y' arid observe that /y(v y ') = 0. In other words, and, in particular, the functions ly,Y C X,#Y = d, are linearly independent. Since #{Y :Y CX, #Y = d} = ("J1) = dim n n _ d + 1 (M d ), we conclude that these set of functions {ly : Y C X, i^Y = d}, span n T j_
valid for all p e n n _ d + i(R d ) (see Fig. 4.11). In particular, we get
Our next lemma follows immediately from this identity. LEMMA 4.5. Let X C JRr/, #X = n + 1, and {0} U X be. in general position. Then for every z = zy. y 6 M d , we have
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FIG. 4.11.
Ten normals determine a cube polynomial.
We are now ready to prove the following result. THEOREM 4.9. Let X C Rd with #X = n + l. Suppose that u e Rd and {u} U X is in general position. Then
Proof. First, let us point out that it suffices to prove this result for u = 0. Quite explicitly, if {u} U X is in general position then so too is the set {0} U X', where X' := X — {u}, since
Also, it follows from the definition of the multivariate B-spline that
Thus one can apply (4.45) when u = 0 to the set X' and replace x by x - u to obtain the general case. Hence we assume without loss of generality that u = 0 in (4AF>\.
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Next let us consider the function g(t) = (-l)n(l+t)~l,t e R. Then/ z (x) : = g ((z,x)) = n!(l + (z.x))-™- 1 and for z = iy,y e R d , we have (n)
That is, we have verified the equation
Specializing this formula appropriately, we have, for every Y C X, #Y = d, the formula
Now set
Hence from (4.46). (4.47), and Lemma 4.5, we get
Lemma 4.4 implies that /i(x) = 0. a.e., x 6 Rd, and Theorem 4.6 gives us the equation
where A 0 ( x ) , . . . , A rf (x) are the barycentric components of x relative to the simplex [{0} {JY}. Specifically, we have the formula
Combining all of these formulas proves the result.
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We remark that (4.45) reduces to (4.1) when d = 1. In fact, if d = 1 and X = {to,..., tn] is a set of distinct positive numbers, then for the set Y consisting of one point, Y = {ti},
4.4.
Multivariate truncated powers: Smoothness and recursion.
We have developed an extensive list of identities for the multivariate B-spline. Let us now provide similar formulas for the multivariate analog of the univariate truncated power function t^T1 = (max(0,i)) n " 1 ,t e M. DEFINITION 4.1. Let X c Rd, #X = n, 0 £ [X], vold[X] > 0. Then the truncated power function is defined as
For X ~ {x 1 ,... ,x n } we introduce the convex cone generated by X,
Note that O(x\X) = 0, if x £ (X} + . The above definition of the multivariate truncated power demands that n > d + 1. For n = d, and vold ([{0} U X}} > 0 we set Let us first identify O(-\X) in the univariate case. Our requirement that 0 ^ [X], X — {x\...., xn}, means that either Xi < 0, i = 1,..., n or Xi > 0, i = 1,.... n. Assume the latter case holds. Then, for n > 2
where g(t] = t
and so
. But it can easily be checked that
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This formula also holds for n = 1. Thus O(-\X) is a constant multiple of the univariate truncated power function. In our next theorem, we summarize some of the most important properties of the truncated power function. THEOREM 4.10. Let X = {x 1 ,... ,x™} c Ed with 0 £ [X]. Then (i)%<W) = E?=(i)%<W) = E?=i CjOUXj),i CjOUXj), where
and
where
(iii) 0(-\X) = lim e _ 0 + t~n+dS(t • |{0} U X). (iv) // every subset of i elements in X is linearly independent then O(-\X) (7"-*-i(]]£d). Moreover, O(-\X) is a polynomial of degree
We remark that Dahmcn [Dah] used this last property (v) as the definition of the truncated power. Proof. We begin with (iii). To this end. we first establish for any y £ Kd the formula
For the proof of this formula, observe that for x e [X] the ray (l-£)y+tx, t e [l,oo) must intersect d[X] at some value t0 [l,oo). Hence for t > t0 the integrand above vanishes and so the integral is clearly finite.
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Let / e C(Rd), set X = {x 1 ,... ,x n } and consider the integral
Using the above formula, with y = 0, we have for any n > d
For n = d, we obtain, directly from the definition of the multivariate truncated power, that
Next we consider (v). Set X := {0} U X. Then for / C0(Rd) we have
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The recurrence formula (ii) is an immediate consequence of the analogous one for the multivariate B-spline given in Theorem 4.4. Specifically, we have
To evaluate this limit we write x in the form
then apply Theorem 4.4 to the set {0} U X and get the formula
because 0 g [X]. We now can use (v) to derive (i). In fact, if / G Cg(R d ) we have for i — l,...,n,
Also, by the definition of the truncated power it follows that
Hence, we have verified that
The general case follows by multiplying both sides of this equation by cl and then summing the resulting expressions over i — I,..., n. There remains the proof of (iv). This is similar to the proof of Theorem 4.2. We use property (i) above, the definition of O(-\X) in terms of the multivariate B-spline, and Lemma 4.2. The next formula for O(-\X) provides a knot insertion identity for the truncated power function, similar to that given in Theorem 4.3 for S(- X). THEOREM 4.11. Suppose that X c Ed and X = {x 1 ,... ,x"} with 0 <£ [X]. If
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FlG. 4.12. A quadratic truncated power.
and either
or
then
Proof. We write y in the form
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arid use the knot insertion formula for the multivariate B-spline given in Theorem 4.3 to obtain
The second term in the bracket above is clearly zero because of (4.49). Using (4.50). we claim that 0^ [{y} \JX]. To see this, we suppose that the hyperplaue {x : x 6 M d , (a, x) = b} separates 0 from (X\. For instance, this is the case if b > 0 and (a, x1) > b, i — 1 , . . . . n. Set bo '•= bmir\(^l"'_1 c,-, 1), then (a, x) > 60 for all x [{y} U X] and so the hyperplanc {x : x e Rd. (a, x) = 60} separates 0 from [{y} U X}. Thus for any x there is an f sufficiently close to zero such that ex is also not in [{y} U X]. This means that the second term in the bracket is again zero, for e sufficiently small. Thus, in both cases we obtain
which, by definition, simplifies to
Our next formula is the derivative recurrence relation for O(- X ) , which is similar to that of Theorem 4.5 for S ( - \ X ) . THEOREM 4.12. For any y e R d and X = { x V ^ . x ™ } with vold ({0} U Xj) > 0. j — 1 , . . . , n, we have
where x = YUj=i cjxJ'Proof. We differentiate both sides of formula (ii) of Theorem 4.10 with respect to Q, f — 1 , . . . . n, and obtain the formula
We write y in the form
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for some scalars a i , . . . , an, multiply both sides of (4,51) by ae, and sum both sides over I = l,...,n. This computation gives us the identity
Next, we use formula (i) of Theorem 4.10 to conclude that
and so we get
Formulas for the truncated power function, obtained from repeating vectors in the set X, are based on the observation that whenever #X — n,
for any k ZTf. with |k|i = m and / £ C0(Rd). This formula can be proved easily by induction on n. The case n = 1 means that for every y G Kd and every / Co«d)
This formula follows easily from the change of variables r = (ro,...,r fac,x e A m ,i e R+, since volmAm = 1/m!. As with the multivariate B-spline, we use (4.52) as follows: for every y G Rd
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Thus, we get
This formula is the truncated power analog of formula (4.40) for the multivariate B-spline. To obtain the companion of Theorem 4.6 for the truncated power function, we suppose that X — {x 1 ,... , xd} and choose the polynomial p <E IIm(R'z) defined by
Specifically, if y 1 , . . . , y d are vectors biorthogonal relative to x 1 , . . . , x d , i.e.,
then for j = ( j i , . . . , j d )
we have
We compute the integer
and so we get
Under certain conditions on X, to be explained below, an explicit formula for the truncated power function is available. The formula we develop next is a version of Theorem 4.9 for the truncated power function. To describe it in detail, we introduce another distribution, from Dahmen and Micchelli [DM2], which is of some independent interest. Given any finite set X C K';, with #X = n, we define the distribution E ( - \ X ) , by requiring that
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for all measurable functions / for which the integrand in (4.53) is absolutely integrable. When rank X = d < n, we can find an n x n nonsingular matrix V such that the d x n submatrix consisting of the first d rows of V is X. We make the change of variables s = Vt in (4.53) and obtain the formula
This formula can be taken as the pointwise definition of E ( - \ X ) . In any case, we see that when rank X = d, E ( - \ X ) is a nonncgative function with integral one. When n = d and rank X = d we can obtain an exact expression for E (-\X). Specifically, the change of variables s = Xi in (4.53) gives us the equation We can rewrite this formula in a slightly different form by noting that det({0) U X) = detX and the vector v x = -X~Te satisfies the property that 1 + (v x ,x) = 0 for all x 6 X. Thus, for any subset Y C X with #F = rank Y = d we have Next we will build E (x.\X) for any set X with #X > d from sums of functions of the type (4.55). Specifically, we shall prove the following result. LEMMA 4.6. Suppose that X C M d , #X - n, rank X = d, and {0} U X is in general position. Then
Proof. The proof of this formula is based on the standard fact that whenever / e L 1 ^ ) and
for all y 6 K d , it follows that h(x) = 0, a.e., x 6 Rd. We will apply this condition to the function h defined to be the difference between the left-hand and righthand side of equation (4.56). Of course, this requires that we show that (4.57) holds for this function. For this purpose, we choose /(x) = e~( z ' x ), x 6 M rf , and z = *Y>y S K d , in equation (4.53). This gives us the formula
According to our partial fraction decomposition given in Lemma 4.5. the righthand side of (4.58) equals
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Now we use (4.58) (with X replaced by V) to replace the function of z in the sum (4.59) by the integral,
This proves that h defined above satisfies (4.57) and so is identically zero. Next we consider the one parameter family of functions
under the same conditions of Lemma 4.6. Combining equation (4.55) and (4.56) with the observation that vp Y = pv v , Y C X, and (4.43), we get
Consequently, for all p > 0, we have the asymptotic expansion
where, for t = 0,1, 2 , . . . ,
Note that each pe is a piecewise polynomial of degree I that is zero off the set (X) + . Hence for any / e Co(R d ) we have the asymptotic expansion
The left side of equation (4.61) also equals, by definitions (4.53) and (4.60), the integral
Suppose now that 0 ^ [X], then by property (v) of Theorem 4.10 the integral above converges as p —> 0 + , to the integral
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Consequently, from (4.61) we conclude that
and
We summarize these facts in the next theorem. THEOREM 4.13. Suppose that X C Rd, #X = n, rank X = d, 0 £ [X], and {0} U X is in general position. Then
a.e., x e M . Moreover, for I = 0,1,.., ,n — d- I and a.e., x e Rd we have
4.5.
More identities for the multivariate truncated power.
Our next formula, from Dahmen and Micchelli [DM2], relates the truncated power function restricted to a hyperplane to the multivariate B-spline. We first develop the formula in a special case. Let X C Md with #X = n and set
We claim that a.e., (t,x) T Rd+1. To prove this formula, we let / 6 C 0 (M d+1 ) and consider the intearal
We make the change of variables t = /iy, where h e [0,oo) and y £ An obtain the equivalent expression
which proves (4.62).
1
and
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FIG. 4.13.
187
A slice of a quadratic truncated power.
The next result is a reformulation of equation (4.62) in its component free form. It states that a truncated power restricted to a hyperplanc is a multivariate B-spline. To make this precise, we begin with any hyperplane // = {x £ Rd : (x,y) = t}, where y 6 Md with (y,y) = 1 and t is a positive constant. We identify coordinates in the hyperplane H by choosing an orthonormal basis { y 1 , . - - , y d } of Rd with y1 = y. Let U be the orthogonal matrix whose columns arc y 1 , . . . , yd. Then x £ H if and only if x = [7w. where w = (t,u) T and u £ R^1. Suppose that (y, x) > 0 for all x £ X. This ensures that H n (X) + is a nonempty bounded convex subset of Rd. We introduce a finite set of vectors in Kd~^ by setting and claim that for x = U ( ( i , u) T ), u £ R''"1. we have
In other words, we obtain the following result. THEOREM 4.14. Let X c Rd. #X = n and suppose that for some y £ Rd with (y, y) = 1 we have (y,x) > 0 for all x 6 X. Then, for every t > 0 the hyperplane intersects (X)+ in a nonempty bounded subset of Rd and, moreover,
where V vs the subset of Rd l defined in (4.63). Proof. As explained above, (4.65) means that
It is easy to see that for any d x d nonsingular matrix T
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Using this formula in (4.66) with T = U shows that (4.66), and hence the validity of (4.65), hinges on demonstrating that To this end, we also need to point out that for every X C M d , t > 0 and an n x n diagonal matrix D = diag (d\, • . . , dn) with d\ > 0 , . . . , dn > 0 that Let and
Then, by (4.63), UTX = WD, and so we can use (4.69) and also (4.62) (with x replaced bv u and X bv V arid d bv d — 1) to get the formula
which reduces to (4.68). The next formula, an identity for the square of the truncated power, is the analog of Theorem 4.8. THEOREM 4.15. Suppose that X C M d , #X = n, rank X — d, and 0 0 [X]. Then
Proof. We use property (iii) of Theorem 4.10 and Theorem 4.8 to prove this formula. Thus we have
Since 0 ^ [X], there is an CQ (depending on x) such that Q X is also not in [X]. Hence, the second term on the right of the equation above is zero and we obtain the desired formula
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Our last formula for the truncated power function is an interesting identity due to Dahinen [Dah], which expresses the multivariatc B-spline in terms of the truncated power function. The background of his formula is the divided difference representation of the univariate B-spline. We recall that when X Q . . . . , xn is an increasing sequence of real numbers, we have for all t G M
We have already pointed out that
whenever x, > 0, i = 1, • • • , n. Thus we get the equation
Notice how we have hidden in the truncated power the fact that for a fixed t, S ( t \ X o , . . . , xn) is a rational function of XQ, - • - , x7l. This notational tour de force leads us to the following mnltivariate formula. THEOREM 4.16. Suppose that X = {x°,... ,x n } C Rd and W := {x^ - x-> : 0 < j < i < n} has the property that 0 0 [W]. Then
where X(j) :=. {x^ - x ° , . . . ,x^' - x.i-i,x.i+i - x - > , . . . ,x n - x°). Geometrically this formula corresponds to a "signed" decomposition of a simplex into cones (see Fig. 4.14). Its proof depends on an ancillary identity which we present in the next lemma. Recall our notation, Dxf for the differential operator n X gx -Dx/ where D y f , y G Kd. denotes the directional derivative of / in the direction of y. LEMMA 4.7. Let X = {x°,... ,x™} C R" and W , X ( j ) , j = 0. L . . . ,n be the sets defined in Theorem 4.16. Then for every f £ C loo (R d ) (functions with continuous derivatives of all orders on Kd) we have
Proof. As in previous proofs, it suffices to verify equation (4.70) for functions of the form /(x) = g ((z, x)), x e K d , where g(t) = e*, t £ K, for all z = iy, y £
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FIG. 4.14. Signed decomposition of a simplex by cones.
R d . In addition, we can assume that ( z , x ° ) , . . . , (z,x n ) are distinct so that the left side of (4.70} becomes
while the right side becomes
These quantities are equal because
To make use of this lemma we need to note that whenever 0 ^ [X]
To prove formula (4.71) we use part (v) of Theorem 4.10 to see that the left-hand side of equation (4.71) equals
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Now, for the proof of Theorem 4.16 we choose any h Cfi°(R(l) and consider the function
Differentiating under the integral sign gives the equation
Thus, appropriately specializing equation (4.71) provides us with the formula
Similarly, we have
Furthermore, by specializing Theorem 4.10 part (i), it follows that for any w W Using (4.74) repeatedly, we get for any set R C W
Hence, (4.73) becomes
We now substitute these expressions for the derivatives of / into formula (4.70) of Lemma 4.7 to obtain
From this formula it follows that
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If in the above proof we used the function
we would have
and
Hence it would follow that
and so we conclude that
Our last comment about the truncated power function is the simple observation that it provides the Green's function for the differential operator Dxfi whenever 0 g X — {x 1 ,.--,*"} and rank X = d. To see this, we choose a nonzero vector y e Rd such that (y,x*) ^ 0, i = l , . . . , n . Set b = min{| (y, x*) | : 1 < i < n}, e, = sgn (y, xl) , i = l , . . . , n , and x* :— ejx'ji = 1,... , n. Then the hyperplane (y,x) — b separates 0 from [X], where X := {x 1 ,..., x™} and so, by formula (4.71), we have
where e := c\ • • • fn. 4.6.
The cube spline.
We now turn our attention to the function associated with the n-cube [0, l]n. There is a useful decomposition of the cube [0, l]n into simplices. The idea of this simplicial decomposition of the cube is to rearrange the components of a typical vector y e [0,1]™ in a nondecreasing order. As we vary over the cube each component of y gets a chance to be first, second, . . . , last in this reordering. That is, if we let Pn be the set of all permutations of {1.2,..., n}. TT a typical element in Pn, and ov the simplex
then the family of simplices {&„ : TT 6 Pn} form a simplicial decomposition of the cube [0,1]™. Any two simplices in this family intersect in a common face and have equal volume.
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FIG. 4.15. Triangulation of the cube..
Recall that the multivariate B-spliiie is an integral over A", what we called the standard simplex. To map an element t — (t 0, /,]...., tn) in A" into av we set
Hence, t —> t ff maps Ara bijectivcly to a^ and, moreover, for any set of n vectors X = {x 1 ,..., x"} C M rf \{0} we have where This leads us to the following definition. DEFINITION 4.2. Let X C Kd\{0} and #X = n. The cube spline C (-\X) is define/I as
THEOREM 4.17. Let X C K d \{0}, #X = n. (i) For any f (Rd)
(ii) 7 / 0 0 [X] then
(iii) If 0 £ [X] and X C Zd\{0} i/ien
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FlG. 4.16. A linear cube spline.
where
(vi) // every subset of i elements in X is linearly independent then C (-\X) Cn~f~l(Hd). Moreover, it is a polynomial of degree < n — d on any region not intersected by X-planes. Proof. The proof of (i) follows directly from the decomposition of the cube [0, l]n into simplices {ov : -K 6 Pn} and the definition of Xv. For (ii) we define the difference operator for every y G Rd and, generally, for any finite set X C R d , we set
We claim that
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This is easily proved by induction on the cardinality of X. Specifically, we consider the set X U {y}, which contains one more element than X, and suppose that V C X U {y}. Then either V C X or V = V U {y}, where V C V, and so we obtain
Next, we observe that for every / £ Co(R d ) we have
When y = x1 this last integral simplifies to
Successively repeating this computation, for each vector in X, we get
For (iii) we again consider any / Co(BLd) and note that we can group the terms in the finite sum
as
Thus we have
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For the proof of (v), it suffices to prove that for any u 6 X
To this end, let / e C(R d ). Then for £ = 1 , . . . , n,
There remains the proof of (iv). First, let us assume that 0 0 [X]. In this case, we can make use of (ii) in Theorem 4.17 and (ii) of Theorem 4.10. In the computation that we are about to perform, it is convenient for us to index the constants a i , . . . , a n in (iv) by an element y e X; that is, we set a i = Ox4)2 = lj • • • in- Therefore, for any set V C X we have
where we define
and also
Now, pick a typical y G X and consider the contribution of the terms ay and ay — 1 in the formula above. These are, respectively,
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and
Hence, in total, we get
There remains the case that 0 £ [X]. { — 1,1}, i = 1 , . . . ,n, such that the vectors
In this case, choose signs Q £
have the property that
It is easy to see, for instance, from (i) in Theorem 4.17, that
where u := S{x* : n = — 1, i = 1,... ,n}. Hence we may apply formula (iv) of Theorem 4.17 to the set X. For this purpose, we note that
where
and, of course, we have by definition that
Thus
We break up the sum into those terms where BJ = Oi and the remainder where &i — I — a,i. Using (4.76)-(4.78) to simplify the resulting expression, we get
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The property (vi) follows easily from (ii) above and the similar property for the truncated power spline given in (iv) of Theorem 4.10. We remark that de Boor and DeVore [deBD] use (i) of Theorem 4.17 to define the cube spline. Most of the results of Theorem 4.17 are due to de Boor and Hollig [deBH].
THEOREM 4.18. Let X = {x 1 ,... ,x n } c Ed\{0} and rank X = d. Then, for any y Rd and
Proof, Differentiate both sides of formula (iv) of Theorem 4.17 with respect to at, 1 < i < n, to obtain the equation
Using (v) of Theorem 4.17 (specialized to the case y = x^) in the above equation and simplifying the resulting expression gives us the formula
Since rank X = d, there are constants u\,..., u
u n such that
Now, multiply both sides of this equation by ut and sum over I = 1,... ,n to obtain the result. The last formula states that the cube spline satisfies a refinement equation (see Chapter 2).
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THEOREM 4.19. Let X = {x 1 ,... .x71} c 1d\{Q}. Then
where
Proof. We prove this result by induction on n. For the case n = 1 and X — {x1} we see that
Moreover, in this case, for any / G C(Kd) we have
Now, suppose that (4.80) is true for all sets X with n elements. Consider the {a?}jez<* be the coefficients corresponding 6Zd,{a"+1}j set X' = XiJ{xn+l}. Let to X' and X, respectively. Then (4.81) implies that
Also we need to observe that
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a fact obtained directly from (i) of Theorem 4.17. Hence for any / C(Rd), we have
The above result was taken from Dahmen and Micchelli [DM2], where it is used to develop a stationary subdivision procedure for computing linear combinations of integer translates of cube splines. 4.7.
Correspondence on the historical development of B-splines.
The development of the multivariate B-spline is an exciting chapter in the history of spline functions. Some of the milestones in the study of B-splines were mentioned in [DM5], especially as they appear in the statistical literature. Perhaps more than anyone else, I.J. Schoenberg pushed the theory of spline functions forward. He described some of his unpublished ideas on bivariate B-splines in a letter he sent to P.J. Davis on May 31, 1965, which we referred to in the first section of this chapter. Below we have provided a copy of this letter. In a seemingly unrelated paper, Motzkin and Schoenberg [MSc] studied a class of multivariate entire functions which, in their terminology, have "affine lineage." In the univariate case these functions were an indispensable tool in Curry and Schoenberg's study of distributions that are limits of B-splines. They presented these ideas in [CS]. Later, in [DM6] we were able to connect the Motzkin-Schoenberg functions of aSine lineage to limits of multivariate Bsplines. Schoenberg's apparent enthusiasm for this discovery is expressed in an open letter he sent to several colleagues. This letter also appears below. Sometime later, with Schoenberg's encouragement, we sent a copy of paper [Mic2] on computing multivariates to H.B. Curry. In his response to our preprint, Curry explains the background of his collaboration with Schoenberg during World War II and his contribution to their discovery of the univariate B-spline [CS].
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A letter from I.J. Schoenberg to P.J. Davis dated May 31, 1965: Professor Philip J. Davis Division of Applied Mathematics Brown University Providence, RI
Madison. Wisconsin May 31, 1965
Dear Phil: Lately I have been busy with cuclidean polyhedra from the point of view of distance geometry, but on and off I return to spline functions. The other day I noticed that the triangle formulae in the complex plane generalize nicely to higher order derivatives. The source is again the Hermite-Gennochi formula
where t0 — I - ti - • • • - tn and rn where the integration is to be carried out over the simplex
A projection onto the real axis allows us to show that the kernel M(x\ XQ, x\,..., £„) in the fundamental formula
may be interpreted as follows: Interpret the x-axis as one of the coordinate axis of H-dimensional space. Erect at XQ,XI, ... ,xn n hyperplane orthogonal to the x-axis at these points and select at will in each of these hyperplaiies a point, with the proviso that the simplex having these points as vertices should have n-dimensional volume unity. If we project the volume of this simplex on the x-axis we obtain the linear density function M(x : XQ, ..., xn) appearing in (2). I now pass to the complex domain. Let ZQ. z-\,..., zn be distinct points (not all on a line) of the complex plane, and let f ( z ) be regular in the convex hull II of these points. Again we erect at 2:0,21,..., zn the orthogonal complements of the plane (these are of dimension n — 2) and select in each of these a point so that the simplex has n-dim. volume unity. We now project the volume of this simplex onto the plane and denote the surface density function by
Then the following formula holds
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For n = 2 this gives the old triangle formula
where A is the area of the triangle T of vertices ZQ,ZI,Z% (P.J. Davis, Triangle formulas in the complex plane, Math, of Comp., 1964, p. 569-577). Joining all parts of points Zj,Zk (j < k) by segments, the polygon II is disserted into disjoint polygons in each of which M(x, y) is a polynomial in x and y of joint degree n — 2, while M(x, y) = 0 outside II. Moreover the function M(x, y) has continuous partial derivatives of all orders < n —3. 1 Moreover, these properties always determine M(x, y) uniquely up to a constant factor. Another property is this: M(x,y} > 0 inside II and
is a concave function. In particular M(x,y] has exactly one maximum point. Thus for n = 3 the surface z = M(x, y) is a pyramid. In case all the points zo,zi,...,zn are on the boundary of the polygon II then M(x,y) can be represented inside F by means of the truncated power function x"~2. Here is an example: For n = 4 and
M(x, y) is up to a positive factor, which I did not determine, identical inside II to
I am very much interested in these 2-dimensional frequency functions M(x, y; ZD, . . . , zn) because I suspect that the limits of much frequency functions (with the Zj depending on n and as n —> oo) in the usual reuse of probability theory, ought to be interesting frequency functions. The similar equation on the line led to the Polya frequency function. What will one get in the plane, if anything? I hope these remarks did not bore you. With greetings and good wishes. Yours, Iso (I.J. Schoenberg)
I add a sketch of the second degree M(x, y) (n = 4) for the case when z0,,..,Z4 are the vertices of a regular pentagon. All vertical sections are, of 1 This assumes that the points zo,...,z n are in "general position," i.e., no three are collinear.
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course, ordinary spline functions. I sent copies of this letter to H.B. Curry, S. Karlin. and T.S. Motzkin. A letter from I.J. Schoenberg to friends sent in 1979: November 3, 1979
Dear Colleague: Let me bring you the good news that B-splincs and Polya frequency functions are now being studied in higher dimensions. This was recently done by Wolfgang Dahmen, of Bonn, and Charles Micchelli, of IBM. Let me remind yon that Polya frequency functions de Bruijn called them "Polyamials"—were characterized in four different but equivalent ways: I. As limits of B-splines with appropriate knots as their degree tends to infinity, II. As totally positive functions (frequency functions), III. By their variation diminishing property on convolution, IV. By their Laplace transforms being reciprocals of entire functions of the Laguerre-Polya class. At the same time a B-spline Mn of degree n — 1, having n knots, was known to be the orthogonal projection onto the real axis of an n-dimensioiial simplex 5, of volume 1. Now Dahmen and Micchelli are projecting S onto the lower-dim, space Rs, obtaining the s-dimcnsional B-splinc Mn_s. Keeping s fixed and letting n tend to infinity, they obtain as limits of Mn,s the s-dimcnsional Polya frequency functions /s, and also their Laplace transforms as the reciprocals of a class Es of entire functions of s variables. This class Es turns out to be identical with the class of "lineaF' functions studied by Motzkin and myself in 1952. Lineal functions were an extension of results of Laguerre and Polya, and B-splines Mn^s and Polya frequency functions /, were not even mentioned in 1952. Now we know where this class Es is coming from. This work extends to Rs the characterizations I and IV. Very likely also II and III will appear by means of restrictions of /, to appropriate lines or planes. I am very happy with these developments. With regards and best wishes. Yours, I.J. Schoenberg We may describe this subject as asymptotic properties of an n-sirnplex as n tends to infinity.
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A letter from Haskell B. Curry dated November 17, 1979: The Pennsylvania State University 215 McAllister Building University Park, Pennsylvania 16802 College of Science Department of Mathematics
Area Code 814-865-7527 November 17, 1979
Dr. Charles Micchelli Thomas J. Watson Research Center, IBM P.O. Box 218 Yorktown Heights, N.Y. 10598
Dear Fellow-worker: The copy you sent me of your report "On a numerically efficient method for computing multivariate B-splines" reached me here about November 1; but I was not able to acknowledge it right away because I am still convalescing from a spell of illness which I had last summer. I still have very little time to tend to scientific matters. However, I am very glad that you sent me this document. To be sure I have not been active in that field for more than thirty years, and my connection with it was somewhat of an accident. During part of World War II Iso Schoenberg and I were both working for the War Department at Aberdeen Proving Ground, attempting to do what we could to help in the war effort. While so engaged a paper by Schoenberg on splines came across my desk. After examining it, I came to the conclusion that the matter could be generalized. Iso had considered only the case where the nodes, or "knots" were equally spaced; I noticed that the approach could be extended to arbitrary nodes (as in the Lagrange interpolation formula) by using algebraic techniques instead of the Fourier transform. Iso suggested that we write a joint paper. I wrote up the algebraic part and submitted it to him. He, however, was evidently interested in other matters; at any rate he did nothing about it for some 15 years. In due time it was published; first in the proceedings of the Army Research Centre at U. Wisconsin, then in Israel. Unfortunately the paper was published with my name first, as if I had been the prime mover; actually, my part of it was very small. However, he would not listen to any other arrangement. When the war ended, I came back to Penn State, and resumed research in my original field, which is a very abstract and specialized field of mathematical logic. I have not done any work in that field since. But I have been quite interested in seeing what you and others have been doing with the field which has opened up. Sincerely yours, Haskell B. Curry
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References [CS]
H. B. CURRY AND I. J. SCHOENBERG, Polya frequency functions IV. The fundamental spline function and their limits, J. d'Anal. Math., 17(1966), 71-107. [Dah] W. DAHMEN, On multivariate B-splines. SIAM J. Numcr. Anal., 17(1980), 179-190. [DM1] W. DAHMEN AND C. A. MICCHELLI, On the linear independence of multivariate B-splines I: Triangulation of simploids. SIAM J. Numer. Anal., 19(1982), 992-1012. [DM2] W. DAHMEN AND C. A. MICCHELLI, Recent progress in multivariate splines, in Approximation Theory IV, C. K. Chui, L. L. Schumaker, and J. D. Ward, eds., Academic Press, New York, 1983, 27-121. [DM3] W. DAHMEN AND C. A. MICCHELLI, On the linear independence of multivariate B-splines II: Complete configuration. Math. Comp., 41(1983), 141-164. [DM4] W. DAHMEN AND C. A. MICCHELLI, Subdivision algorithms for the generation of box spline surfaces, Com put. Aided Geom. Design, 1(1984), 115-129. [DM5] W. DAHMEN AND C. A. MICCHELLI, Statistical encounters with Bsplines, in Function Estimates, J. S. Marron, ed., Contemporary Mathematics 59, American Mathematical Society, Providence, RI. 1985. [DM6] W. DAHMEN AND C. A. MICCHELLI, On limits of multivariate Bsplines, 3. d'Anal. Math., 39(1981), 256-278. [deB] C. DE BOOR, Splines as linear combinations of B-splines: A survey, in Approximation Theory II, G. G. Lorentz, C. K. Chui, and L. L. Schumaker, eds., Academic Press, New York, 1976, 1-47. [deBD] C. DE BOOR AND R. DEVORE, Approximation by smooth multivariate splines, Trans. Amcr. Math. Soc., 276(1983), 775-785. [dBH] C. DE BOOR AND K. HOLLIG, B-splines from parallelepipeds. J. d'Anal. Math, 42(1982/83), 99-115. [H] K. HOLLIG. Multivariate splines, SIAM J. Numcr. Anal, 41(1982), 1013-1031. [Micl] C. A. MICCHELLI, A constructive approach to Kergin interpolation in Rk: Multivariate B-splines and Lagrange interpolation. Rocky Mountain J. Math, 10(1980), 485-497. [Mic2] C. A. MICCHELLI, On a numerically efficient method for computing multivariate B-splines. in Multivariate Approximation Theory, W. Schempp, and K. Zeller, eds, Birkhaiiser, Basel, 1979, 211-248. [MS] V. D. MILMAN AND G. SCHECHTMAN, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics, Vol. 1200, Springer-Verlag, Berlin, 1980. [MSc] T. S. MOTZKIN AND I. J. SCHOENBERG, On lineal entire functions of n complex variables, Proc. Amer. Math. Soc, 3(1952), 517-526. [NW] G. L. NEMHAUSER AND L. A. WOLSEY, Integer and Combinatorial Optimization, John Wiley and Sons, New York, 1988.
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A.W. ROBERTS AND D.F. VARBERG, Convex Functions, Pure and Applied Mathematics, Vol 57, Academic Press, New York, 1973. A. SCHRIJVER, Theory of Linear and Integer Programming, John Wiley and Sons, New York, 1986.
CHAPTER
5
Recursive Algorithms for Polynomial Evaluation 5.0.
Introduction.
Properties of the univariate Bernstein polynomials motivate the material presented this chapter. Specifically, it has been known for a long time that, given any point on the Bernstein-Bezier curve, there is a simple recurrence that terminates at that point. This observation naturally connects to the recurrence formula for the Bernstein polynomials themselves and also to dual bases, the binomial theorem, and polarization of polynomial identities. All of these ideas are amplified in this chapter to include univariate B-spline representation of a polynomial curve over an interval between two consecutive knots. But it is in the multivariate context that these ideas blossom into a diverse mathematical theory for the recursive computation of polynomials by pyramid schemes. Several examples of such schemes are given and one example is singled out by its unique symmetry properties. This scheme, for H.P. Seidel's B-patches, specializes in the univariate case to the B-spline bases on an interval between consecutive knots. At the end of this chapter, using the recurrence formula for the multivariate B-spline developed in Chapter 4, we identify B-patches with B-splines.
5.1.
The de Casteljau recursion.
The results presented in this chapter closely follow the material in Cavaretta and Micchelli [CM] and Dahmen, Micchelli, and Seidel [DMS]. Specifically, we study recurrence formulas for the generation of polynomial curves and surfaces and explore their relationship to the multivariate B-spline of Chapter 4. We begin as in Chapter 1, with the Bernstein-Bezier polynomials. In Chapter 1 our concern was with the de Casteljau subdivision and its generalizations. Here, we focus on the de Casteljau recurrence formula given in (1.68) as
It was pointed out without proof, after our discussion following (1.68), that this recurrence formula terminates at the value of the Bernstein-Bezier curve at x,
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corresponding to the control points cr := c°, r = 0,1,..., n, viz.
where
(See Fig. 5.1 for the case of cubic planar curves.) Actually, we shall now show that
so that, in particular, (5.2) holds. The proof of formula (5.4) requires the recurrence formula for the Bernstein-Bezier polynomials, namely
initialized with b^x) := 1. This recurrence formula follows directly from (5.3). We prove (5.4) by induction on t. Thus we suppose that t > 1 and that (5.4) has been proved for t replaced by t — 1. Using the recurrence formulas (5.1)
FIG. 5.1.
de Casteljau evaluation of a cubic curve.
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and (5.5) (with n replaced by £) we get
This computation advances the induction step to i and proves (5.4). In particular choosing I = n in (5.4) gives us the formula
that is, CQ is the value of the Bernstein-Bezier curve Cb at x. A convenient way to visualize the recurrence formula (5.1) is in a recursive. triangle; see Fig. 5.2 for the case of polynomials of degree three.
FlG. 5.2. Up recurrence formula for the Bernstein-Bezier cubic curve.
Note that at the base of the triangle are the control points of the Bernstein Bezier curve Cb. The upward pointing arrows each have a label, either 1 — x or x depending on whether it leans to the right or left. One proceeds up the triangle
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by means of formula (5.1). Thus the vector at any location in the triangle is a sum of the two vectors at the base of the arrows pointing into its location and weighted by the labels associated with these arrows. At the apex of the triangle emerges the curve Cb at x. Now, let us reverse these steps. Place the value one at the apex of the triangle, reverse the arrows, and proceed down the triangle, and at the base out pop the Bernstein-Bezier polynomials (see Fig. 5.3). This is just the recurrence formula (5.5).
FlG. 5.3. Down recurrence formula for the Bernstein-Bezier cubic curve.
The upward and downward recurrence formulas are tied together by a conservation law. By this we mean that the quantity
is independent of I, i = 1 , . . . , n + 1. The proof is straightforward. We compute the above expression by using first downward recurrence formula (5.5) and then the upward recurrence formula (5.1), viz.
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In summary, we have shown that
5.2.
Blossoming Bernstein—Bezier polynomials: The univariate case.
Much of this chapter is devoted to multivariate extensions of recursive triangles. Before we get into this matter, we continue to highlight, in the context of the de Casteljau recurrence formula, other basic principles for recursive triangles. The two that we have in mind are blossoming (or polarization) of polynomial identities and duality. We begin with blossoming. A function P : K" —* K is said to be multiaffine provided that for each .7 = 1 , . . . , n, P(x\, X2, • • • , xn) is an affine function of Xj. Let En be the set of all vectors in M" whose components are either zero or one. ETI consists of 2™ vectors and for each a = (aj, a%, • • • , a7i.)T G En the equation
defines a multiaffine function. Moreover, each multiaffine function P can be written as a unique linear combination of these functions. In fact, we have the (multilinear) interpolation formula
This formula follows by induction on n and the univariate interpolation formula
Specializing formula (5.7), we see that P(x,x,...,x) is a polynomial of at most degree n. The next lemma provides a converse to this fact. LEMMA 5.1. Given any polynomial p of degree at most, n there exists a unique multiaffine fiyrnme.tric function P such that
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Proof. When P is symmetric, formula (5.7) becomes
where
and
a symmetric multiaffine function. A little thought yields the fact that #{a : a En, #{i '- di = 1} = k} = (£), from which it follows that
Hence, if P is a symmetric multiaffine function such that P(x, x,..., x) = 0, a; R, then P is identically zero. This proves that given any polynomial p of degree at most n, written in Bernstein-Bezier form
the unique symmetric multiaffine function such that P(x, x,..., x) = p(x) is given by
We say that the symmetric multiaffine function P described in Lemma 5.1 is the blossom of the polynomial p. There is a corollary to Lemma 5.1 that should be emphasized here. Namely, given a polynomial curve p of degree n in Bernstein-Bezier form, viz. p = Cb. its control points are given by
because it was proved above that
Next we describe a recursive triangular scheme to blossom a polynomial curve given in Bernstein-Bezier form. To this end, we go back to the recursive triangle
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FlO. 5.4. Blossoming de. Caste.ljau's triangle, for a cubic curve.
in Fig. 5.2 and relabel all the labels at each level of the recurrence as indicated in the recursive triangle above. The claim is that the apex of the triangle in Fig. 5.4 is the blossom of the Bernstein-Bezier cubic obtained as the vertex of Fig. 5.2. Generally, corresponding to (5.1) we consider the companion recurrence formula
We shall show that dg is, indeed, the blossom of CQ . For this purpose, we need only verify that dp is a multiafnne symmetric function of x\,....xn, since it obviously agrees with CQ when x\ = • • • = xn = x. Obviously, each djr is an affine function of Xk and so the core of the matter is to show that they are symmetric functions. For t = 1 there is nothing to prove. Now. assume the result is true for all k < i with (. > 2 and consider the function df (a.'i,... ,Xf). From the recurrence formula (5.10) and the induction hypothesis it is clear that Ar is a symmetric function of x\,... ,.T£_I. Hence it remains to show that, for each j = 1 , . . . , I — 1, it is a symmetric function of Xj and X(. As a function of these two variables it is a bilinear function and therefore its symmetry is equivalent to showing that its value at (xj.Xf) — (0,1) and at (xj.xi) = (1,0) are the same. That is, the proof hinges on showing that
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We evaluate the left-hand side of (5.11) by specializing (5.10) to xg = 1 and obtain
In the last two equations, we first used the symmetry guaranteed by the induction hypothesis and then the recurrence formula (5.10) at level t—\. Computing the right-hand side of (5.11) in the same manner gives
Comparing (5.12) and (5.13) proves (5.11) and therefore our claim that djf (x\,..., xn) blossoms CQ (x). Next we explore the role of duality in the present setup. Associated with every sequence of n + 1 linearly independent polynomials PJ, j = 0 , 1 , . . . , n, of degree at most n are unique polynomials QJ, j — 0,1,... ,n, also linearly independent, and at most of degree n such that
We will call Q = {qj : 0 < j < n} the dual basis associated with P = {PJ : 0 < j < n} and (P, Q) a duality pair. The philosophy behind dual pairs is that sometimes it is easier to study properties of a dual basis Q through the binomial identity (5.14) and then obtain properties of P. For the BernsteinBezier polynomials, the dual basis is given by
This follows from the binomial theorem and the formula x — t = (1 — x)(—t) + x(l-f). A nice application of dual pairs is an alternate proof of (5.9). The blossom of the polynomial Pt(x) := (x - t)n is Pt(x\,..., xn) = (xi — t) • • • (xn - t) and, consequently, P t ( l , . . . , 1 , 0 , . . . ,0) = Q^(t), where one occurs j times. Therefore, we may rewrite (5.14) in the equivalent form
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For any n + 1 distinct points t0,...,tn the polynomials pto,... ,ptn arc linearly independent. One way to see this is to notice that their dual bases are the linearly independent Lagrangc polynomials
since
Now represent any polynomial p as
for some constants ao, a\,..., o n , arid obtain its blossom as
Hence we get from (5.15) that
which is equation (5.9). 5.3.
Blossoming B-spline series: The univariate case.
Let us apply the above ideas to the B-spline basis. Throughout our discussion we concentrate on the dual basis for B-splincs and only at the end will the identification with B-splines be made. This is the approach that we shall follow later in the multivariate case. We begin with any set of In scalar values £ - „ , + ] , . . . , tn such that
These points are neither arranged in any particular order nor do they need to be distinct. Encouraged by the identity (3.77) we consider the polynomials of degree n
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First let us verify that these polynomials are linearly independent. To this end, suppose there are constants a 0 ,01,..., an such that
Since N0(ti) ^0, Ni(ti) = • • • = Nn(ti) = 0, we conclude that a0 = 0. We delete £_ n+ i and t\ from our set {t_ n + i,... ,tn} introduce the subset of points defined by
and the functions Nj(t) = (f_ n + j + 2 - t) • • • (ij - t ) , j = 0 , 1 , . . . , n — 1. Then one can easily check that Nj(t) = (t\ - i)7Vj_ 1 (t), j = 1,..., n, and so
Since the new points i_ n +2i • • • , tn-i satisfy (5.18) with n replaced by n — 1 we can now proceed, by induction on n, to conclude that QQ = • • • = an = 0 and consequently, NQ,Ni,...,Nn are linearly independent when (5.18) is satisfied. With this result in hand we consider the dual basis for NO, . . . , Nn, which we denote by MO, . . . , Mn. Our goal is to develop upward and downward recurrence formulas for the dual basis, develop blossoming relations, and finally identify the dual basis with B-splines. We begin by observing that the common zeros of the polynomials Nf+l and •^i+i1 are exactly the zeros of AT", i = 0,1,... ,n. For this reason, we get the following formula:
Let us use this formula to find the recurrence formula for the dual basis. We compute as follows:
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where in the last equation we have set M^(x) = M"+l(x) = 0. Comparing both sides of the first and last sum above we obtain the recurrence formula
initialized with MQ(X) = 0. For the upward recurrence formula we conserve sums, as with the BernsteinBezier polynomials (see (5.6)) and compute as follows:
Thus the upward recurrence formula is
where i = 0 , 1 , . . . , n — I , I = 1 , . . . , n. Moreover, since
the apex of (5.22) is
To express the control points of the polynomial on the right-hand side of (5.23) in terms of its blossom we note, using the same notation as employed in formula (5.15), that
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Hence, as in the proof of (5.9) for the Bernstein-Bezier representation, it follows that for any polynomial p with blossom P we have
Therefore, if dg (x) is the blossom of CQ(X) in (5.23). we have
Moreover, the fact that dJJ(x) can be computed by the recurrence formula
where r = 0 , 1 , . . . , n — I, i = 1,.... n can be proved in the same way that we proved (5.10) blossoms (5.1). Our final remark about the duality pair (Af, M) identifies MO, . . . , Mn as a set of B-splines. From now on we assume the points in the set ( t _ n + i , . . . ,t n } are ordered,
and we add two more points to it, one to the left of t-n+i and one to the right of tn, viz.
Corresponding to this larger set there are n + I B-splines of degree n. In the notation of Chapters 3 and 4 they are S(x t.-n+i, • • • , tt+\), i = 0 , 1 , . . . . n, and each is nonzero on the interval [to,ti]. Next we express each x & M in the form
and specialize Theorem 4.4 to the B-spline S(x\t-n+i,.. - , *i+i) to obtain the recurrence formula
Let R?(x) := nl(ti+i-t-n+i)S(x\t-n+i, • • • , U+\}-, i = 0 , 1 , . . . , n, and rewrite the above identity for n > 1 in the form
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Note also that R%(x) = 1, and R°^i(x) = R^(x) = 0 for x e (t0,t-[). Thus, it follows by induction on n that
for x in the interval (to,ii). Consequently, we have established that (jV,7£) form a duality pair on (to, £1). This observation is nothing more than the identity of Marsden, (3.77), restricted to the interval (£0^1)- So what have we accomplished? At the least we have again confirmed the point of view, also central to Chapter 3, that one way to understand B-splines is through duality. Chapter 4, on the other hand, approaches B-splines through geometric principles. This gave us a technique to develop a multivariate theory of B-splines. Can these approaches be unified for multivariate functions? To answer this question we need to identify the appropriate multivariate recurrence formulas and dual polynomials. The principles of polarization will be our guide and Theorem 4.4 will allow us to make the connection with multivariate B-splines. This is our agenda for the remainder of the chapter. Let us now use the univariate material presented so far as a foundation and turn our attention to recurrence formulas for multivariate polynomials. 5.4.
Multivariate blossoming.
It is convenient to work with homogeneous polynomials. All results have inhomogeneous versions, most of which follow in a straightforward manner from their homogeneous counterparts. The space of real homogeneous polynomials of degree n in s variables will be denoted by Hn(Rs). We make use of the set F = r.,in C Zi given by where k = (ki,.... ks)T and |k|i — ki + • • • + ks. to index the monomial basis of Hn(W). Thus every p 6 # n (R- s ) can be written in the form
for some scalars Pj^k r.,i7J, and
The notion of lineal polynomials is basic to our study of multivariate polynomial recurrence formulas. We define this concept next. For every set of n vectors we define the lineal polynomial determined by X as
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where (x, y) denotes the standard inner product of x and y in R s . Clearly a(-\X) Hn(Rs) and a fundamental question is to decide when a set of lineal polynomials spans J/n(lRs). To answer this question, we need to first extend our previous discussion of blossoming of univariate polynomials to the multivariate case. Let GI , . . . , es be the standard basis for R s , that is, (e,)^ = Sij, i, j = 1,..., s. Then every linear function Q on Ks can be written as
We need some additional notation to extend the univariate interpolation formula (5.7) to multilinear functions in n vector variables y i , . . . , y n in R s . We use / for the set {i = (ii,..., in)T : 1 < if < s , £ = 1 , . . . , n } and also set y = (yij • • • >yn}T for the column vector in R™s whose components are successively built from the components of y i , . . . , yn by setting (y)(k-i)s+i = (y/Oi, k 1,..., n, i = 1,..., s. A basic multilinear function is defined by the formula
where yr = ( ( y r ) i , . . . , (y r ) s ) T , r=l,...,n, and i = (ii,...,i n ) T is atypical vector in /. Also, it is nice to have available the vector
in R KS , associated with each such i. Then, every multilinear function has the form
This can be proved, inductively on n, using (5.30) as a starting point. A simplification occurs in (5.32) when P is symmetric. To this end, we let PS}n denote the space of symmetric multilinear functions in n vector variables in Rs. For every k e rSjT1 we define a vector in Rns by setting
where k = ( f c i , . . . , ks)T. For every i e I, we let dj(i) be the number of times that an integer j, 1 < j; < s, appears as a component of i, that is,
Then d(i) = (dj(i),... ,ds(i))T Ts,n and for every P Ps,n
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We also remark that the mapping r.,]fl : k —> r(k), given by
is a right inverse of d, that is. d(r(k)) = k while r(d(i)) is the permutation that puts the components of i in nondecreasing order. Hence, for P 6 "PS;n (5.32) becomes
where L^ is the multilinear function defined by the equation
Moreover, since Fj(e:) = <5|: for all i.j G /. it likewise follows that L^(vi) = 6^: for all k,j 6 I\n. Since, for every i 6 /,
appropriately specializing formula (5.36) gives us the fact that
Moreover, a little thought shows that
and so
(These monomials on homogeneous space play the role of Bernstein-Bezier polynomials on affine space.) Hence we obtain from (5.35) the formula
from which we conclude that whenever P PSiH and P(x, x , . . . , x) = 0 then P = 0. This leads us to the following extension of Lemma 5.1. LEMMA 5.2. Given any p 6 H n (R s ), there exists a unique P P.,)Ti such that and, moreover, dim Pn<s = dim Hn(M.s).
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Proof. There is not much more to say except that given a typical element p of Hn(Rs) written in the form
the function
is the unique P 6 Ps,n, which satisfies (5.38). As an application of these ideas, we present a proof of the formula in Lemma 1.5 of Chapter 1. We restate it here, since we will use it often in the remainder of this chapter. LEMMA 5.3. Let C be an n x s matrix. Then
Proof. We let GI, ... ,c n be the rows of the matrix C. These are vectors in 9.s. Call the function on the left-hand side of (5.39) R(CI, ..., c n ). Of course, it depends on x, but as a function of C i , . . . , cn it is in Ps,n- Referring back to definition (5.33), we see that for every k F Sjn
and hence comparing (5.39) to (5.35) shows that (5.39) is equivalent to
Note that both sides of (5.40) are multilinear symmetric functions on Rs. Therefore, by Lemma 5.2, to verify this formula it suffices to check it for n x s matrices C, all of whose rows are equal to some vector c. Suppose C has this special form. Then, by the definition of a permanent it follows that per (7(k) = n!c . Thus, using equation (5.37), we see that (5.40) indeed holds when Ci = f • . • — t**^n — ^-
5.5.
Linear independence of lineal polynomials.
We are now ready to answer the question we raised earlier about when a family of lineal polynomials span Hn(Rs). PROPOSITION 5.1. For every k e Ts>n let V^ — {v^v £ = l,...,n} be a set of vectors in Ks and define
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Then the set of lineal polynomials spans Hn(M.s) if and only if whenever P "P.,,,,. and P(uj t ) = 0 for all k e rs,n then P = 0. Proof. Let V* be the n x s matrix whose rows are vk i' • • • ' vk rr Then from (5.39) we have
Therefore, the linear polynomials o(-|Vj t ), k e Ts
Using formula (5.40), we see that this is the case if and only if there are constants /k,k r S j f l , not all zero, such that the multilinear symmetric function
vanishes at u^.. Since we have already established that span {L^ : k £ r SiH } = Hn(W) we see that there is a nontrivial P e 7\n with P(u^) — 0, all k r s , n if and only if there are constants /i,j e I\ n , not all zero, for which Q in (5.41) vanishes on all ui,. k e r s . n . A simple example of the above result is the case that each set Vj^ consists of one vector vj^. e M5 repeated n times, that is, In this case and these lineal polynomials span Hn(W) if and only if there is no p e //n.(Ki's)\{0} that vanishes on all the points v^-, k £ TSjn. To obtain a more substantial example we begin with a rectangular array X of vectors in W,
(see Fig. 5.5). For each k e Ts.n we choose a subset X^ of X consisting of n elements by setting It is to be understood that if k, — 0 for some i = 1 , . . . . s no vector of the form x zj . j = l , . . . , n appears in X^. We can visualize X^ as a histogram by associating every element in X with an integer vector (i, j) in the rectangle [l,,s] x [l,n] (see Fig. 5.6).
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FIG. 5.5. Rectangular array of vectors.
FlG. 5.6. Histogram of vectors.
Thus, every k FSin corresponds to the histogram in [l,s] x [l,n] given by {(*, j) '• 1 < j' < &j, i = 1,..., s} and .X^ selects all vectors in this histogram. THEOREM 5.1. Let X — {x1'-7' : 1 < i < s, 1 < j < n} be an array of vectors in W with s > 2 and for every k = ( f c i , . . . , fcs)T e r.,iTi set Xk = {x^': l < j < f c i ; t = !,...,«}. Suppose that for every j = (ji,..-,js)T with |j|t < n - 1 tfie vectors x 1>J ' 1+1 ,... 5xs'-'s+1 are linearly independent. Then the lineal polynomials a^ := a(-|Xj t ), k 6 r Sin are linearly independent.
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Proof. The proof is by simultaneous induction on s and n. The easiest case to consider is n = 1 and any s > 2, since it is obvious then that the linear functions (x.x 1 ' 1 ),..., (x.x 5 ' 1 ) arc linearly independent on W if and only if x 1 ' 1 ,... ,x- 5 ' T arc linearly independent. The next case to consider is s = 2 and any n > 2. Here the lineal polynomials are given as
Suppose that there arc scalars cj^.k G r2,n such that
There is a y G K2 such that (y,x li:l ) = 0 and (y,x 2 ' n ) ^ 0, since by hypothesis x1'1 and x 2>n are linearly independent. Therefore, aj c (y) = 0 for all k 6 F 2 ,n except for k0 := (0, n)T. Moreover, for that choice of k. a^ (y) ^ 0 because x1'1 and x2'J are linearly independent for j — 1, 2 , . . . . n. Thus, evaluating (5.43) for x = y gives ckf> = 0. We now remove the two vectors x 1 ' 1 ^ 2 '™ from our collection and introduce the new vectors
Every lineal polynomial a^ with k ^ ko contains (xjX 1 ' 1 ) as a factor so that
where ei = (1,0)T. Substituting this formula into (5.43) and cancelling the factor (x, x1'1) gives the formula
Thus, by the induction hypothesis applied to the array y = {y1'-' : i — l , 2 , j =
1,... ,n - 1}, we obtain c- = 0 for all |j|i < n - 2 and so c^ = 0 for all
k G r 2 , n -i. This proves the linear independence in the case that s = 2 and any n > 2. We now deal with the final case when s > 3 and n > 2. Again, we suppose for some scalars c^, k G rs/,,,
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We let F° n = {k 6 I\n :, ki = 0}, F+n = Fs,n\F° n and break up the sum in (5.44) as '
Let T be any s x s nonsingular matrix. Our hypothesis on the array X remains valid for the array
where T is any nonsingular s x s matrix. Therefore, we may rotate x1'1 into 61 = ( 1 , 0 , . . . , 0)T and assume, without loss of generality, that in fact x1'1 = ei. We let P : Rs'1 -> Rs be the projection denned by setting
Then (Py, x1'1) = 0 and so a k (Py) = 0 whenever y R5"1 and k F+ n . This means that (5.45) implies that
Next we let Q : Rs -> Rs-1 be denned as
and introduce the vectors in Rs
1
From our hypothesis on X it follows that for every k = (k\,..., fes_i)T Z+ 1 with |k|i < n — l the vectors u l i f c l + 1 ,..., u lifcs ~ 1+1 are linearly independent, and consequently, by the induction hypothesis the polynomials {a(-|fjc) : k F s _ 1]n } are linearly independent on R8"1. Moreover, since
(5.46) implies ck = 0, k e F° n . Now let us look at the other scalars c^, k 6 F+ n , appearing in equation (5.45). For k . F+ n , the lineal polynomial a(-|Xj £ ) contains (X)X 1 ' 1 ) as a factor. Hence, as in the case s — 2, we define the vectors
and observe that for k 6 F+n
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where ei = ( 1 , 0 , . . . ,0) r . Therefore (5.45) reduces to the equation
and, since our hypothesis implies that the vectors y 1 > f c ' + 1 ! . . , ; ys,k,+i are imeariy independent whenever k — ( f c i , . . . . fcs)T with k i < n-2, our induction hypothesis gives us the linear independence of the lineal polynomials a(-\Y-), j e Ta.n-iConsequently, we also conclude that c:+e — 0 for j e I\ n _-i. This means all the scalars appearing in equation (5.45) are zero since F+ n = r s , n _i +ei. D Let us pause a moment in our multivariate development and look back to the univariate case and connect the theorem above to what was said there concerning B-splines. Let {t_ n+1 , . . . , £ „ } be a set of scalars. We create from the elements of this set the vectors in R 2 , given by
Clearly, for any k — (fci, k^Y Z+ with k\ + k^ < n - 1 the vectors x l i f c l + 1 and X 2,fc 2 +i are }ineariy independent exactly when tk^+i ^ ^-fc 2 ' which is precisely condition (5.18). The corresponding lineal polynomials are given by
which are precisely the polynomial Nk(t) defined in (5.19). We proved in this case that the dual polynomials are B-splines. Our goal is to extend this result to R s ,,s > 2.
5.6.
B-patches.
We begin by introducing, under the hypothesis of Theorem 5.1. the dual polynomials B = {bfc : k rs,n.} for the set A = {a(- X-^) : k g F s . n } of lineal polynomials. In analogy with the univariate case, they are defined by the formula
and therefore are guaranteed to be linearly independent. We call each 6^, k e l\n a B-patch and the collection B a B-basis for Hn(W). To obtain a recurrence formula for B-patches we use the linear independence of the vectors x l i f c l + 1 ,.... x s ' fcs+1 for |k|i < n — 1 to define linear functions ui j^. on E™ by requiring that
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for all x Rs. The explicit form of the lineal polynomials ajt(x) leads us immediately to the recurrence formula
Thus, from the duality relation (5.49) and equations (5.50) and (5.51), we have
To ensure the validity of our change of indices in the sum above we impose the convention that b^ = 0 for k e Z s \r Sjn . Now, comparing coefficients of a?,+1(x), k G r SiTl+ i above and assuming their linear independence, we get the formula
We state this computation formally in the next proposition. PROPOSITION 5.2. Let Xn = {xij : 1 < i < s, 1 < j < n} be an array of vectors in Rs such that for each k r S j n _i the vectors X 1 > f c l + 1 ,... iX s ' fcs+1 are linearly independent. Define homogeneous polynomials b|. k e Ts,ii @ = 0,1,... ,n by the formula
where {al : k e ^s,e} are the lineal polynomials corresponding to the array Xt ~ {x*"3' : 1 < i < s, l<j<£} with the convention that b^ = 0, k e Z s \r S]ra and OQ — 6r> = 1. Then we obtain the recurrence formula
In the sense used in the univariate case, this is the down recurrence formula for B-patches. The up recurrence formula is given next.
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PROPOSITION 5.3. Let Xn be an array of vectors in Ms which satisfies the hypothesis of Proposition 5.1. For any scalars c^, k e r s>n , we define homogeneous polynomials cl (x), k £E r Si7l _^, I = 0 , 1 , . . . , n, by the recurrence formula
where Then the sum
is conserved for t = 0 , 1 , . . . , n. In particular, we have
Proof. We compute for £ = 0, l , . . . , n — 1 the sum below using (5.53), viz.
Proposition 5.2 states that the sum in the parentheses is 6? in summary, the equation
(x) and so we get,
The recurrence formulas (5.53) and (5.54) gives a pyramid scheme for computing any homogeneous polynomial of degree n expressed in terms of Bpatches. It generalizes the corresponding univariate recurrence formulas for Bernstein-Bezier polynomials and B-splines presented earlier. Also, when the array Xn = {XM : 1 < i < s, I < j < n} has the special property that XM = x, for all i, j we get aj c (x) = (x, Xi ) fcl • • • (x, xs)ks. So, by the multinomial theorem,
These are the (homogeneous) multivariate Bernstein-Bezier polynomials and s (5.52) and (5.53)-(5.54) are the familiar up and down recurrence formulas fo Bernstein-Bezier polynomials (see Figs. 5.7 and 5.8, respectively).
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FlG. 5.7. Up recurrence formula for quadratic bivariate Bernstein-Bezier polynomials.
FIG. 5.8. Down recurrence formula for quadratic bivariate Bernstein-Bezier polynomials.
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Our next result explains how to express the B-patch coefficients of a polynomial in Hn($Ls) in terms of its polar form. PROPOSITION 5.4. Let X = {x1'^ : 1 < i < s, 1 < j < n} be an array of vectors in Ks that satisfies the hypothesis of Proposition 5.1. Suppose p is an arbitrary homogeneous polynomial given in B-patch form
for some scalars cj^k FSitl. Define the vector q^ e Rsn by the equation
Then
Proof. For a fixed x 6 M s , the polynomial Px(y) — (x, y) n has the polar form
and therefore a(x|Xjc) = Pxi^)- Substituting these equations in (5.49) gives us the formula
Since span {pK : K R s } = # n (R s ), we get (5.58). The polar form of a polynomial in B-patch form can be computed by an up recurrence formula; this is presented next. (See Fig. 5.9 for the special case of Bernstein-Bezier polynomials.) THEOREM 5.2. Let X = {xlj : 1 < i < s, 1 < j < n} be an array of vectors that satisfies the hypothesis of Proposition 5.1. For any scalars cj^k 6 F Sin and vectors X i , . . . , xn £ K s , we define C"^(xi...., x^), for I = 1,.... n, by the recurrence formula
where
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FIG. 5.9. Blossoming the up recurrence formula for quadratic Bernstein-Bezier polynomials.
Then Cfi Ps,n and
Proof. The only thing that is not immediately obvious is the fact that CjUxi,... ,x.() is a symmetric function of its arguments, (. = 1,... ,n. Using (5.59) we will prove this by induction on t. By (5.59) it suffices to prove that C^ +1 (xi,..., x^+i) is a symmetric function of Xj and x<, for j = 1,..., f, that is,
We first evaluate the left-hand side of (5.61) by using our recurrence formula (5.59), viz.
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The right side of this equation has the same form with the variables Xj and x^ + i interchanged since the term C,K4-Gi+6 ~l is symmetric in i and m. Thus, it suffices m to show that the bilinear expression
is symmetric in x and y. Equivalently, if we let
then we must show that
To this end, we need to relate the linear function u1 ,l\.-rtJ \..0 ,•i = 1 , . . . ,' s to u.2,t\. i,. i = 1,... ,s. Since the sets of vectors that determine them (see (5.50)) have all the vectors x l ' fc ' +1 , i = 1,.... s, i 7^ m in common, it follows that m
r
where We prove equation (5.63) by showing that both sides of this equation agree on the linearly independent vectors x j ' >fcj ' +1 . j = 1 , . . . ,s. The details are straightforward and arc based on the fact that
and
which are both obvious from our definition (5.50) of the linear functions u^ ^ andw
' z ,k+e m ' Now, on the left side of the equation in (5.63) for x = xj''fcj' + 1 , j ^TOwe get, by (5.65), 6 t j . and on the right side, using (5.64), we get (1 — 6im)f>ij = &ij — Simfiij = fiii-, which are the same. When j = m on the left side we get A,m, while on the right side we get (1 - 8im)Sim + Aim = Sim - 6im + Aim = Aim. This establishes (5.63) and we can use it to compute Z7j jm (x, y). A substitution of (5.63) into our definition of C/iim and a simplification gives us the formula
from which (5.62) is obvious.
234 5.7.
CHAPTER 5 Pyramid schemes.
Among a large class of polynomial recurrence formulas Theorem 5.2 characterizes B-patches. To explain this, we begin with a general collection of vectors
from which we will define a pyramid scheme. We require that for each i = 0 , 1 , . . . , n, k Ts 'n-e, the vectors x *1 ,K , , . . . , x*, are linearly independent. Then, s ,K
we can define linear functionals ue i,(x), i = I,..., s by requiring that for all x e R8
Associated with X is a pyramid scheme, or up recurrence formula, given for (. — 1,..., n, by the formula
The apex CQ(X) of this pyramid is clearly a homogeneous polynomial. corresponding down recurrence formula is given by the formula
The
where, by our usual convention, pf = 0 if k e Z S \F S ^. The up and down pyramid recurrence formulas conserve sums, that is, as before the function
does not depend on i — 0 , 1 , . . . ,n. Therefore, in particular, the apex of the pyramid is
A basic unsolved problem is to describe the totality of homogeneous polynomials, which can be an apex polynomial of a given pyramid scheme in terms of the array X that defines the scheme.
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Every pyramid scheme has an associated pyramid scheme for generating multilinear functions, namely for I = 1 , . . . . n.
As it turns out. the B-patch scheme is the only one that is symmetric. This result follows next. THEOREM 5.3. Let X - {xf^ : i — 1 , . . . , s,f. — 1 , . . . , n,k
Proof. The sufficiency of this result is covered by Theorem 5.2. For the necessity, we first show that for each i = 2 . . . . , n, k e Ts , n _^, i,j — 1,... ,s and x. y e R's we have the formula
To this end, call the left-hand side of equation (5.71) Vf' i.(x, y). Thus equation (5.71) says that this function is symmetric in x and y. Using the up recurrence formula twice, as in the proof of Theorem 5.2, we get the equation
The left-hand side of (5.72) is a symmetric function of y and x. Thus, to prove (5.71), we must show, given any s x s symmetric matrix T , that for some choice of x i , . . . , Xf-2 and scalars c^., k e r., i7l,
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By definition, we see that for each level t— 1,..., s
Hence, it follows that for any k G I\ n _^
for any x i , . . . , x^_! e 1RS. Repeating this equation i — 1 more times, we get f any i\,..., ig G {1,..., s} the formula
Thus we can fix any z ' i , . . . , z^_2 and choose ig-i = i,i( = j and set r := k + e^ + • • • + Qif-2- Then we choose scalars so that (cr+ei+ej)i,j=i,...,s is the prescribed s x s symmetric matrix. This establishes equation (5.71). Our next assertion is that
For the proof of (5.74), we distinguish between two cases. Case 1. i = j. In this case, (5.71) simplifies to the equation
Now choose x = x^ }
m,K-\~Gi
and y = x^, 1
i,K+Gi
in the above equation. By the
definition of these linear functions (see (5.67)) we have u^T1 (y) = 1 and t^T1 (x) = 0, since ra ^ i. Hence, we get
Case 2. i ^ j. Here we choose x = xfm,K+Gj i. „ and y = x^,;,K-t-Gi -1 so that
Also, from (5.75) of Case 1, we have
Substituting these equations into (5.71) and simplifying gives us the equation
which agrees with (5.74). Thus, in all cases, equation (5.74) holds.
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Now fix an m ^ j. Then definition (5.67) (with x = xf~^+e ) and (5.74) yields the formula
Taking the inner product of both sides of this equation with a, we conclude that l um , (x^",1 ) = 1 and therefore we obtain the relation , k ^ m,K+e/
We use this equation repeatedly in the following way. We choose k = kmem above and, for every m £ {ji,..., js-i}, we have
Choosing {jj,... tja-i} = {!,..., s}\{m}, it follows that for k F Sin _^
Now we define
and obtain
5.8.
M-patches.
Before we return to B-patches we wish to describe another example of a pyramid scheme. The scheme is selected by assuming the labels u^, , i = 1,... ,s are all the same at any given level £, that is, are not dependent on a location k within the level. Specifically, we start with the array
as with B-patches, but now we choose the set of vectors
assumed to be linearly independent, and define our linear forms by the requirement that
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This leads us to the down recurrence formula
(See Fig. 5.10 for the case of bivariate quadratic M-patches.) We call the homogeneous polynomials M£(x), k Ts,n, M-patches and, as before, we set M£ = 0 for all k <= Z s \r s > n _f. To obtain the generating function for M-patches, it is convenient to introduce for each f, linearly independent vectors y 1 ' ^ , . . . , ys'e such that
Let Re be the matrix
and observe that u\(x) = (flj"x)j,
FlG. 5.10.
i = 1,..., s. Then, by equation (5.77), Re l
Down recurrence formula for quadratic M.-patches.
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is given by
This setup leads us to the next proposition. PROPOSITION 5.5. Let X = {xij : 1 < i < s, 1 < j < n] be an array of vectors in Rs such that the vectors x1-^,..., xs'f are linearly independent for each d=l,...,n. Then
Moreover, the following are equivalent: (i) p is an apex polynomial for the M-patches, that is. for some scalar s c^, k e Ts,n
(u)Pespan{Ul=l(x,R3-):xtE.s},
(iii) p e {P(Rr,R-2-,...,Rn-) • P e Ps,n}
Proof. First we prove equation (5.79). To this end, we multiply both sides of (5.78) by y* and sum the resulting expressions over k e r. s , T ,._f. This gives us the formula
With this formula, it follows easily, for £ = 0 , 1 , . . . . n, that
and, in particular, (5.79) follows. Claim (ii) follows directly from (5.79), while for (iii) we recall by Lemma 5.3 and formula (5.40) that
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We already proved that span [L^ : k TStH} = Pa^n and so the above formula shows that
which establishes (iii). From the fact that dim Ps,n = dim Hn(Rs) we obtain the next corollary. COROLLARY 5.1. Let X = {x1^ : 1 < i < s, 1 < j < n} be an array of vectors that satisfies the hypothesis of Proposition 5.4. Then the M.-patches {Mjj. : k rs,n} Q-TZ linearly independent if and only if whenever P G Ps,n and P(Rix,..., Rnx) = 0 for all x e K s , it follows that P = 0. We apply this corollary to prove two facts about M-patches. First, we go back to formula (5.79) and observe that the M-patches corresponding to matrices RI ,..., Rn are unchanged if we replace these matrices by the symmetric matrices ^(Ri + R'I), • • • , \(Rn + RH)I respectively. Also, observe that if for some i, I < i < n, Ri is a singular matrix, so that ^y = 0 for some y e Rs\{0}, then by (5.79)
Thus the nonsingularity of the matrices RI ,..., Rn is a necessary condition for the linear independence of the corresponding M-patches. The next proposition provides necessary and sufficient conditions for the linear independence of quadratic M-patches on Rs. PROPOSITION 5.6. Let R\,R^ be two s x s matrices and suppose AI, . . . , As are the eigenvalues of the matrix Q = RI R% l • Then the corresponding quadratic M-patches are linearly independent if and only if
Proof. The set of symmetric bilinear functions on Ms correspond to s x s symmetric matrices. Thus, Corollary 5.1 says that the quadratic M-patches for RI and RI are linearly independent if and only if whenever B is an s x s symmetric matrix such that (RiX, B#2x) = 0 for all x IRS then B = 0. That is to say, whenever BQ + QTB = 0 it follows that B = 0. By a theorem of Schur, cf. Stoer and Bulirsch [SB, p. 328], there is a nonsingular s x s matrix A such that Q = ADA~l where
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is an upper triangular matrix. Now suppose that B is an s x s symmetric matrix such that BQ + QTB — 0. Multiply both sides of this equation, from the left by AT and from the right by A to get the equation
where G = ATBA. Since D is upper triangular it follows that G = 0 (or equivalently B — 0) if and only if (5.80) is valid. The next theorem extends this result to M-patches of degree n. THEOREM 5.4. Let R\,..., Rn be nonsingular s x s matrices such that there are nonsingular s x s matrices L and K such that the matrices Qi = LRiK are lower triangular for i = 1,... ,n. Let W be the s x n matrix, with columns w i , . . . , w n given by
Then the M-patches {M^ : k G ^s,n} are linearly independent if and only if
Proof. First, observe that if P G Ps,n and P(.Rix,... ,Rnx) = 0 for all x G Rs then G ( x i , . . . , x n ) :— P(L~ 1 xi,..., L -1 x n ) has the property that G(QiX,..., Q n x) = 0 for all x G R s . Moreover, G G Ps,n and is zero if and only if P is zero. Thus we may assume without loss of generality that RI = Qi^ ? = 1 , . . . , n. Now, let any P G Ps,n be written in the form
where
and
(see equations (5.31) and (5.36)). Therefore, we conclude that
and, since Qj, i = 1,..., n are lower triangular,
where ./^(x) is a homogeneous polynomial containing only monomials of lower order than x relative to the lexicographic ordering on F Sjri . Thus we see that
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the polynomials Lj^Qr,..., Q n -), k r S;n , are linearly independent if and only if
But in view of equation (5.40), this is precisely the condition (5.81). 5.9.
Subdivision by B-patches.
We now return to B-patches and present a result concerning subdivision using these functions as a basis for polynomials. The observation we want to make is best presented for affine (inhomogeneous) B-patches. For almost all of the remainder of the chapter we will work in the space II n (R s ~ 1 ) of all polynomials of total degree at most n in s — 1 variables with s > 2. We will not clutter our exposition with new notation special to the inhomogeneous case. Recall that any p n n (E s-1 ) gives a homogeneous polynomial q, by the familiar device
Conversely, every q 6 Hn(Rs) when restricted to (l,x) T , x e Rs~l yields a polynomial in n n (]R s ~ 1 ). For the inhomogeneous B-patches, we start with an array in Rs~1
apply our previous homogeneous theory in Rs to the array
and then restrict the homogeneous B-patches to the vector (l,x) T ,x e Rs-1 to form the inhomogeneous B-patches, fcj^x^x Rs~1. Also, the corresponding lineal polynomials aj c (x),k £ TS:H are similarly obtained. To ensure that they are linearly independent we will always demand that for every k 6 F S)n , the vectors x l j f c l + 1 ,... ,x s>fcs+1 are affinely independent, that is, the simplex they generate has positive volume. Thus the lineal functions associated with the array in Rs when restricted to the vectors (l,x) T ,x e R*"1, become the barycentric components of x relative to the simplex generated by x l i / C l + 1 ,... ,x s ' fcs+1 . This means that equation (5.49) becomes
Also, since a^(0) = 1, k G F S)Tl , we get, in particular, that
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The result on subdivision for the B-patch basis that we are about to describe deserves some preliminary explanation. To motivate our presentation, let us suppose for the moment the array has the special property that XM' = x l , i = 1 , . . . , s, j — 1 , . . . , n. Therefore, it follows that for x £ R s-1
where w i ( x ) , . . . , us(x.) are the barycentric components of x relative to the simplex generated by x 1 , . . . , xs (see equation (5.56)). These functions are, of course, the Bernstein-Bezier polynomials. We wish to compute a given polynomial p II^R8"1) expressed in Bernstein-Bezier form
at some point y e Rs l. For this purpose, we pick some affine bijection T on R8"1 that is contractive and has y as its unique fixed point. For each t— 1 , 2 , . . . , we represent p in its Bernstein-Bezier form relative to the simplex generated by the points T ^ x 1 , . . . , T£x6'. We claim that the new control points will all converge to p(y) as i —>• oo (see Fig. 5.11). This fact is proved next.
FlG. 5.11.
Subdivision by ^-patches.
THEOREM 5.5. Let T be a contractive affine bijection on Ms l with fixed point y. Suppose that X = (x zj : ! < « < « , 1 < j < n} is an array of points in M*"1 such that for each k e r S)Tl , x l i f e l + 1 , . . . ? x s ' f c s+1 are affinely independent. Given p G Il ri (R' s ^ 1 ) in the B-patch form
For each i — 0,1, 2 , . . . , we. write p as
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where b k ^, k e T Sjn , are B-patches for the array X^ := T^X.
Then for all
k £ 1 s,n
The proof of this result is best given in the homogeneous setting. The basic idea of the proof is to identify the form of the matrix that takes the initial control points c = (ck : k G Ts,n) into GI = (c^e : k £ r s>n ). Therefore, all our matrices will be N x N where TV := #rSjTl. It is convenient to organize all the (homogeneous) B-patches and lineal polynomials into vectors in RN. For this reason, we define
and Here X is any array in Rs that satisfies the hypothesis of Theorem 5.1. We want to see how these vectors transform when X is replaced by the new array TX. It is the monomial basis in Hn(Rs) that is the most convenient for the study of this question. Therefore, we define
where
Now let us express a(x) and b(x) in monomial basis by introducing N x N matrices A and B, by setting a(x) = Am(x) and b(x) = 5m(x). Obviously, A = A(X) and B = A(X} depend on the array X. We also need for every s x s matrix R the matrix £(R) defined by
Glancing back at the formula preceding Theorem 1.3 we see that
In the next lemma, we describe the transformation rules that pertain to sending the array X to the new array TX. LEMMA 5.4. Let R be an s x s matrix and X an array in Rs satisfying the hypothesis of Theorem 5.1. Then
and
RECURSIVE ALGORITHMS FOR POLYNOMIAL EVALUATION
245
Proof. From the definition of lineal polynomials we have the equation
while by the definition of £(RT), it follows that
This computation proves the first equation (5.88). For the proof of (5.89), it is convenient to first express the duality .relation (5.49) in matrix notation, viz
(Keep in mind that the inner product on the left lives in R^ and on the right it lives in R 5 " 1 .) Expand the right-hand side of equation (5.90) by the multinomial theorem to obtain the equation
where
Thus, replacing X by RX in this identity and likewise using (5.88), we see that
However, from (5.87) we obtain
and so equation (5.89) follows. Next we identify a useful form for the control points Ci obtained after one step of the subdivision algorithm. Specifically, we have, by Lemma 5.4, the equation
Hence, we conclude that
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where
For the £th stage of subdivision, it follows in a similar manner that
where
because the arrays Kg are generated by the recurrence formula, Xi = TXi-\,t — 1 , 2 , . . . . Thus, the proof of Theorem 5.5 demands an analysis of lim^oo 8(RYThis is done next. But first we need an auxiliary fact about the N x N matrix £(R). LEMMA 5.5. Let R be an s x s matrix with eigenvalues r i , . . . , rs. Then 8(R} has eigenvalues k!mj ( (r), k 6 F S ) n > where r — ( r i , . . . ,r s ) T . Proof. From definition (5.86), specialized to the choice R = /, it follows that 8(1} = I and also that 8(R\R'2) = 8(Ri}8(R2}. Hence, we have the equation
This means we can assume that R has been reduced to its Jordan normal form, viz.
where each Jj is a Jordan block. With a little thought, we see that for any k e r S)Tl and x e Es
where /^ is a homogeneous polynomial of lower order than m^, relative to lexicographic ordering in F S)Tl . We remark that this lemma leads to the formula
The next lemma is central in our proof of Theorem 5.5. LEMMA 5.6. Let R be an s x s matrix whose largest eigenvalue is one and is simple. Suppose that x, y are the corresponding left and right eigenvectors of -R, that is, Rx. = x, RTy — y, normalized so that (x, y) = 1. Then
Proof. From the definition of 8(R}, equation (5.91), and our hypothesis it follows that
RECURSIVE ALGORITHMS FOR POLYNOMIAL EVALUATION
247
Moreover, from Lemma 5.5 and our hypothesis, we conclude that the largest eigenvalue of £(R) is also one and simple. We also have by the multinomial theorem the formula
Consequently, the lemma follows from the following well-known fact, cf. Stoer andBulirsch [SB]. LEMMA 5.7. Let G be an N x N matrix whose highest eigenvalue is one and simple. Suppose that Gu = u and GTv = v, where u,v are vectors in RN, normalized so that (u, v) = 1. Then
We have assembled all the pieces needed to complete the proof of Theorem 5.5. It follows next. Proof of Theorem 5.5. We introduce the s x s matrix R by setting
By our hypothesis on T, J?((l,y) T ) = (l,y) T and one is a simple eigenvalue of R. Thus it follows from Lemma 5.6 and formula (5.93) that for all k, r E F S)n
The scalars hr depend on the right eigenvector of R, and also, of course, the array x- We can identify them by using the following computation. According to (5.82), we have the equation
In other words, it is the case that 5e = e where e := ( 1 , 1 , . . . , 1)T e RN. Thus, summing both sides of equation (5.94) over k E I\n, we get h? = 1, for r E r S;Tl . This means that
and so combining this equation with (5.92) proves the result.
5.10.
B-patches are B-splines.
In our next result, we identify (inhomogeneous) B-patches with multivariate Bsplines of Chapter 4. To this end, we recall the notation used in that chapter which states that for any set V = ( v ° , . . . , v n } C Rn
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Here we also use, for j = 0 , 1 , . . . , n, the affine functions of x Rn given by
In particular, it follows that det(V) = detj(v j |y), for j = 0 , 1 , . . . , n . Also note that the barycentric components of x E W1 relative to the simplex V = [v°,..., vn] are given by the formula
Thus, starting with a set X in Es 1 with ra points , ra > s, the recurrence formula for the multivariate B-spline presented in Theorem 4.4 may be rewritten in the following way. For any V = {x n ,..., x*s} C X
ma for ra > s
Recall that in the univariate case, when we identified B-splines with the dual polynomials Nj, j = 0 , 1 , . . . , n , corresponding to some set of scalars {t-n+i,..., tn}, we constructed the B-splines from the larger set {£_ n , • • • , tn+i}We do the same in the multivariate case. Starting with an array
with corresponding B-patches, frj^x), k e F S)n , x e Rs 1, where x 1 ' f c l + 1 ,... ,x s ' fcs+1 are assumed to be affinely independent, we enlarge X to the array From this larger set, we select subsets
much like we selected subsets from X to form B-patches. In this case for any k E TSjH it follows that #Y^ = n + s. Therefore, the corresponding B-spline Sr(x|yjc) is a piecewise polynomial of degree n (see Theorem 4.1). We will show that on a certain region of R8^1 5'(-|Fjc) is proportional to 6^. To this end, we let V^ = [x 1 ) f c l + 1 ,... ,x s>fc5+1 ] be the simplex generated by the set of vectors x 1 > f c l + 1 ,... ) x s>/es+1 and Qn the intersection of all such simplicies obtained from points in the array X, viz.
(See Fig. 5.12.) Note that the set V^ represents the vectors across the top of Y^.
RECURSIVE ALGORITHMS FOR POLYNOMIAL EVALUATION
249
FlG. 5.12. The set f} 2 .
THEOREM 5.6. Suppose that QO ^ 0. Then for every k e r SjTl
We begin the proof of this theorem with the following lemma. LEMMA 5.8. Suppose ft° ^ 0. Then there is e e (-1,1} SMC/I t/m£
/or a// k = ( f c i , . . . , fcs)T, 0 < fcj < n — 1, i = 1 , . . . , s. When k FS;n and fcj > 0 then ^-e = ^Mx1'^"1"1}, otherwise we use the convention that ^ic_e. contains no vectors of the form x * ' 1 , . . . , x*'n. Proof. Suppose k G F Sjri and for some i, 1 < z < s, /c^ > 0. By hypothesis the intersection of the two simplices [Vj^] and [Vj c _ e .] has a nonempty interior. The hyperplane
is a common face of both simplices. Since [VjJ and [Vj c _ e .] have a nonempty interior, the vectors x l ' fci+1 e [V^] and x l>fei e [Vj c _ e .] must lie on the same side of this hyperplane. Thus we conclude that
From this equation, applied repeatedly to the positive components of k, the result follows. LEMMA 5.9. Suppose that the hypothesis of Lemma 5.8 holds. Then for every k G ^s,n such that for some i, 1 < i < s, ki = 0 we have
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The proof of this lemma can be found in [DMS]. Proof of Theorem 5.6. The proof is by induction on n. When n = 0, 60 ( x ) = 1 by convention, and YQ = [x 1 ' 1 ,... ,x s>1 ] = VQ. Thus by equation (5.98) 5(x|y0) = l/|det(Vo)| on 00 = VQ. Therefore, (5.100) is true in this case. Now suppose n > 0 and call the left-hand side of equation (5.100), JVj^x), that is,
From the recurrence formula (5.99), specialized to the sets X = Y^ and V = Vj^, we get the equation
and therefore it follows that
Using Lemma 5.8, this formula simplifies to the equation
On the other hand, from the down recurrence formula for the B-patches (see Proposition 5.1) we have the formula
In this equation, we used the formula for barycentric components given in (5.97). Keeping definition (5.96) in mind, we have that detj(x|Vjc_e ) = det^x^) whenever kj > 0. Now suppose the result holds for n — I where n > 1. From Lemma 5.9, the sets [lj c _ e .] and £ln have a disjoint interior, when ki = 0 and so, in this case •^k-e-( x ) = ^' for x E £ln. Moreover, 6jc_e (x) = 0, when ki — 0. On the other hand, when ki > 0, the induction hypothesis implies that 6jc_e (x) = Arjc_e (x) for x G Jln. Thus in all cases 6jc_e (x) = A^j(_e (x), i = 1 , . . . , s, x 6 O n , and therefore, by the above two recurrence formulas, ^(x) = N^(x). COROLLARY 5.2. Suppose that fin has a nonempty interior. Then
and {Nfc : k 6 ^s,n} are linearly independent on any region contained in £ln.
RECURSIVE ALGORITHMS FOR POLYNOMIAL EVALUATION
251
Proof. Both results follow immediately from the identification of b^ and N^ on Jln made in Theorem 5.6. Theorem 5.6 provides a method for construction of linear spaces spanned by B-splines. The interested reader can find a discussion of this important issue in Dahmen, Micchelli, and Seidel [DMS] as well as in the recent papers by Gormaz [G], Gormaz and Laurent [GL], Sauer [S], and Seidel [Se]. References [CM]
[DMS] [G] [GL]
[S] [Se]
[SB]
A.S. CAVARETTA AND C.A. MICCHELLI, Pyramid patches provide potential polynomial paradigms, in Mathematical Methods in Computer Aided Geometric Design II, T. Lyche and L.L. Schumaker, eds., Academic Press, San Diego, 1992, 69-100. W. DAHMEN, C.A. MICCHELLI, AND H.P. SEIDEL, Blossoming begets B-spline bases built better by B-patches, Math. Comp., 59(1992), 97-115. R. GORMAZ, Floraisons polynomials: Applications a I'etude des B-splines a plusieurs variables, These, Docteur de L'Universite Joseph Fourier-Grenoble 1, June 1993. R. GORMAZ AND P.J. LAURENT, Some results on blossoming and multivariate B-splines, in Multivariate Approximation and Wavelets, K. Jetter and F. Utreras, eds., World Scientific, Singapore, 1993, 147-165. T. SAUER, Multivariate B-splines with (almost) arbitrary knots, preprint, August 1993. H.P. SEIDEL, Representing piecewise polynomials as linear combinations of multivariate B-splines, in Mathematical Methods in Computer Aided Geometric Design II, T. Lyche and L.L. Schumaker, eds., Academic Press, San Diego, 1992, 559-566. J. STOER AND R. BULIRSCH, Introduction to Numerical Analysis, Springer-Verlag, New York, 1980.
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Index
Bernstein-Bezier basis, 2 Bernstein-Bezier curve reparameterization, 26-27 Bernstein-Bezier manifold, multivariate, 27 Bernstein-Bezier polynomials blossoming, univariate case, 211—215 dual basis for, 214 multivariate B-splines and, 169—171 of degree n, 27 on affine space, 221 recursion, 208-210, 229-230, 232 Bernstein operator, 8-9 ^-splines, 106-107 Bias, 106 Binary fractions, 7 Binomial theorem, 139-140, 141 Biorthogonal vectors, 183 Bivariate B-splines, 154-155, 156, 200, 201-202 Blending functions, 115-116 Blossoming B-spline series, univariate case, 215-219 Bernstein-Bezier polynomials, univariate case, 211—215 de Casteljau's triangle for a cubic curve, 213 multivariate, 219-222 polynomials, 211-212 up recurrence formula for quadratic Bernstein-Bezier polynomials, 232 Brunn-Minkowski inequality, 154 Budan-Fourier lemma, 127-128
Affinely independent points, 155 Affinely independent vectors, 156, 242, 243 B-patches, 227-233 as B-splines, 247-251 pyramid schemes and, 234-237 subdivision by, 242-247 B-splines B-patches as, 247-251 basis, 123-131 dual, 215-219 variation diminishing property of, 143-146 bivariate, 154-155, 156, 200, 201-202 correspondence on historical development, 200-204 fine knot quadratic, 61 geometrically continuous, 133 multivariate Bernstein-Bezier polynomials and, 169-171 denned, 155 degree-raising formula for, 169—171 multiple points and explicit formula, 169-176 smoothness and recursions, 155-169 square of, 171 truncated power functions and, 189-191 nth degree forward, 58 Polya frequency functions and, 203 quadratic, 59, 61 Schoenberg's geometric construction of, 150-153 series, blossoming, univariate case, 215-219 sum of products of, 171 univariate, 153, 200-201, 203-204 as volumes, 149-155
Cardinal spline functions, 58—61 Cauchy-Binet formula, 38 Cholesky factorization of matrices, 113 Common supporting hyperplane, 161 253
254
INDEX
Connection matrices, 116-121, 123-131 Connects, defined, 34 Continuity Prenet frame, 113-115, 116-117 geometric B-splines, 133 of order n, 110 reparameterization matrices and, 105-110 modulus of, 14 Control point, control polygon paradigm, 1-10 Convergence de Casteljau algorithm, 10 de Casteljau subdivision to cubic curves, 4 matrix subdivision, 10-20 stationary subdivision, 67-83 Convex polyhedrons, 160 Corner cutting, 3, 34-38, 40 Correspondence on historical development of B-splines, 200-204 Cube splines, 192-300 Cubic curves blossoming de Casteljau's triangle for, 213 convergence of de Casteljau subdivison to, 4 corner cutting, 34-36 de Casteljau evaluation of, 208 Curvature vector for planar curves, 106 Curvatures Prenet equation and, 110-115 of vectors, 112 de Casteljau recurrence formula, 207-211 de Casteljau subdivision, 1-10 de Casteljau tableau, 3 de Rahm-Chaikin algorithm, 55-58 Degree raising formula for multivariate B-splines, 169-171 Dual basis for B-splines, 215-219 for Bernstein-Bezier polynomials, 214 Dual functionals, 135-143 Exceptional point, 160 Faa di Prenet Frenet Frenet
Bruno formula, 109 equation, 110-115 frame, 111-117 matrices, 110-115, 117, 121-122
General position, sets in, 156 Geometric continuity B-splines, 133
of order n, 110 reparameterization matrices and, 105-110 Gram matrix of vectors, 113 Hermite-Gennochi formula, 201 Histogram of vectors, 223-224 Holder continuous functions, 17 Holder's inequality, 16-17 Hurwitz matrices, 86-87, 89-90 Hurwitz polynomials, 83-95 Jordan normal form of matrices, 246 Knots fine-knot quadratic B-splines, 61 insertion identity for truncated power functions, 179-181 for multivariate B-splines, 163-166 and variation diminishing property of the B-spline basis, 143-146 knot regions for bivariate quadratic B-splines, 157 lifting, 151 Lagrange polynomials, 215 Lane-Riesenfeld subdivision, 61-67 Laurent polynomials, 68, 83, 84, 95, 98, 102 Lifting knots, 151 Lifting of curves, 115-122 Lineal polynomials, 219-227 Linear cube spline, 194 M-patches, 237-242 Mask of stationary subdivision, 68 Matrices Cholesky factorization, 113 connection, 116-121, 123-131 connects, defined, 34 Frenet, 110-115, 117, 121-122 Gram matrix of vectors, 113 Hurwitz, 86-87, 89-90 Jordan normal form, 246 permanent of, 28-29 permutation, 5 reparameterization defined, 109 examples, 20-29 and geometric continuity, 105-110 signum of, 30-31 stochastic, 29-34 strictly totally positive, 41-43
255
INDEX subdivision, 6-10 totally positive defined, 37 product of two totally positive matrices, 37-38 strictly totally positive, 41-43 variation diminishing curves and, 38-53 Matrix subdivision. See also Stationary subdivision convergence criteria, 10-20 corner cutting, 34-38 de Casteljau subdivision, 1-10 example, 33 introduced, 1 reparameterization examples, 20-29 stochastic matrices, 29-34 total positivity and variation diminishing curves, 38-53 Modulus of continuity, 14 Multiaffine functions, 211-212 Multiple points of multivariate B-splines, 169-176 Multivariate B-splines Bernstein-Bezier polynomials and, 169-171 denned,155 degree-raising formula for, 169-171 multiple points and explicit formula, 169-176 smoothness and recursions, 155-169 square of, 171 truncated power functions and, 189-191 Multivariate Bernstein-Bezier manifold, 27 Multivariate Bernstein-Bezier polynomials, of degree n, 27 Multivariate blossoming, 219-222 Multivariate partial fraction decomposition, 172—176 Multivariate truncated powers identities, 186-192 smoothness and recursion, 176-186 One periodic functions, 22 Partial fraction decomposition, multivariate, 172—176 Patches B-patches, 227-233 as B-splines, 247-251 pyramid schemes and, 234-237 subdivision by, 242-247 M-patches, 237-242 Permanent of matrices, 28-29
Permutation matrix, 5 Piecewise polynomial curves B-spline basis, 123-131 connection matrices and, 123-131 curvatures and Frenet equation, Frenet matrices, 110-115 dual functionals, 135-143 geometric continuity, reparameterization matrices, 105-110 introduced, 105 knot insertion and variation diminishing property of the B-spline basis, 143-146 projection and lifting of curves, 115-122 Piecewise polynomial surfaces correspondence on historical development of B-splines, 200-204 cube splines, 192-200 geometric methods B-splines as volumes, 149-155 introduced, 149 multivariate B-splines multiple points and explicit formula, 169-176 smoothness and recursions, 155-169 multivariate truncated powers identities, 186-192 smoothness and recursion, 176-186 Planar control polygon, 2 Point zeros of multiplicity n, 124 Polarization. See Blossoming Polya frequency functions and B-splines, 203 Polyhedrons, convex, 160 Polynomials. See also Piecewise polynomial curves; Piecewise polynomial surfaces; Recursion Bernstein-Bezier. See Bernstein-Bezier polynomials blossom of, 211-212 Hurwitz, 83-95 Lagrange, 215 Laurent, 68, 83, 84, 95, 98, 102 lineal, 219-227 Roth-Hurwitz, 88 Polytopes, 160 Pre-wavelets, 100 Projection of curves, 115-122, 116 Proper triangle, 158 Pyramid schemes, 234-242 Quadratic B-splines, 58, 61
256
Quadratic M-patches, 238 Quadratic truncated power, 180, 187 Recursion B-patches, 227-230 Bernstein-Bezier polynomials, 208-210, 229-230, 232 blossoming and. See Blossoming de Casteljau, 207-211 introduced, 207 M-patches, 238 multivariate B-splines, 155-169, 166-169 multivariate truncated powers, 176-186 pyramid schemes, 234 truncated power functions, 181-182 Recursive triangles, 209-210 Regular curves, 108 Reparameterization matrices denned, 109 examples, 20-29 and geometric continuity, 105-110 Reparameterization of the Bernstein-Bezier curve, 26-27 Ripplets, 96 Rolle's theorem, 128 Roth-Hurwitz criterion, 87 Roth-Hurwitz polynomials, 88 Schoenberg operator, 64 Schoenberg's geometric construction of B-splines, 150-153 Sets in general position, 156 Signum of matrices, 30-31 Smoothness multivariate B-splines, 155-169 multivariate truncated powers, 176-186 Splines B-splines. See B-splines /3-splines, 106-107 cardinal spline functions, 58-61 cube splines, 192-200 Standard d-simplex, 27 Stationary subdivision. See also Matrix subdivision cardinal spline functions, 58-61 convergence, 67—83
INDEX de Rahm-Chaikin algorithm, 55-58 Hurwitz polynomials and, 83-95 introduced, 55 Lane-Riesenfeld subdivision, 61-67 mask, 68 symbol, 69 wavelet decomposition, 96-103 Stochastic matrices, 29-34 Strictly totally positive matrices, 41-43 Subdivision by B-patches, 242-247 matrix. See Matrix subdivision stationary. See Stationary subdivision Subdivision matrices, 6-10 Subdivision operator, 68 Sylvester's determinantal identity, 43 Symbol of stationary subdivision, 69 Tension, 106 Totally positive matrices defined, 37 product of two totally positive matrices, 37-38 strictly totally positive, 41-43 variation diminishing curves and, 38-53 Truncated powers, multivariate identities, 186-192 smoothness and recursion, 176-186 Univariate B-splines, 153, 200-201, 203-204 Variation diminishing curves and total positivity, 38-53 Variation diminishing property of the B-spline basis, 143-146 Vectors affinely independent, 156, 242, 243 biorthogonal, 183 curvature, for planar curves, 106 curvature of, 112 Gram matrix of, 113 histogram of, 223-224 Wavelet decomposition, 96-103 Zero counting convention, 124-125
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