Applied Mathematics and Mechanics (English Edition, Vol. 21, No. 4, Apt 2000)
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Applied Mathematics and Mechanics (English Edition, Vol. 21, No. 4, Apt 2000)
Published by Shanghai University, Shanghai, China
Article ID: 0253-4827(2000)04-0437-10
4TH-ORDER SPLINE WAVELETS ON A BOUNDED INTERVAL* Duan Jiwei ( ~ j ) l ,
Peter Kai-kwong Lee (~j~J~mS~)~-
( 1. Department of Civil Engineering, Zhejiang University of Technology, Hangzhou 310014, P R China; 2. Department of Civil Engineering, The University of Hong Kong, Hong Kong, P R China) (Communicated by Wu Qiguang) Abstract: The 4th-order spline wavelets on a bounded interval are constructed by the 4 thorder truncated B- spline functions. These wavelets consist of inner and boundary wavelets. They are bases of wavelet space with finite dimensions. Any function on an interval will be expanded as the sum of finite items of the scaling functions and wavelets. It plays an important role for numerical analysis of partial differential equations, signal processes, and other similar problems.
Key words: B-spline; wavelet; bounded interval CLC numbers: 0242.29; O241.82 Document cede: A
Introduction Wavelet analysis has experienced an enormous development in recent years. One of the important fields is the numerical analysis of differential equations. The advantage of adopting wavelets has been reported [1-4] Classical approaches to wavelet construction deal with multiresolution analysis ( M R A ) on the whole real axis, but recently, interests have been developed to construct wavelets on interval. Any function on an interval can easily be expanded into wavelet series as long as these wavelets are obtained. Some techniques have been developed in dealing with wavelets on an interval TM 6] One simple technique is to extend a function, given on the interval [ 0, 1 ] , to the real line by setting its values outside [ 0, 1 ] to zeros. This will introduce discontinuities at the end points but can be solved by increasing the number of wavelets. Another method is to periodize a given function. By using the scale and wavelet function in a periodic MR_A, a function can be expressed approximately into an interval similar to a function in Fourier expansion. The focus of the present study is to construct wavelets on the interval. Since the scale and wavelet space are finite dimensional at a given level, any function on an interval can be represented by a t-mite wavelet series for the interval. The wavelet developed can play an important role in partial differential
Received date: 1997-09-22; Revised date: 1998-09-i0
Biography: Duan Jiwei (1964 ~ ), Associate Professor, Doctor 437
438
I)uan Jiwei and Peter Kai-kwong Lee
equation, signal processing and similar problems. A method for the construction of spline wavelets on the interval is proposed. Once constructed, it becomes easy for wavelets to be applied in problems involving computational mechanics.
1
M u l t i r e s o l u t i o n A n a l y s i s a n d W a v e l e t s o n [0, 1]
D e f i n i t i o n 1 ( M u l f i r e s o l u t i o n analysis on [0, 1]) A f u n c t i o n f ( x ) E L2[0, 1] is said to be able to generate a multiresolution analysis (MRA) if it can generate a nested sequence of closed subspaces V~~ t] that satisfies 1)
V[oo, t] C V[~ 11 C "";
2) 3)
eIoSL2[0,l](.UaVJ~ j;,,0
) = L2[0, 1];
~
4) f ( x ) E V~~ 1]<:,f(x + 2 -j) E V~~ 1]r where Z+ is the set of positive integers;
E Y~~ 1] , j E Z + ,
5) { f ( x - k)}kel.a~,forms a Rieszbasis of VCo~1] on [0, 1], i . e . , there are constants Aj and Bj, where 0 < As ~< Bj < ~ , infAj > 0, sup Bj < ~ , such that kE l~dta
for all { ct } E f-. If a function ~ ( x ) generates an MRA, then ~ ( x ) is called a scaling function. The different integer translation and the dyadic dilation of ~ (x) will give a new family of the functions, namely ~ii(x) = 2J/"q)(2Jx - i) which are the base in V5~ 1] Definition 2 {Wavelets on [0, 1]) A function r E L'-[0, 1] is called a wavelet if it generates the complement arc orthogonal subspaces W5~ 1] of an MRA, i . e . ,
nz o, ll = in which Cj,(x) = 2JP-r
- i) E WS~
E Z+), E Z+).
Dilation and translation of a wavelet also generates a family of wavelet similar to Definition 1. If the inner product is given by
= oCJk(x)r
= 8irSkk,(j,k,j',k"
E Z+)
(1)
then r is called an orthogonal wavelet. Otherwise it is called a semi-orthogonal wavelet (such as spline wavelet) or a non-orthogonal wavelet (such as Mexican hat wavelet).
2
Spline and Scale Space
Any function, f ( x ) , on [ a , b ] can be changed to f ( t ) on [ 0, 1 ] by substituting x = (t - a ) / ( b - a ) . Since the even order spline functions are often applied in the field of computational mechanics, the ruth B-spline function is therefore adopted in this paper and can be defined as
4th-Order Spline Wavelets on a Bounded Interval
~,(x)
-
_ (m
1)!,.0
x
k
-
k
439
-
+
,
(2)
where m is an even integer,
- k!(m2
k)!'
O,
and the compact support of ~,, ( x ) is on [ -
m/2,
m/2]
x < O, .
Let the knot sequence on the bound [ O, 1 ] be defined by 0 = x 0 < x 1 < "" < xn_.,/2 = 1. N + 1 is the total number of knots. By extending the sequence. The two knots beyond the left and right ends are respectively, X-m/2+l
<
X-m/2+2
<
"'"
<
X:-I
1 < X,N_m/2+ 1 < "'" xk
= khj,
hj
= 2 -j
,
k
=-
< <
~;0
=
0
<
X1 <
"'"
<
~'N-m/2
=
" ~ N + m / 2 - 1 ~,
m ~-~-+1, - ~ -m+ 2 ,
"-', N + - ~m- - 1 ,
N = 2j .
By such extension, a m-th order spline space is obtained and its base functions are: r
) = @,, - i f i --
i
=
m
@,,,(2-Jxm
2 +1,
--~-+
i),
2,-",
h
-
1, m
N +-~--
1.
(3)
} 1, N = oJ. ,
(4)
The above functions form a linear space which is denoted by, S.,,/ =
{
~,,.i(x),
m m m i = - - ~ - + 1, _-~- +o,~ " " , N + - ~ - -
where tj represents the spline knots of m-th order. Its dimension is, dim(S.,,/)
= 2j + m - 1.
(5)
Taking the spline function as scaling function, the spline forms a multiresolution analysis in terms of the spline theory and the dimension of the scale space V5~ 11 = S . , , / m a y be determined by ( 5 ) . 3
Spline Wavelet
S p a c e o n [ 0 , 1]
Having obtained the scale space, the wavelet functions can be obtained by the scaling functions. Some theorems on wavelet construction are presented in bibliographies [6 - 8 ]. These include the following three steps:
1)
To construct the 2mth-order spline space by incre .asiug one more time over the original
knots $2,~./" = { ~ 2 = ( 2i+1x -
2)
i):i
= -
m,
-
m + 1, " " ,
N + m -
To ensure that the 2mth-order spline space satisfies the end conditions
1};
(6)
Duan Jiwei and Peter Kai-kwong Lee
440 -o , Sz,,,.t"
3)
= {s(x)
= 0, p = 0 , 1 , " " ,
E Sz,./" :s(')(q)
m - 1;q
= O, 1 } ;
(7)
To ensure the 2ruth-order spline space satisfies the wavelet conditions s,_%,m
= o, k = o, .-., 2 J } .
=
Note t h a t t / = t/~, = { • , - i_4+~_i =+1 , ~ = k2J, k = -
m + 1, " " , 2 j + m -
(8) 1.
The following theorems can be summarized: Theorem 1
For any m E Z + , the ruth-order derivative operator D " maps the spline space
SO.,,/'' to the wavelet space W~~ 1] Theorem 2
For all j E Z . ,
if 2 j I> 2 m - 1, there exist 2 j +1 _ 2 m + 2 linear independent
wavelets which can be expressed as 3m-2
~@(x) .
~b(2Jx .
i)
2.I - z ' ~ ( -
. 1)k-'~2,,(k
+ 1
2m)~2,,,.2i+l, ( x )
k=m
i = 0 , " " , 2y - 2 m Theorem 3
+ 1.
(9)
For all j E Z + , if 2 j I> 2m - 1, there exist m - 1 wavelets on the 0-boundary
which can be written as -I
dbii(x)
=
21-2"[
2
- o(=) ,2re,j+1 ( x ) +
~k'~
k= - m * l 3~-2~-21
1)
r
+ 1
--
2i
--
2 m ) m('~) "*"2m
._J+l
- k)]
X
,
k=m
i =-
m + 1, " ' ,
- 1,
(10)
where [B]{r
[B]u
(10a)
{r},
= [B~.z~.j+l(212-J'l)].
{r}=
{q
r,.'"
r,,_~} r
k = - m + 1 . ' - - , - 1, {a}=
{a~, a,,'"
a,,_l} T
3m-2.2~
rl = (-
~
(- 1)~2~,(k
+ 1 - 2i - 2 m ) ~ z = ( 2 /
- k)),
k ffi m
l = 1,'",
m-l,
its compact support is supp~bq(x) = supp~b(EJx - i) = [0, (2m - 2 i =-
+ i)2-J],
ra+l,
-'-, -1.
The 1-boundary wavelets can then be easily obtained through the symmetry relationship of 0boundary wavelets as follows:
~.(x)
= ~l('~Jx . ~b~
-
i) =
o _2~+1_i(1 ~bj,',
- X)
-- x) - ('~J - 2m + 1 - i ) ] .
=
4th-Order Spline Wavelets on a Bounded Interval
441
i =21-2m+2,21-2m+3,'",21-
m,
(11)
its compact support is suppr
= [i2 - i , 13,
= supp~b~.~_2,~§
i = 2i-2m+2,
"",2i-
m.
4 4th-Order Spline and Its Wavelet Using the above theorems, the 4th-order spline wavelet on the interval can be constructed. Commencing with the scale space of the 4th-order spline ( i . e . m = 4 ) , S,,i = {r
i =-
1, O, 1 , 2 ,
" - , 2 i + 1}.
(12)
In order to obtain the wavelet space, the spline space of the 2ruth-order is first constructed for m=4
Ss,i" = { ~ s , , ( x ) , In the above equation, the xi
i =-3,
khj+l , hj+l
=
=
-2,
- 1,0,1,2,'",2/§
2 -j-1
,
k
=
-
3,
-
2,
+3}. -
(13)
1 , 0 , 1 , " " ,2 i+1 + 3, and
N = 2 j+l .
At the 0-boundary, there exist seven truncation functions. They are
~8,_3(x), r
O~,_1(x), r
~8,x(x), O~,(x), Os,3(x);
At the 1-boundary, there are also seven truncation functions. They are
Os,,_J"_3(x), ~s,zJ"_,_(x), ~8./"~_~(x),Os,,."(x), ~8,/'t+l(X),
~8./*1+2(X),
~8,2J*t§
The above functions are the boundary base functions but it is not convenient to be used directly for given boundary conditions. So, it is necessary to first convert the seven O-boundary functions to seven linear-independent base functions with the following boundary conditions 3
Bi(x)
3
= 2120. f~bs --if- i = ~ a o . ~ s , i ( x ) , j . -3
i =-3,-2,
]= -3
"",3.
(14)
The converted base functions are
B_3(x) B_,_(x)
12-3,-3
ff-3,-2
12- 3 , - 1
12-3,0
12 -3,1
12 -3,2
12-3,3
r ~8,-:
0/-2. -3
12-2, -2
t/-2,- l
12 -2,0
12-2,1
12-2,2
12-2,3
(1~)8, _
12-1,-3
12-1,-2
12-1,-1
t~-l,0
if-l,1
12-1,2
ff-1,3
1~8, _
120,-3
120,-2
0/0,- 1
120,0
if0,1
ff -3,2
120,3
(-258,0
Bl(x)
121,-3
121,-2
t~l,-1
121,0
~1,1
al,2
17/1,3
1~18,1
B2(x)
122, -3
122, -2
0~2,-1
122,0
C/2,1
122.2
122,3
(158,2
B3(x)
123.-3
123,-2
G'3,-1
123.0
a3,1
123,2
t23.3
B_l(X)
80(x)
=
k ~8,3
,
(15)
442
Duan Jiwei and Peter Kai-kwong Lee
in which B; ( x ) must satisfy the following boundary conditions
8_3(0) ~ o, B_2(0) = 0, e_~(0) = 0, 80(0) = 0, 8~(0) = 0,
8;(o) = o, B3(0) = o,
B(12 . . . .("'~ , ~ o , ~ _ ~ .R(1) (o)~0, 8~ 1) (0) =
B(!I _ ) (0)=o,
B(oI) (0)
=
0 , B~I) (0)
0,
=
0, B3(I) (0) -- 0, (16)
8%>(o) ~ o, 8t~ (o) ~ o, 8~_'-?(o) ~ o, B(o2) (0) = 0, 8~;) (0) = 0, B~ ~-) (0) = O, 8~~-)(o) = o,
8c3~ (o) ~ o, Bc?~(o) ~ o, 8(~) (o) ~ o, B(J) (o) ~ o, 8~~) (o) = o,
B?~ (o) = o, 8~~) (o) = o. The following results can then be obtained 1
'B_3(x) ~
B_~.(x)
8x(=)
1
0
0
0
0
O
1
- 12---0
0
0
0
0
0
1
473 18648
4 2 331
0
0
0
0 i ~s,-z
1
125 2 394
47 5 985
13 352
0
B_x(x) Bo(~)
0
~8,-1
1
9 5 320
0
0
1 352
0
13 352
352 279
685 279
901 43 155
11 126
0 i r
83(x)
0
0
0
352 279 24 4 795
I
339 38 360
(17)
i ~s,1
Bz(x)
685 279
,
~)8,2
0 k r
1.
It can be seen from (17) that the presented base functions satisfy the following zero-boundary conditions, i . e . 5 ~ = 0, p = 0 , 1 , 2 , 3 . It can then be proved immediately that the boundary base functions are
B I ( ~ ) , B~_(~), 8 3 ( ~ ) . These functions are linear-independent and can be expressed in the figures below. When the three base functions are obtained, the wavelets can be constructed from the following procedure: 1) The interior wavelets from Theorem 2.
4th-Order Spline Wavelets on a Bounded Interval
443
10
~bj,(x) = 2 - 7 ~ ( -
1)~@s(k + 1 - 2 m ) @ s(4) , 2 , + , ( )x,
(18)
kffi4
where i = 0, " " , 2 j - 7 and the compact support set supp[~bq(x)] = [i2-J,(7 + i)2-i]. 2) d,b~
The 0-boundary wavelets from Theorem 3
fiB(24)(2J+Xx)
B,(x) k
2-7(aB~4)(2J+ax) +
= (bO(2Jx - i) =
0.4 0.3
+ 7B~ 4) ( 2 J + ' x ) +
0.2
10+21
~(-
1)k~s(k
+ 1-2i-6
8)~I 4)(2j+'x - k)), i = - 3, ..-, - 1,
-4
-2
0
2
4
6
(a)
(19)
0.4 0.6
0.3
0.4
0.2 0.1 I
-4
-2
0
2
4
2
0
4
(c)
(b) Fig. 1
The boundary base functions
where a , / 9 , y can be determined by 10+21
r, = ( -
~(-
1)k~bs(k + 1 - 2 i
-
8)~s(2l-
k)),
kffi4
i =-3,-2,
l = 1, 2 , 3 ;
r~(l
= 1), l
~ B , ( 4 ) + f i B 2 ( 4 ) + 1B3(4) =
rt(l
= 2),
aB,(6)
rz(l
= 3);
aBl(2) +
flBz(2) + 7 B 3 ( 2 )
-1,
+ ~B2(6) + rB3(6)
=
=
(19a)
(19b)
The support set supp~bj,(x) = s u p p ~ b , ~ ( 2 i x - i) = [ 0 , (7 + i ) 2 - J ] , i = - 3, - 2 , ..., - 1.
(19e)
Duan Jiwei and Peter Kai-kwong Lee
444 3)
The 1-boundary wavelets similar to the 0-boundary wavelets
r
= r
-
r176
i) :
C jo, s,- z . + l - i ( 1
-
x ) - (2 i - 2 m + 1 -
x)
=
i)]
= r176
i = 2 i - 2 m + 2 , "'", 2 i . m = 2. j - 6 ,
- x ) - (2 i - 7 .2 i - 5 , . oJ _ 4
(m
i)], 4) (20)
The support set supp r
( x ) = suppCj.s'-7-i 0 (x)
= [ i2
-j ,
1].
(20a)
In order to separate the effects of the two endpoints of the interval and to ensure that there is at least one inner wavelet, without the loss of generalization, attention can be restricted to the levels j ~> J0 with J0 = m i n { j E Z . :2 i I> 2 m - 1 } .
(21)
With the wavelets obtained for level j 0 , scaling functions and wavelets for the j level when j I> J0 are given [7] . When m = 4 , j0 becomes 3. Using the Mathematics software, wavelets can be presented in the following figures:
0.010
O.OLO
0.005
0.005
0
t
0.8
- 0.005
o
-o.oo5!
0.010
-0.010
- 0.015
-0.015
-
0:2
(b)
Ca) Fig.2
The inner wavelets
1)
the
a)
the translate integer, i = 0
b)
the translate integer, i = 1
2)
the 0-boundary wavelets ( j = Jo = 3)
a)
the translate integer, i = m -
inner wavelets ( j = Yo = 3)
1 =-3
b)
the translate integer, i = - 2
c)
the translate integer, i = -
3)
the I-boundary wavelets ( j = jo = 3)
a)
the translate integer, i = 2
b)
the translate integer, i = 3
c)
the translate integer, i = 4
1
445
4th-Order Spline Wavelets on a Bounded Interval
A
0.015 0.010 0.005 0 - 0.005 -
000
0.5
v0 V
0.010
015
1 0.005
016 - 0.005 -0.010
- 0.015
-I 0.015 -
(a)
0.03 O.02 0.01 0
••
0.015 0.010 0.005
0 ~ " 2
-0.01 -0.02
(a)
0~
014
0
015
I_ 0.005 - 0.010 - 0.015
- 0.03
h)
(b) 0.03
0.010
o:/o.9t
O.005 0 -
0:005
-
0.010
.
,
.
,
0.5
0.02_
\
0.6
-0.015
(c)
(c) Fig.3
5
The 0-boundary wavelets
Fig.4
The 1-boundary wavelets
Conclusion The 4th-order truncated spline has been used successfully for consu-ucting wavelets on the
interval. These wavelets may have applications in differential equations, signal processing and other similar p r o b l e m s . It is considered that these wavelets can help to o v e r c o m e the difficult treatment o f any function on the interval.
References: [1]
Chen Mingquayer, Hwang Chyi, Shih Yenping. A wavelet-Gale"rkin method for solving population
[2]
balance equations[J]. Computers Chem Engng ,1996,20(2) :131 ~ 145. Chen Mingquayer, Hwang Chyi, Shin Yehping. The computation of wavelet-Galerkin approximation on a bounded interval[ J] . International Journal for Numerical Methods in Engineering, 1996,39 :
[3]
2921 ~ 2944. Williams John R, Amaratunga Kevin. Introduction to wavelets in engineering[J]. International
446
Duan Jiwei and Peter Kai-kwong Lee
Journal for Numerical Method in Engineering, 1994,37 (14) : 2365 ~ 2388. [4]
Chen W H, Wu C W. A spline wavelet element method for frame structure vibration[J]. Computa-
tional Mechanics, 1995,16(1 ) : 11 ~ 12. [5]
Daubechies Ingrid. Two recent results on wavelets: wavelet bases for the interval, and biothogonal wavelets diagonalizing the derivative operator[ A ] . In: Larry L Schumaker, Glenn Webb Eds. Recent Advances in Wavelet Analysis, Academic Press, Incorporeted, 1993.
[6]
Quak Ewald, Weyrich Norman. Wavelets on the interval[A]. In: Singh S P E d . Approximation Theory Wavelets and Applications [ C ] . 1995. Quak Ewald, Weyrich Norman. Decomposition and reconstruction algorithms for spline wavelets on a bounded interval[J]. Applied and Computational Harmonic Analysis, 1994,1:217 ~ 231. Guan Liitai. Truncated B-spline-wavelets on a bounded interval and its vanishing moment property [J] . Acta Scientiarum Natrualium Universitatis Sunyatseni, 1996, 35(3) :28 ~ 33. (in Chinese)
[7] [8]