Основы компьютерной томографии

, 19.05 | c , 19.06...

Author: Кравчук А.С.

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, 19.05 | c , 19.06 | ! -# , | ! # # 553400 | ! , , & ! .

. .

1999

1.

iv 1

2.

9

1.1. . . . . . . . . . . . . . . . . . . . . . . . . 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. . . . . . . . . . . . . . . . . . . . . . . . 2.1. ! " # . . . 2.1.1. ! " # . . . . . . . . . . 2.1.2. ! " % . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. ' () * . . . . . . . . . . 2.2.1. ,! -*" -. # . . . . . . . . . . . . . . . . . . . . . . 2.2.2. . "( . . . . . . . . . . . . 2.2.3. % 0( . . . . . . 2.2.4. # . . . . . . . . . . . . . . . . 2.2.5. *"(. . . . . . . . . . . . . . . . . . . . 2.2.6. *"(. " # 0( . . . . . . . . . . . . . . . . . . 2.2.7. 0"(. .#. . . . . . . . . . . . . . . . . . 2.3. 3 . () # * . . . . . . . . 2.3.1. 3 0( . . . . . . . . . . i

1 2 3 9 9

14 17 17 20 23 25 27 29 31 33 34

ii

2.3.2. 4" 0( (6 0) " () (Cooley & Tuckey) . . . . . . . . . . . 2.3.3. "( | : . . . . . . . . . . 2.3.4. 4" .. . . . . . . . 2.4. 0" ' *. . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. 0" ' " *. ! !. . . 2.4.2. 0" ' " *.# "(! !. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. < % ' . . . . . . . . 2.5.1. # "! !

. . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. =** 3" " . . . . .

3.

3.1. 0 ># * . . . . . . . . 3.1.1. , . . . . . . . . . . . 3.1.2. ' ( . . . . . . . . . . 3.1.3. " * " .#

. 3 . . . . . . . . . . . . . . . . 3.2. . . . . . . . . . . 3.2.1. , . "" . . . . . . . . . . . . . . . . . 3.2.2. - = . . . . . . . . . . . . . . . . . . . . . 3.2.3. "(. . . . . . . . . . . . . . . . . 3.3. ># () # * . . . . . . 3.3.1. "( . . . . . . . 3.3.2. * > > ."( ' . . . . . . . . . . . . . . . . . 3.3.3. 4" . "( 0( . . . . . . . . . . . . . . . . . 3.3.4. % 0( . . . . . . . 3.3.5. *"(. .. . . . . . . . . . . . . . 3.4. > . . . . . . . . . . . . . . . . .

35 40 42 53 54 56 67 67 73

78 78 78 80

82 86 86 89 91 94 94 97 99 101 103 105

iii

4. - -"

4.1. "( @' . . . . . . . . . . . . . . . . 4.1.1. A 6"! . . . . . . . . . . . . . . . . . . . . . 4.1.2. >** 4.1.3. 0 @'- ", >!. . . . . . . . . . . 4.2. 0 @'- # . . . . . . . . . . . . . . . 4.2.1. D". , " %() "# . . . . . . 4.2.2. ,,,E. # *# . . . . . . . . . . . . . . . . . 4.2.3. A " ! # . . . . . . . 4.2.4. ' . " # . 4.2.5. E "(# T1 # T2 ". . . . . . . . . . . . . . . . . . . . . . . . .

5. %

5.1. : % . . . . 5.2. "(. . . . . . . . . . . 5.2.1. # . 5.2.2. "( "(. . . . . . . . 5.2.3. 3 .. . . . . . . . . . . . 5.3. 4" G . . . . . . . . . . . . . 5.3.1. <(). . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

107 107 107 112 115 118

118 121 125 129 131

133 133 135 135 136 138 140 140

= ! 0 "(# . " # "E ." ("I

. G * "(# 1997-2000 "), "(# -279-01J "( ", " ! # N o 426 (", -# " ! "), '0 "0 "( " " ", "0 "( " " " ( N o 97-03-4.3-47) '# * * "(! " # ( N o 99-01-0021).

iv

1. 1.1. <<() *>> ( ( | ) M " * ! #, ! ! ! , "( ! " " # "! M G ! . " #. M " ( ! .| >" , G #, M | " , * , "% . ( *. " .# , M " " " " . = " ) .* " .) *.- "(! , . #. ! ! # ". # "

! ! "!. , ! ! .# %() ! . G) G)% " . " * # G) * " , >" . " ! " ! G) " ! !, #" . ," << *>> M ! " | oo (c"#) 1

2

1.

o ((). , " << *>> <<( "#>>, . . "# M . , ", (, *.) * " " %! >" ! . # " (.) # " M .

1.2. 3" G # () # * | " # "( # ># ( "( ") * " " . "G( | # " (.) # " G( . " | " ( | " "( ! ! . , , ** ! . , )%! "(( . % " G # | " ) " )% ## : 1) " ( -)J ( # # | ( - ! " . ! , ! | " " M J 2) " J > * (% '0 ) | % " " | % M .# " % "(# J G " '0 " " # ! # . = ! " "( # )% " , ( . ) # " >**. "% - M J

1.3.

3

3) >" " J > "( * >** (@'->** ). @'>** )% # " J # "(, ( "( " "" " >" " , " ! () "(! ( <<)>>, ", ).

" % # . !)% >" " , " ( ! , ( )) M . | ! # > " " (" -" >" ) M ". 4) A"( " J ( ( " " " M , ! %() "(# ! G( , , " !J .# "() ## # | # ( *), " " , # " >" , " " ( *), ! >" ! (>" *, >" >. *" *). N( >! , ) %) ! , .

1.3.

# # " ! G ! ! | # #G * !, ")%! "( ) " " M . 1) , . * . . ' )%! ( M --" # 8 1895

4

1.

" ). " ").) .# . " > " " . . ' 1895 , . . ' ."( " , ". % ")! ! , ! "( --" #, , " " . ," > " ( ' | 1896 ,:4 # "( () 15$. . ' " " 6 ( ") " " )% " " " . , ! . ! ! "( " % ! # | G "( "( " ", "( .. .# " , ( ! ! ! "( , " > , ! . ' "). , "G ( > , "( " 70-! () * | > "" -*" - (G. N. HounsSeld & A. McCormack), > < " # *" . 1979 . () * -*" -, # W % ) , "( # (<<">>) ! -, )%# " # M " # ". "" # " " " *. >**. "% "( ", > " ", ")% # "% -, ! ) *. >**. "% " #! " . "( " , " " G . ' % 1917 . -*" - ! " " " G ' , -

1.3.

5

! " G # , "()%! # " "(# !. , , " -*" -, ! " ( " " ! ", , ( G " # .. = % ". ") 6. E., "( ,. E. ) 4. 4. (X1]) " *. "( * "" -*" -. ( ! () ! *) % ") ( > , , ) ( " # : I " | M "" "(! " #,

" ! , II " | ! %! " # ( ) "-"" "( % , , % , III " | "( G " , # " ( ( "" "( , " "( % , IV " | "( %)% . "(. , V " | ", " >" # ! #J >! *! % >" ) % , ) "(G! " * " ". 6"(G " W " ' % IV- " , ) * Siemens, Hewlett Paccard . , * ! " # "( ,,,' | . , G" | (< <<, >>).

6

1.

2) @" , "( @' *!, 6"! "" (Bloch F. & Purcell F. M.) 1946 . "(, > " ") # >" ! ! (), % ! " , # # >" # ( -) ". @'- * " %() >" ! ! " ")%, G # " ! M "( # " # M >" # " ( )) . " @'- * "" * D (P. C. Lauterbur) 1973 . ," (, "" "(( " @' " .! . " # # # 3 (R. V. Damadian) 1972 . W "( @'- * "( . 4. E ( . D ) 1959{1960 . X?]. () @'- * " .! # "( # | ! % " " " ! . 3) E ( % " ! " ! # ", - , ! , ! - - " . " ")"( , # <<>> " <<">> | # " * -. <" " " " ! ! ! "! (1946 ). 1948 = "" '" (G. Ansell, J. Rotblat) "" % # " , ) .) " " . # % ! " ># * " " . -" "

1.3.

7

" , ( .# . = " " G = (H. O. Anger) 1952{53 , # " - ."". "" # , ")% *.) " ", , % ) *>" ". 1968 . = "" "(( ) - *)% "" | "# " - " . "() " * . 3"( #G ># * # *# ># * (0=), # ) -, )% "( # # " )% % ># #, # # ># * ( =), *) , )% "( ". >" " )% "! " !. = "( ! *! "! >, ,. A> X1], ( !. W "(# " " ># () # * " )"" A> " (G. Muehlenner, R. A. Wetzel, 1971 .), 6" . (A. R. Bowley et all., 1973 .) . ! ! " .# ># * I. 4. 0 ,. 4. % X?]. 4) "(! ( !) ! ! # .# ) ! " ! ! ! " # "! # ! M !. % "( " . # | ! " , | (G # G)% # . A"( , , G "() # "., " ! . " #, . " " "( "(

8

1.

>" # "(# ")% # ! " # # #. 6"(G "(! ! ! . # "(! " | <<>!- ">>, ! ! "(! | " ! . ". " "( " .! . " # " 50-! , ! # # " " " 0# % 1940 . 3 >!- " , " % , " " A#"( '# (J. J. Wild, J. M. Reid) 1952 . - 6" (D. H. Howry, W. R. Bliss) "" "( . "( #G "(# ! ! " G . " " #, , )%! " # # " , *. , " - . % "(G " ( G ! , # ! @, ", I , 0., ,:4, ' ! !.

2. 2.1. ! " # 2.1.1. "'() * (+, ,! " ! " #, !G G"( *, . 2.1. ' # " ! >" .

'. 2.1 9

10

2.

, " , " >" " # () .". =" , G( > " , ( , . "( >" " , . = E )%! - # >" # " G : E = h h = 2 h} (2.1)

h} | ". " # , . . ) "( # "(# b (" "(# M ). ~" ( ." U , >" | e, "(# " " ."(# > >" eU > G " :

hb = eU

(2.2)

" )% # " b = c=b, c | ( . 3" ! . " # " ( > " I = c= " " ,

" "G "(G! # U , " # . 2.2, . X4] ( " | "(*). 0", )% . 2.2, *" :

I = (c2Aiz ( ; b))=b3

(2.3)

i | # , z | # , A | , " > .

" ), " > "

+ d dI = I d (2.4)

2.1.

11

'. 2.2 " ( " , U , Z1 I = b I d = C1izU 2 (2.5)

C1 | . < *. I() "( ( I( )J " ) I d = I d (2.6)

I=

Z b 0

I d = Cizb2=2

C = const:

(2.7)

! "! # ." "() "( )%) ( ."J

12

2.

> " > # ! . ) " # > " .

" ." " ! !, " ! . 2.2, ) . <" >! M " ! " , )% # !. M ! " . A > >" ! ", ", >" ! " ". <" "# ( > ) K | ", " )%# " > L | " . ., " () ) "(G " "*. ~" >" # " " G >" " *, ! , ! >" G! " G ". ~" > >" # "# E1, > "#, ! ! )) ", E2, # . " ! " # = (E1 ; E2)=(}h). - " , )% ! K - (, ) K - ) , L- ( | L- ) . . N K " # " K - " ": (2.8) K = 43 R(z ; 1)2

, , z | # % , R | ' . , K , L, M . . | # J ( " # " *" I = Ci(U ; Ue )n (2.9)

C , Ue, n | , Ue | ." " #

2.1.

13

"J 1:5 6 n 6 2: ") > ". = " >" -"(! (=), ">" "(! (=), >" "(! (=)J 1 = = 1:6 10;19 , 1 = = 103 = 1 = = 106 =J h} = 1:05 10;34 : " " " | 10;4 10 ( A)J 1 A = 10;10m: ." | 25 150 ""( (KV). > ! - | 20 150 =. " | 0.3 I # " 9.2 I * . 3" : . )%! " #. 1) = )%! .: E = L2MT ;2 L | ", M | , T | ( .). ~ "!J 1 . " 1 () (<), %)% # 1 1 . %( ! ()J 1 = 1 1 . 2) > )% " !( ! | 1 > 1 1 . 3) "% )% " D D = dm=dw

dm | >" , w | "( > . ~ . | >#J 1 I (1 Gy) "% # )% " , # % # 1 > 1 . 1 I = 100 J

" ! 1 I = 114 (').

14

2.

4) =" )% " H :

H = kD k | >**. )% " . ~ . | 1 (1 = 1 Sv). >**. k ! ) ( | ( "%( )% " J 1 = 100 6> (" # >" ).

2.1.2. *) "(+ +" -

! " % ! # " % , "( ( I " , . "% !, "# " ) , (G . " ! "! (, > (G ."( # . 3" # *" " # I0 . , "%)% # )% # | (. . 2.3).

'. 2.3 ( !, . 2.3, I (x) (

2.1.

15

" J " ) dI (x) = ;X~x]I (x)dx: (2.10) >**. ."( *" (2.10) >**. " # "% " "" J <<>> " " "( . = >**. " *. # ! ! (x y z ) (x1 x2 x3 ), )%! - ~x. A " *" (2.10) ), " ! " (#), "" "( !. >**. (~x) " # ! # % , " () # * " # .# . , (~x) ") " % ~x. E (2.10), Zx

I (x) = I0 exp(; 0 Xx] dx) (2.11) % 6 (" D | 6 ).

!, ")%! "% ("" ) " , , " (, " . ! ! | *>" "% | ") ! > " Er >" % >" # > E = me v2=2, me | >" , v | ( " . ! , %# ! >" "(! " " ( ), ! " | > "( G . '( > # # G # "! ! ( "( ! " , >" G! " . = >** >** , >" | >" . . > # >" * ! ( ! , ! > (G

16

2.

> >" . " *>" "% % "% " " )%! ! " # () (" "( ."( " ") % , ) " (> ( " "( ). # # ! "% | > ! " #. # >" " " " >" % . ' "( # " " "( >" # )% " , , " >" " # #. E ( " "# " )% # # "# *" :

I = I0e4(1 + cos2 )=(2m2e c4R):

(2.12)

># *" , %!J R | # ") .

"(G# > )%! " , * ( # > >" , >" , " (G # > # , " , " # "# " % . " ! . , "( ! ! , # X4], "! ! * .

2.2.

17

2.2. $ %& ' 2.2.1. .* /( 0-1. , "+ ,! *. M , " " -*" -, . 2.4. E

'. 2.4 " " ( ) % "( ")% AA0, | "( BB 0 . E" "( %) !, M *. | ( " ! " " ! G . D * G ! "(# . #. <")% AA0 , BB 0 " % # , %( " # O. 3" " -

18

2.

.#, )%! "" "(! " #J > ) . 3" " # M ! | | (

" ! " # J * M *. " " G . , # # " *., ! # " , "( (4{5) , > "( #G " " (G # " " , "( " # " . . | > "( ". 1. N" " " (* #) " " ! ! *. "), ( " " , ! " *., * # ! "" "(! " #. ># "( ! "" "(! " #. " " "( #G " , . ", " )% " ( " " . " ! ( () I (~x) # ~x " . @, * ( # ), > ( I (~x) ( " *" (2.11) ". E I (~x) "( " "( "( >**. " # "% (~x) *"# 6 (2.11). < # M " (~x) " ), . " ) (")) . % (. 2.5). , -

2.2.

19

'. 2.5

# (%)% #) # ) O , ( O # " " " "( . "( (! % ( % ) ". ( O " , . 2.5. " # "( # " " , = x cos + y sin = ;x sin + y cos (2.13) x = cos ; sin y = sin + cos : (2.14) *"# (2.11) (: ZR

I ( ) = I0 exp(; ;R Xx y] d ) (2.15)

, , ( Xx y] , >**. " # "% " , % " " " " % " . "( (. . 2.5)J I0 | " ! " ", 2R | (, ! # ".

2.

20

< " (, " M ( ! ) = 0, " "(, " *" (2.15) "( " . , , , " > " ( , "(

"( #G . A G " .: . # p( ) " )%# ": Z1 p( ) = ; lnXI ( )=I0] = ;1 Xx y]: d (2.16) # () # * ") , ( " (x y) .# p( ). , >**. " # "% (x y) ! )) " M , .# "( # M ( << G)%#>> "(). > ) ) # . " . #.

2.2.2. 2 3 + ,( ># , 0( ! .# 0( >**. " # "% . 3" # *" "( " )% " . .1

( f (x) | % *. # % # # xJ 0( (" | " | 0( ) *. f (x) " - *. f (!) % # # !, " *" Z1 ; 1=2 f (!) = (2 ) (2.17) ;1 f (x) exp(;i!x) dx: 1

, " | $, "% | &, ' % (

% | ( " .. | * +" | " ( " 3.

2.2.

21

% 0( Z1 ; 1=2 f (x) = (2 ) (2.18) ;1 f (!) exp(i!x) d!: . "( " # n 0( *. f (~x) n % ! ! ~x = (x1 x2 : : : xn) " %() n- " *" Z1 Z1 f (~! ) = (2 );n=2 ;1 : : : ;1 f (~x) exp(;i~! ~x) d~x (2.19)

~! ~x = !1x1 + !2x2 + + !nxn | " ~! ~x, d~x | >" M n- . n- Z1 Z1 ; n=2 f (~x) = (2 ) (2.20) ;1 : : : ;1 f (~!) exp(i~! ~x) d~! :

3

< *" (2.19) (2.20) "( " " n = 2. 2 . ( . "( ). E ( ) = (2 );1=2p( + =2) (2.21)

( ) | " " ~! = (!1 !2 ) = (u v):

u = cos v = sin " *" (2.21) | | ( ). 3 "(. G *" (2.16) " .

# ": ZZ 1 p( ) = ;1 (x y)( + x sin ; y cos ) dx dy 2 ,

(2.22) 0( , (2.23)

( " "" -" ( . , % "" " ( %" " (2);n=2 . 1 ( 2 "" -", % " 2 ( * %" %" % " | * -" " .

22

2.

(x) | "(-*. 3, " *" : Z1 ;1 f (s)(t ; s) ds = f (t):

(2.24)

3" *. ! ! f (x y), " # "( f Xx y] # ; ; (x y) = 0, = const, " )% " : ZZ 1 Z (2.25) ;1 f (x y)( ; (x y)) dx dy = ; f Xx y] ds

s | " # ;. A % ", " " " "(# . "( ", = ;x sin + y cos (2.26) ) *" (2.23){(2.25) " >" ( " # (2.16) (2.23). " # 0( - p(! ) . p( ) # : Z1 p (! ) = (2 );1=2 ;1 p( ) exp(;i!) d = (2 );1=2 (2.27) Z 1 ZZ 1 ;1 ;1Z Z(1x y)( + x sin ; y cos ) exp(;i!) d dx dy = (2 );1=2 ;1 (x y) exp(;i!(;x sin + y cos )) dx dy = Z1 Z 2 ; 1=2 (2 ) r dr (r ) exp(i!r sin( ; )) d: 0

0

WG ( 0( " (x y) "# (r ), > " (u v) ! 0( ") u = cos v = sin : E Z1 Z 2 ; 1 ( = ) = (2 ) 0 r dr 0 (r ) exp(;ir cos( ; )) d: (2.28)

2.2.

23

," " *" (2.27) ) ( (2.28), " (2.21), " "( ( ! = , = : 3 , "(, ( ) >**. " # "% (x y) * = ) . p + =2 " . W 0( ( ) = const " # << >> ! z = ( ) "() = = const, M .

2.2.3. 0 '- '" 4(

" . "( (2.21) " # " G " # G # | ! " >**. " # "% M M .. 0"( G ( # *" % 0( (2.20): ZZ 1 ; 3=2 (x y) = (2 ) (2.29) ;1 p ( + =2) exp(i(ux + vy)) du dv:

"( *" (2.29) ( ) 0( - X?] ) " . - !, "(# *. ( ) p ( ) "( "! # "# (. 2.6), ) " # , ! !

. *" (2.29) (, . # ) ! " # " )%! # *.# " "(! !, " ! ). ~ G # " " ". # *. , *. , !, ( > ). 6" ( # " # " , "( *. *" (2.29) # (ux + vy) ."" ,

24

2.

'. 2.6 - *" ) (. "() ! G ) # , ! ) " () " " ."")%! *.#J " X10], . 125. # ! "(, ")% G # ( #) " " # *. , ", G ( G) . ," "(, " > ")% (, "(( =.

2.2.

25

2.2.4. 0 ' , 3 = " . (2.23), " # "( " G (2.29). " # # ., " *" Z 2 ; 1 g(x y) = (2 ) p(;x sin + y cos ) d (2.30) 0

-% , > " ! .# p. , "( ! # *. *" (2.30) "( "G ) "( "( | * , *"()% ( G ) G ) # . ..

" " (2.30) (2.23), # ZZ 1 ; 1 g(x y) = (2 ) ;1 (x0 y0) dx0 dy0 (2.31) Z 2

(;(x ; x0) sin + (y ; y0) cos ) d: # # | " )% #! "(-*.: A. ( (x) | ** . *., )% " ( ) " xi J X ((x)) = (x ; xi)=jd=dxjx=x : (2.32) 0

i

i

3 "( " "(*. Z1 3 (2.33) ;1 f (x)(x ; x0) dx = f (x0) 8f (x) " *. (x) #" " " xi , (kx) = jkj;1(x) 8k 6= 0 (2.34) 3 " 8 (" << >> << +>>

26

2.

" "( ( >

"( " ).

" (2.32) " ",

= () = ;(x ; x0) sin + (y ; y0) cos : (2.35) <" *. () : = 1 = arctg((y ; y0)=(x ; x0)) 2 = 1 +

(2.36) "( # d=d ! 1 2 : q j = (x ; x0)2 + (y ; y0)2 j~r ; r~0j (2.37) j d =1 2 d

~r = (x y) r~0 = (x0 y0).

" (2.36), (2.37) " (2.32), # , (;(x ; x0) sin + (y ; y0) cos ) = (( ; 1) + ( ; 2))=j~r ; r~0 j: (2.38) ," "(, . g(x y) : Z Z 1 (x0 yo) 1 g(x y) = ;1 j~r ; r~ j dx0 dy0: (2.39) 0 . 3 # # *.# ! ! f (x y) h(x y) " )%# ": ZZ 1 q(x y) = ;1 h(x ; x0 y ; y0)f (x0 y0) dx0 dy0 h f (2.40)

3

( ). 0( ) ! *.#, . .

q = hf : (2.41) 3 "( ! ! G # , . *.# " . 3

2.2.

27

"( . " # > *. p( ), (2.41) " ( ( ") G # # # # . g. 3" "( > " # # (2.39) 0( : 1 ZZ 1 g (~!) = g (u v) = 2 ;1 g(~x) exp(;i~x ~!)d~x = 1 h d~x = dxdy (2.42)

h - 0( *. (x2 + y2);1=2, : ((x2 + y2);1=2) = (j~rj;1) = (u2 + v2 );1=2 = j~!j;1: (2.43) E (2.42) *" (2.43) ! G # 0( : (~! ) = g (~!)=h(~! ) = j~! jg(~! ) (2.44) " 0( , ! "() * G : 1 ZZ 1 p 2 2 (x y) = 2 ;1 g (~!) u + v exp(i(ux + vy)) du dv: (2.45)

2.2.5. 0 3 " , 0( , " # ". ) "(G , G, , " # " " # | " ! ! ! ( "(G G . < ( "( .: ( G( ( "( ., " " " " G . .), ! "-

28

2.

( G . " . G ". 4. <. ! X13, 12]. ! " ! "" , ( " - % - *"(.) ! ")%! G , ")% # ".) " ! . , # ! ( " , " . " , >** . 0"(.) " 0( "( " G ) .) * (2.29), " "( G (2.45)J *"(. ! .# (2.16) # . 0"(.) "( " # . ) ;*"(.. .

( A(u v) - *., # *.) g(u v) " .) *"(. ( " ) . #- . " )% ..)J *.) A(x y) ) )% #. ~" G # ) .), *"(. " )% ( , " ) G ): ZZ 1 p 1 (x y) = 2 ;1 A (u v)g (~!) u2 + v2 exp(i(ux + vy)) du dv

(2.46)

" ". " % )% # *. " # # - " G ! !

! ( " " G .

2.2.

29

2.2.6. 3 6+ '" , 4(. ( *"(., "( G * (2.29). # " " (u v), " (x y): x = r cos y = r sin u = cos v = sin : (2.47) "( (2.47) ( (2.29) G " )% : Z 2 Z +1 ; 3=2 (r ) = (2 ) d p ( + =2) exp(ir cos( ; )) d: (2.48) 0

0

W , cos( ; ) = ; cos( + ; ) p ( + =2) = p( + 3 =2): (2.49)

*" (2.49) | > >" , ! * ! #. , >! *" " # (2.48) ( : Z Z1 ; 3=2 ( + =2) exp(ir cos( ; )) d: (2.50) ( ) = (2 ) d j j p 0 ;1 # " " # *" " # ( " (2 );1=2 ) 0( ! *.# jj p( + =2)J " " *. jj " *. p, . . . # p.

> " (, | " " > *" (2.50) | " r cos( ; ). , *"(. ( .) ") "( ."( ) *.) A(), )%) ".). "( " " " (2.50) "G *. *" , " # " ) # .,

2.

30

. " ) . *"(. #. )%! "() *. " )% ) 8< max A = : j0j jjjj 6 > max

) )

8< max A = : + (1 ; ) cos( =max0) jjjj 6 > max 8 < max A = : cos( =2max0) jjjj 6 > max

.

"# " " # "( )% # *. ).

# G ") " " A *. A()J " r cos( ; ) x^, ( Z max Z max ; 1=2 (2 ) A(^x) = ;max jj exp(ix^) d = 0 exp(ix^) (2.51) Z0 ; ;max exp(ix^) d: E"( *" Z max ;1 ; 1=2 ; 1 max (2 ) A(^x) = X(ix^) exp(ix^]0 ; 0 (ix^) exp(ix^) d ; (2.52) Z0 ; 1 0 X(ix^) exp(ix^)];max + ;max (ix^);1 exp(ix^) d = max(ix^);1Xexp(imaxx^) ; exp(;imaxx^) + (^x);2Xexp(ix^)]omax ; X(^x);2 exp(ix^)]0;max sin(maxx^) + 2(^x);2Xcos(maxx^) ; 1] = 2max(^x);1 x^

2.2.

31

N )% " "(# *. sin x=x "" ."( sin x (2.53) sinc x = x : , "( (2.52) ( : (2 );1=2A(^x) = 2maxX2 sinc(maxx^) ; sinc2( maxx^ )]: (2.54) 2 E " A(^x) *. A(), # " *" (2.50) ( Z1 + p (r cos( ; ) + =2) = ;1 p((r ; r^) cos( ; ) + =2)A(^r) dr^ (2.55) " "( G # : Z (r ) = (2 );1 p+r cos( ; + =2) d: (2.56) 0

2.2.7. 43 3,.

" )%) : # *.) H = H ( ) # #, ), :

:

(x y) = (2 );1

Z 2 0

Z1

G(;x sin + y cos )d

(2.57)

G( ) = ;1 p(0 )H ( ; 0)d0: (2.58) E " # *" (2.58) ) *"( ( # ), " %) " ) *.) H - *. # *"(. . (2.57) # " # " # .) ., " % ""( # . # - '

32

2.

(2.16), ( *"(# # . G( ). , " *. *"( G 0( , " ! % # . - " " (" G , , " ). 3" "( >! # (2.58) *" (2.57): # Z 2 "Z 1 ; 1 (x y) = (2 ) 0 ;1 p(0 )H (;x sin + y cos ; 0)d0 d: (2.59)

"G) *" " . (2.23) # ":

(x y) = (2 );1

Z2(Z 1 "Z Z 1 0

;1 ;1 (x0 y0)(0 + x0 sin #

)

(2.60)

; y0 cos ) dx0 dy0 H (;x sin + y cos ; 0)d0 d

# 1 Z2"Z Z 1 (x y )H (;(x ; x0) sin + (y ; y0) cos )dx0dy0 d = 2 0 ;1 0 0 "Z2 # ZZ 1 1 = (x y ) H (;(x ; x0) sin + (y ; y0) cos )d dx0dy0: 2 ;1 0 0 0

" "(-*., ") , : 1 Z 2 H (;x sin + y cos )d = (x)(y): 2 0

( H () - 0( *. H ( ): Z1 1 H ( ) = p H () exp(i )d 2 ;1

" > *" (2.61), " :

(2.61) (2.62)

2.3.

33

1 Z1 ( Z2"H () exp(i(;x sin + y cos )#d)d = (x)(y): (2.63) (2 )3=2 ;1 0

E 0( , :

ZZ 1 1 (x)(y) = 4 2 ;1 exp(i~x !~ )dudv Z1Z2 1 expXi(x cos + y sin )]dd: = 2 4 0 0

(2.64)

," (2.63) (2.64), ") , : H () = p1 jj (2.65) 2

! # "( *. *"( . 3" ! " - *. H ( ) )%) *.) A(), " # jjA() " # 0( . , " : 8 < jj 6 max A() = : 10 " " jj > max # , : 1 X sin( ) + cos( ) ; 1] H ( ) 2 (2.66) max max 2 max - " *. *"( "() G .

2.3. ( ) %& # '

"( * " () " " *. , ! *.

34

2.

" , ! " | >

. | ! ! " # *. ! . > * |

0( , "( | : , ")% ! # ., ". . . #.

2.3.1. 7 '" 4( 6 ( "( 0( J %# "# ! ! " .

( *. f (t), ") X0 a]J "(, %# "# ( *"(.. " > *.) # # t a. # ( . <<' 0( >>) " ) *.) f (t) "( 0( 1 X f (t) = cn exp(2 int=a) (2.67) n=;1 >**. 0( 4 Za cn = a1 0 f (t) exp(;2 int=a) dt: (2.68)

( ( N # *. f (t) X0 a], = a=N . *" (2.67){(2.68) , " ! "("( "( f (ka=N ), k = 0 1 2 : : : N ; 1: 4 1" " ( *" % ), ( . " " ( -"8

, ( ( * " 2 *" %" " 2 .

2.3.

35

" " " (2.68) *" "( G ", )% " . : NX ;1 (2.69) n a1 f (ka=N ) exp(;2 ink=N ) Na Yn: k=0 " exp(2i =N ) !N f (ka=N ) yk: (2.70) (2.69) # NX ;1 ;nk Yn = N1 yk!N n = 0 1 : : : N ; 1: (2.71) k=0 . (2.71) # " "( fykgNk=0;1 " "(( fYngNn=0;1 0( . 3 0( " G N # (2.71) "( N ! yk "( " )% # "( *.# f!Nn gNn=0;1: NX ;1 (l;k)n 8< N l = k !N = : 0 l 6= k n=0 , " (2.70). A " ) ) ( (2.71) " !Nnl , "( n "! 0 N ; 1 "( # "(, # NX ;1 nk yk = Yn!N (2.72) n=0

> ( 0( .

2.3.2. 8 '6 '" 4( (:4) 1( 2 (Cooley & Tuckey)

". *" (2.71) " (2.72) () ! "( (N ;1)2 " ! # N (N ;1) " ! "-

36

2.

# ( . ) # ! | "). "( | > " " 1965 " () | "( # *.# !Nnk n k " " ! # ( (N log2(N ))=2, " "(G! # N (G ) G . 3# " "" << 0( >>, % | <<" 6 0>>, "## . | FFT: Fast Fourier Transform.

# ) " 6 0. , , (, # G G" (, " (2.70): !N(n+mN)(k+lN) = !Nnk 8m lJ m = 1 2 : : : J l = 1 2 : : : (2.73) ~ " " 6 0 " " , " N . "# " 2, > , ")) " "(( "( # " ". ' " " . " # *" (2.71)J " *" (2.72) " " . E, ( N = 2mJ (2.71) ! "(! | (k = 0 2 : : : N ; 2) (k = 1 3 5 : : : N ; 1)J N=2 N=2 X;1 X;1 y2k!N;2nk + N1 y2k+1!N;n(2k+1) (2.74) Yn = N1 k=0 k=0 N=2 ; 1 N=2 X X;1 1 1 ; 2nk ; n ;2nk y y = 2k !N + !N 2k+1!N : N k=0 N k=0 0. !N;2k (!N;2 )k = g(k) . "" # # k N=2, " " (2.70). ~" ( "( " Pn = m1 (y0 + y2!N;2n + y4!N;4n + : : : + yN ;2!N;(N ;2)n ) (2.75)

2.3.

37

In = m1 (y1 + y3!N;2n + y5!N;4n + : : : + yN ;1!N;(N ;2)n ): # g(n + N=2) = g(n) (2.76) , Pn+N=2 = Pn In+N=2 = InJ n = 0 1 2 : : : N=2 ; 1: (2.77) , ! ( (2.71) " ! # n | *( " Pn In !N;n " n = 0 1 : : : N=2, " # " Yn = 21 (Pn + !N;n In) (2.78) Yn+N=2 = 12 (Pn ; !N;n In) " * ! ## (G " "( . , " " Pn In *" (2.75) ") # 0( " "( # fy0 y2 : : : yN ;2g fy1 y3 : : : yN ;1g " m, (G # ! #J > "() ! ), # !N !N=2 , > " ! ". ( # G " . " # ! s 0( " "( # " 2, s | "( 2 N = 2s . 3" "( " N = 2 ) "( . " (): (2.79) Y0 = 21 (y0 + !2;0y1) = 12 (y1 + y2) Y1 = 21 (y0 + !2;1y1) = 12 (y1 ; y2): A G . " .# N log2(N )=2 " " # " | N ! 1.

38

2.

3" ""). " . " # N = 23 = 8. ( . # " # ( *) 2.7. "( ". |

'. 2.7 " "( ( "), > | I, II, . . . , VI. ! I ) II

2.7 ! 0( " "( " N = 8 ) ! -

2.3.

39

" "( # " N=2 = 4. ! II ) III ! # " "( # " 4 ) " "( # " 2. III " " #, ( " " "( # (G ", 2. I"( " " ") " . "! "(! " !, " ! " ", " ) " | " ! " *" (2.75).

! > IV > V, ")% #

! # 0( " "( # " 4, "( *, " (Y0 Y1), # " > IV, ) " (P0 P1) (I0 I1) > IV. 4" " ! " )%! >. , ", ! % # " Yk , ), ! " * ". < > VI " " "((, >" # " " % . ") > . " "( #. 1) 3" , ( " , ! ) " "(( fy0 y1 : : : yng ! ( ,

2.7. 3" G # % " ' # , "( "(, , ! # " "( , " # " "( #, # . ". < , " ! 2.7 " " )% :

40

2.

3# ! # " "( 0 = 000 1 = 001 2 = 010 3 = 011 4 = 100 5 = 101 6 = 110 7 = 111

<<W "(>> % #

# 000 = 0 100 = 4 010 = 2 110 = 6 001 = 1 101 = 5 011 = 3 111 = 7

'. 2.8 2.7, ( . " # ! ! . , " ( ( "" "(! " #. = " "), , . ! () !,

" "() ( "( (G( G , , "( G # | " # | " ". #.

2.3.3. 2 1 | >

"(, > " ( " )%# : G " ( *. f (t) X;a a] " , ( *.) ! ! X;a a]. > " G . 4. "( "( " " G : "( % # *..

"( ># " ". . 6 (, *. f (t) ) G b, " 0( - ># *. " " "( X;b b]J 0( - " ( .

2.3.

41

. ( *. f (t) ) G b ( " h " ")

h 6 =b: (2.80) *. f (t) " ! f (hk) k = 0 1 2 : : : :

*. f (t) 0( Z1 1 p f (t) exp(;i!t) dt (2.81) f (!) = 2 ;1 Z1 1 f (t) = p ;1 f (!) exp(i!t) d!: (2.82) 2

, " ), G b, Zb f (t) = p1 b f (!) exp(i!t) d!: (2.83) 2

" *. f (!) " ( " " 0( 1 X f (! ) = fk exp(;i !k=b): (2.84) k=;1 " >**. 0( fk: Zb (2.85) fk = 21b ;b f (!) exp(i !k=b) d!: ," (2.85) (2.83), ") , p (2.86) fk = 22b f ( k=b):

" h = =b (2.86) *" (2.84), : p X (2.87) f (!) = 22b f (kh) exp(;i !k=b): k

42

2.

" (2.87) *" (2.83): Zb X (2.88) f (t) = 21b f (kh) ;b exp(;i !k=b) exp(i!t) d!: k E " *" (2.88) ) *.) sinc x: Zb (2.89) ;b exp(i!(t ; kh)) d! = 2b sinc b(t ; kh): ," "(, X f (t) = f (kh) sinc b(t ; kh) (2.90) k

"( (. 3 A" (2.80), "( , " <#. ~" > " G , )%) *.) "( ". # (. " . .). > " ) " . G, ! G " .

2.3.4. 8'+ 6 (3.

( *. f = f (x y) # " D " Oxy, ")% * # ! # " M " , )% "() D, , f = (x y), | >**. " # "% " . # "( *. f (x y) #) "! | ( , *" (2.81)), ** ., . . ". >! () ! ! | > "( | ! ( ! *.#, %! )%! , *. ! ( ( .)). # ! ! *.# , # ". #, ! , , " >**. " # "% ! ! . p( ) "! !, , *" (2.47).

2.3.

43

1(+ -

3 ?3 , - . ' " #G# % "(-

# ! "( (2.16) " #! " ! #. G "( > # * :

p( ) =

ZR

;R

( cos ; sin sin + cos ) d

(2.91)

R - . E" ( N Tk , M >! " # "() - .2.9.

'. 2.9

44

2.

W # . " " k = 1 2 ::: N # # Tk >**. " # "% ( #:

(x y) k = const: (2.92) "() " ! ( ), " # * ": i = h i i = 0 1 2 ::: N J h = 2L=N j = h j j = 0 1 2 ::: N J h = =N

(2.93)

N N - " # # J h h - G J ;L 6 6 L 0 6 6 . (ij ) " " " Oxy )%) " .) p(i j ) p(ij) - ) " ) ( > ) ". D (ij ) " k1(ij) k2(ij) :::, ! # ks(ij) ( h(ij) s . ," "(, "( (2.91) " )% # " #! " ! #: X k

(ij) k h(ij) k = p i = 0 1 ::: N J j = 0 1 ::: N;1

(2.94)

(" = 0 = ) .) p). N" # (2.94) M = (N + 1)N , " ! N . % " "( #, ""( M > N , , (2.94) " " #. W , ! " " ( ") M > N , (2.94) " #J ># . . (x y) " . )%! " " ! G #. 3" , ( (2.94) # * :

2.3.

45

XA]~ = ~p (2.95)

XA] - ., ~; " . !,~p - " . # ) "( # " *" " G , - ".:

~ T = (1 2 ::: N ) (2.96) T (00) 10) N (N ; 1) p~ = (p p ::: p ) ( "T " .) )J . XA] G " )% : h(00) h(00) ::: h(00) 1 2 N (10) (10) (10) h2 ::: hN XA] = h1 :::::::::: :::::::::: ::: :::::::::: (N ;1)) (N (N ;1)) (N ;1)) h(N h2 ::: h(N 1 N , h(ij) k = 0, " " (ij ) # "k".," "(, "( " "! " p>" . p XA], , ! N , "( N ; N >" "). ~" ! # " 20 , "% " - 0.1 , " ( ! # >**. " # "% ) N = 4000000, " , " " ! >" . XA] ! " 2000 M (, M - " .#). E , ")% " >" . XA] ") " G # ! " ( ! .. ( (2.94) ") " . " "! " G " ! - (G! . ,( ") . :

jj~cjj2 = (XA]~ ; p~)2

(2.97)

46

2.

N (1 2 ::: N ). W " > *. jj~cjj2 ") ! c k , " )% # " #! " ! #: XA]T XA]~ = XA]T p~ (2.98) # # . # XA]T XA] N N . 3" G (2.98) ( " " , )% ")! " ! ! (MATCAD, MATLAB . .)J ! .! . . @6 ''- . ( ".) # *. f = f (x y) " D 2 R2. W " D - r~i , i = 1 2 : : : N J ( > ". fi *. f " r~i . '" *.) f # *.# fuk (x y)g: X f (x y) f~(x y) = Ak uk (x y) (2.99) k

>**. Ak ) " " ")% # *. f~ fi: f~(xi yi) = fi (2.100) " " " ")% # *. # | % " " X10]. s fuk gNk=1 "() *., ", G "! # . . ' - ! >" , # " " # "(# *, ! -" ! " #. ,( ". % ! >" ") " )% . "( D " |

2.3.

47

>" , # ! " *. f (x y) ".. #G " ( " ! ) >" ) "(, ! . # >" ( ")) *. ' "(# "(# # >" G fr~i r~j r~k g (. . 2.10): E"( ""() .) *. f (x y)

'. 2.10 "( J " # , ! " = "( ! * ! ##, "! . , () # * ), ", " " # " 1. " *. f~(x y) "( , " D -# "G( " (, "( ! # . 3" " 1 ( ) # " D -" #) ".), " (, ".# "(!, )%! %) G, " ># G . 3" "(, . 2.10, " f (x y) f~(x y) = a + bx + cy: (2.101) >**. a b c *. f (x y) G-

48

2.

!, " a + bxi + cyi = fi (2.102) a + bxj + cyj = fj a + bxk + cyk = fk

" G (2.102) # ! # " a b c *" (2.101) ": f~(x y) = Nifi + Nj fj + Nk fk (2.103)

Nl = (al + bl x + cl y)=(2) l = i j kJ (2.104) ai = xj yk ; xk yj bi = yj ; yk ci = xk ; xj 1 xi yi 2 = det 1 xj yj 1 xk yk " aj , bj , cj , ak , bk , ck ") # (2.104) ." # # i ! j ! k ! i:

" (2.103) ( *"( )% " (2.99), *. uk (x y) 2 T T | "( G (r~i r~j r~k ) ") *" (2.104).

"( " ! " ! >" ( # X?]). ) N ! "( >" ". ", . " D c" !, "" "(! Ox Oy. "( D ( ! "( >" , "" "( Ox Oy: W G ! "( " . " " 1 Ns "(# # >" , G i j k l. ( *. G! r~i r~j r~k r~l fi fj fk fl . E ". f~(x y) ( f~(x y) = a + bx + cy + dxy (2.105)

2.3.

49

! #, ( 4 ! , ! fi fj fk fl . ," # (2.102) G "( >**. a b c d # f~(x y) = h 1h X(xj ; x)(yl ; y)fi + (x ; xi )(yk ; y)fj + (2.106) 1 2 (x ; xj )(y ; yl )fk + (xk ; x)(y ; yi)fl ] h1 = xj ; xi xk ; xl h2 = yl ; yi yk ; yj : ", % fi fj fk fl , ") *.# , " (2.99), % ! "( G ri rj rk rl : ) >" . ( *. f = f (x) XA B ] "# #. > " xi, i = 0 1 : : : Ns *. "! fi = f (xi). -" # ". f~ *. f (x) Xxi;1 xi ], ! , ( : f~(x) = x ;1x X(xi ; x)fi;1 + (x ; xi;1 )fi]: (2.107) i i;1

( ( ".# # x "( ". # f~(x) = a + bx + cx2 (2.108) Xxi;1 xi ] " " ! a b c ! ( *. ! "!. " Xxi;1 xi] . ) xi;1=2 *. fi;1=2 #. , " G # (2.102), # : f~(x) = Ni;1 fi;1 + Ni;1=2fi;1=2 + Nifi (2.109)

1 Nl = 2 (al + bl x + cl x2)

l = i ; 1 i ; 1=2 i

(2.110)

50

2.

ai;1 = xi;1=2x2i ; xix2i;1=2 (2.111) bi;1 = x2i;1=2 ; x2i ci;1 = xi ; xi;1=2 1 xi;1 x2i;1 2 = det 1 x1;1=2 x2i;1=2 1 xi x2i "( >**. a b c ") *" (2.111) ." # # i ; 1 ! i ; 1=2 ! i ! i ; 1. ) >" # #. < ) ".#, ! " ( # ) " ! " . G ! >" #G ( ) " "( >" Xxi;1 xi ] ! ( | *. fi;1 = f (xi;1 ) fi = f (xi)

# fi0;1 = (df=dx)x=x ;1 fi0 = (df=dx)x=x . 3" ".# "() ", . . f~(x) = a + bx + cx2 + dx3 (2.112) 3" ! >**. " (2.112) " )%) " #! " ! #: 8 >> a + bxi;1 + cx2i;1 + dx3i;1 = fi;1 >< a + bxi + cx2i + dx3i = fi 2 = f0 >> b + 2 cx + 3 dx i ; 1 i;1 i;1 >: b + 2cxi + 3dx2i = fi0 G # " #, , " J G ( . "(, ". ( # #) " G " *.# "(G " G)%! ! ! G 2. A , "., )% ( ! , ) "# J i

i

2.3.

51

! >" "( <<>" = >>. E! % "# ! " ! # X?].

") " ! .! " G (2.98)J # " .! " # X10]. W , "

" # G)% # ! "(( "( # (x y), > . jjAij jj ( 1010 >" . E > M , " G (2.98) "() ") "( . . A , . ) " G " ! " ! #, " *.) " " " , ! () " # "( # # . a) . (2.98) A^, # - p~. " #G .! # . - G " )% : X (k) X (k) (k+1) = (k) i i + k Hij (pj ; Ajl l ) j

l

(2.113)

k | .J #G " Hij = ij , ij | " (. . Hij | .). k . Hij(k) ) "# "G # ! . . , " "(G max (G min " .Aij , "G# :

= (max + min)=2:

(2.114)

) #G . = # " , (2.98) >" ! " )% # # *.

52

2.

! !:

J (~) = 21 (~p ; A^ ~) (~p ; A^ ~)

(2.115)

.) "

! , " A^ ~ . . ( "(, , ). . ! *. (2.115) " . : # . % G ~(k) " k ~z(k), ~z(k) | " #G *. J (~) ~(k) : ~z (k) = ; grad J (~(k)) = A^T (~p ; A^ ~(k) ) (2.116)

k | " G ~(k) " ) ~z(k), *. ( ) = J (~(k) + ~z(k)) : ^ ~z(k))T (~p ; A^ ~(k) ) ( A k = ^ (k) T ^ (k) (2.117) (A ~z ) (A ~z ) 4" (2.116){(2.117) #G . ) 4" ART (<>). = "( *.#, ! : X pi ; ~aTi ~(k) (k+1) (k) ~ = ~ + k (2.118) ~aTi ~ai ~ai i

~ai | -" ., )% ( i) . jjAjjJ k | "# , # " "G # ! . , 0 < < 2: < > ) "( , ) # . pi ( %() " # Aij , " ! # . # ".).

2.4. *.

53

4" ART *., ")% ( ) *.) ! *.!. , " ( " >**. " # "% , , 0 6 6 max

max | >**. "% " " "! #. ~" # # ~(k+1) ! )) ., " , " !)), " "# max: <" " (2.118) " , ( " # # . # " ~, )% * # .. ! .! " "( G ! " " G, " ! , "G # . .J ", > . # " # ! ! *.# ! # ! !.

2.4. + " $*. > " 5 G # () # *. G " ' X7] % 1917 . ," % . " "" " G # - " *. ! ! " # " " " , G , " ' . 3" " "( " " "(# n ( "( * X5])J , " " # "( n=2 n=3. 5 :""

" " (

, "% ( ( " " ", '( 2" "".

54

2.

2.4.1. 4( ( 3 (*

6*. "() *.): Z2 1 (2.119) }( ) = 2 ( cos ; sin sin + cos ) d 0 " ) ) ( # *" "! 0 21: 1 Z1 Z2 (2.120) I ( ) = 2 d Xx y]d: 0 0 <, " , ") *. , ), > *. " ! # " Oxy, :

= ;x cos + y sin (2.121) ( *). " . p( ), , : 1 Z2 I ( ) = 2 p( ) d (2.122) 0 - ># *" "(, . " # " >**. " # "% # ", "( # " ! . , # , *": q

= x2 + y2 ; 2 = arctg(y=x) (2.123) )% (2.13)-(2.14), ! " (2.120), ! " )% ) " " I ( ):

2.4. *.

55

Z1 Z1 (x y)dxdy 1 px2 + y2 ; 2 : (2.124) I ( ) = 2

;1 ;1

# " (2.124) "# :

x = r cos y = r sin : (2.125) , (2.122) *, . p( ) > ", " " )% "( " >**. " # "% (x y): 1 Z1d "Z2(r cos r sin )d# p rdr = I ( ): (2.126)

0 r2 ; 2 : Z2 1 }(r) = 2 (r cos r sin )d (2.127) 0 (2.126) ) 4 " X?]: Z1 }(r)dr 2 p 2 2 = I ( ) (2.128) r ; "( # }(r). ' G ( " : 1 Z1 dI ( ) : (2.129) = ;

0 3 G "( " *. " # *. " " " ' X7]. 0" (2.129) G (0, 0), G ")# # " " ) .

56

2.

2.4.2. 4( ( 3, *

6*. a) 8 6 3 + .

" ) *., ! # " " # " . 1. ' *. f (~x) ~x 2 Rn, ) " )%# ": Z

(Rf )(~ s) = f (s~ + ~y)d~y ~y 2 ? ?

(2.130)

~ - # Rn, ! %# " J . ! ) ) * S n;1 Rn. N ? "(, "( ~, s - " # # ")% ~ . 2. D *. f (~x) ~x 2 Rn, ) " )%# ": Z1 ~ (Pf )( ~x) = f (~x + t~)dt (2.131) 0

< (, n = 2 ' " ). 3. *. f (~x) ~x 2 Rn, ) " )%# ": Z1 ~ (Df )(~a ) = f (~a + t~)dt ~a 2 Rn ~ 2 S n;1 : (2.132) 0

" # ) "( . *. *! , ~a > " G# (). " " "( #G! # " :

2.4. *.

57

(R f )(s) = (Rf )(~ s)

(2.133)

(P f )(~x) = (Pf )(~ ~x): (2.134) W , "(Pf )(~ ~x) " ~, > (, ~x 2 ?. " *! # ~a : (Daf )(~) = (Df )(~a ~): (2.135) E ) " )% % . "( :

2.4.1.

(R f )() = (2 )(n;1)=2f (~) 2 R1

(2.136)

(P f )(~) = (2 )1=2f (~) ~ 2 ? (2.137)

, , " " 0( J *" (2.136) * 0( # s, - n - 0( # ~x. . " ) 0( 1 R : +Z 1 ; 1=2 (R f ) () = (2 ) exp(;is)(R f )(s)ds (2.138) ;1 "Z # +Z 1 ; 1=2 = (2 ) exp(;is) f (s~ + ~y)d~y ds: ;1 ? W , " "(# ~x 2 ? " " : ~x) = ~y + s~) (2.139) " ,: s = ~ ~x d~x = d~yds: (2.140)

58

2.

# (2.139)-(2.140) (2.138) : Z ; 1=2 (R f ) () = (2 ) f (~x) exp(;i~ ~x)d~x Rn = (2 )(n;1)=2f (~)

(2.141)

- # . 3" "( *" (2.137) " # # (2.131) (n ; 1) - 0( : Z (P f )(~) = (2 );(n;1)=2 exp(;i~ ~y)(P f )(~y)d~y ? " +Z1 # Z ; (n ; 1)=2 = (2 ) exp(;i~ ~y) f (~y + t~)dt d~y: ;1 ?

(2.142)

, (2.139), " , ( (2.142) # (2 )1=2f () 2 ?, "( (. " G , ' (2.130), " (2.131) (2.132) - > *., " > . # # () # * " G # (2.130), (2.131) (2.132) "( *. f (~x), ! )% # )) M .

') .)

6 6 (6 '- .

"( , ! " G # (2.130)-(2.132)J *"( > R R P P . ' " )% - " " L1(;1 +1 , ! *.#: # +Z 1 +Z 1" Z ~ ~ (R f )(s)g(s)ds = f (s + ~y)g(~x )d~y ds: (2.143) ;1 ;1 ?

2.4. *.

59

% (2.139), " , ( *" (2.143) : Z f (~xg(~x ~)d~x (2.144) Rn

, "( " ) *."( ", : (R~ g)(~x) = g(~x ~) (2.145) *"( # R J "# . . (,, : +Z 1 ~ (R f )(s) = (Rf )( s) = f (s~ + ~y)ds (2.146) ;1 (2.143) (n ; 1) - # # * S n;1 Rn, # (2.144), (2.146), " " )%) . : # # Z " +Z1 Z "Z ~ ~ ~ ~ ~ (Rf )( s)g( s)ds d = f (~x)g( ~x )d~x d~ S n;1 ;1" S n;1 Rn # Z Z = f (~x) g(~ ~x ~)d~ d~x (2.147) Rn

S n;1

" # ! , " # # , " : Z (R~ g)(~x) = g(~ ~x ~)d~ (2.148) S n;1

*"( # R. 3" P~ P~ , ! P P , ": Z (P f )(~y)g(~ ~y)d~y: (2.149) ?

" (2.131), " )% : # Z " +Z1 Z ~ f (~y + s)ds g(~ ~y)d~y = f (~x)g(~ O~x)d~x: (2.150) Rn ? ;1

60

2.

" " O ~x = ~y 2 ? . ~x "( ?. ," *" (2.149) (2.150), , : (P~ g)(~x = g(~ O~x): (2.151) E " ) ) ( (2.150) # * S n;1 , "( P~ , # P : Z ~ Pg(~x) = g(~ O ~x)d~: (2.152) S n;1

# ( G ) # (2.136), (2.137) "( *. f . 3" ># . " "(# : Z ; n=2 (I f )(~x) = (2 ) exp(i~x ~)j~j; f (~)d~ (2.153) Rn

")%# # 0( *. j~j; f (~), f (~) - 0( *. f (~x)J - "# , " ( " *" % " ! " # ".. E"( " )%# "( 2.4.1.: f (~) = (2 );(n;1)=2(Rf )(~ ): (2.154)

~ ~, " ) ) ( (2.154) " (2 );n=2j~j; exp(i~x ~ "( ~: Z ; n=2 (2 ) f (~)j~j; exp(i~x ~)d~ = (I f (~x) (2.155) Rn Z = (2 );n+1=2

Rn

(Rf )(~ )j~j; exp(i~x ~)d~

! ! ~ " *" : ~ = ~ (2.156)

2.4. *.

61

(2:155) # , : Z +1;n 1 ; n+1 I (Rf )( ~x ~)d~ (2.157) (I f )(~x) = (2 ) 2 n ; 1 S 1 ~ +1;n(Rf ))(~x = (2 )(1;n)(RI 2 , " I ; " G # : ~ +1;n(Rf ))(~x): f (~x) = 21 (2 )(1;n)I ; (RI (2.158)

( ( " ! .# (Pf )(~ ~x): 3" ! f (~x) (Pf )(~ ~x) "( *" (2.155), ) 0( f (~x) ( f (~), " *" (2.137): (I f )(~x) =

Z ; n=2 (2 ) exp(i~x ~)j~j; (2 );1=2(Pf )(~)d~: R

n

(2.159)

, ~ 2 ? ! " (2.159) ) S n;1 ?, ! , : 1 Z Z ; (n+1)=2 exp(i~x ~)j~j1; (Pf )(~)d~d~: (2.160) (I f )(~x) = (2 ) n ; 2 jS j Sn;1 ? , " "( " )% *" * X5]: ! Z Z Z 1 h(~)d~ = jS n;2j j~jh(~)d~ d~: (2.161) n n ; 1 ? R S E"( " I , % ) ?: Z (I h)(~x) = (2 );(n;1)=2 exp(i~x ~)j~j1; (h)(~)d~ ?

(2.162)

62

2.

(2.160) : Z ;1 I (Pf )(~ O (~x))d~: (I f )(~x) = (2 );1 n1;2 jS j Sn;1

(2.163)

" " # # (2.163) I ; "( " (2.152) " *" % : ~ ;1(Pf ))(~ ~x) (2.164) f (~x) = jS1n;2 (2 );1 I ; (PI

) 4(6 .

E # ! G # (2.158), (2.164) ( " " *" % , " *" ' , ! " .. ) ' . ' " G (2.158) = 0: ~ 1;n(Rf ))(~ s): f (~x) = 21 (2 )(1;n) (RI (2.165) E"( " I R1 " = 1 ; n: +Z 1 1 ; n ; 1=2 I (Rf ) = (2 ) exp(is)jjn;1(Rf )(~ )d: (2.166) ;1 , ( (2.166) " # 0( *.:

jjn;1(Rf )(~ ) = (sgn())n;1n;1 (Rf )(~ ):

(2.167) ' "( *", % "( )% n = 2 n = 3J " "(! # n # X5].

" *" (2.167) n = 2, (: (I ;1(Rf ))() = (sgn())(Rf )(~ ): (2.168) E % ! *.# X2], *. sgn() " " 0( ":

2.4. *.

63

1 +Z1 f (y) (2.169) Hf (x) = x ; y dy ;1 I"( *. f (x). E " (2.169) " " G, 0( : (Hf )() = ;i(sgn())f (): (2.170) E"( ( ) 0( , " # " ) # k *. f ) 0( ># *. " (i)k . 3" *. n !: @ jkjf (~x) ! (~) = (i~)jkjf (~) (2.171) k 1 k n @1 :::@n

k = (k1 k2 ::: kn ) - "( , jkj = k1 + k2 + ::: + kn. 3" *. # #: dk f (s) ! () = (i)k f (): (2.172) k ds A , ( , (2.168) 0( # # I"( *. (Rf )(s): (sign())(Rf )() = iX;isgn()(Rf ) ()] = i(H (Rf ))(): (2.173) " 0( , ! , : (2.174) (I ;1)(Rf ))(s) = @ X(H (Rf ))(~ s)]: @s

" # ) *" (2.165) " (2.148) R~ , " :

64

2.

1 Z @ f (~x) = 4 @s X(H (Rf ))(~ s)]d~: S1

( ( n = 3, :

(2.175)

(I 1;n(Rf ))() = (I ;2(Rf ))() (2.176) = (sgn())2()2(Rf )() = ;(i)2(Rf )():

"( " *. sgn() , % 0( " ) ) s: 2 @ ; 2 ; 2 ~ (2.177) (I (Rf ))(s) = (I (Rf ))( s) = ; 2 X(Rf )(~ s)]: @s "(# "( " *" (2.177) "( R~ :

Z @2 ~ ~ f (~x) = ; 8 1 2 @s (2.178) 2 X(Rf )( s)]d : S2 W 1. " " . ** . "(! # (2.175), (2.178) ! s = ~x ~: W 2. 3" "(! # n G X5]: Z 1 1 ; n (n ; 2)=2 f (~x) = 2 (2 ) (;1) (H (Rf )(n;1))(~ s)]d~ (2.179) ; 1 S

" ! # n, Z 1 1 ; n (n ; 1)=2 f (~x) = 2 (2 ) (;1) (Rf )(n;1)(~ s)]d~ (2.180) ; 1 S

" ! # n. ! *"! !# (n ; 1) ) (n ; 1) # s "(! *.#J ~ " s = ~ ~x: n

n

2.4. *.

65

E"( G (2.175), (2.178) " ! % (2.179), (2.180), ( *", E.' . 1) ( ( n - "J " G (2.179) " H , ! , : " Z1 (Rf )(n;1) (~ t) # Z dt d~: (2.181) f (~x) = (2 );n(;1)(n+2)=2 s ; t n ; 1 ;1 S

q = s ; t, " : Z " Z1 (Rf )(n;1)(~ s + q) # ; n n=2 f (~x) = (2 ) (;1) dq d~: (2.182) q n ; 1 ;1 S

"( - " G - # " " " G, , , "( *" :

f (~x) = (2 );n(;1)n=2 21 (2.183) Z " Z1 (Rf )(n;1)(~ s + q) ; (Rf )(n;1)(~ s ; q) # dq d~ q S ;1 ;1 # 1 Z1 1 " Z n=2 ; n (n ; 1) (n ; 1) ~ ~ ~ = (;1) (2 ) ((Rf ) ( s + q) ; (Rf ) ( s ; q))d dq: 2 ;1 q S ;1 3" ! # n (n ; 1) *. # #, " "(: n

n

f (~x) = (;1)n=2(2 );n :

Z1 1 " Z

;1 q S ;1 n

((Rf )(n;1)(~

Z

Fx(q) = jS1n;1 (Rf )(~ ~x ~ + q)d~ S ;1 n

#

~x ~ + q))d~ dq:

(2.184)

(2.185)

66

2.

c(n) = (;1)n=2(2 );n jS n;1j (2.186) ( *. Fx(q), " *": Z1 Fx(n;1) dq (2.187) f (~x) = 2c(n) q 0 " ) E.' .

n = 2 (2.187) " *": Z1 dFx(q) ; 1 f (~x) =

(2.188) q 0 # ..) ' . 2) 3" ! # n "( "# n = 3. W , , *" (2.171) " , : n 2 X I ;2f = ;f = @@xf2 : (2.189) i=1 i

" G (2.158) = n ; 1, ! : f (~x) = 21 (2 );2 (I ;2R~ (Rf ))(~c) (2.190) , *" (2.189) " R~ " *" ' " n = 3: Z (2.191) f (~x) = ; 81 2 (Rf )(~ ~x ~)d~: S2 W 3. % *" % , . "()% " 0( , " # ( .#) G * X5]J > *" " . ,* ") # , ! ". # * % " %() , " ! (* ).

2.5. # $ .

67

W 4 ( G ). ( " - "( % "(! G #, )%! # ' , " ! " ! .#, " )% # *" " G . % "( ") % ! G *" , ! >**. " f (~x) ") " # >**. " .# ( #), ") " ! ") G . 3" ' " ( "( # " , , "( G # . 3"

" "( % " ~a + t~, )%

) "(.

2.5. . / $. 2.5.1. 0 +, 6* * . # , # " G " " ! " ! !, " # ! >** ! . " >! ") # % ' "#, - !, " #! #, -!, " "# .

! % # ..

( # # M " # ) ", ) " " : U =

1 @ 2U c2 @t2

(2.192)

68

2.

*. U = U (~x t) ( " , "(), M , >" ." . .: c - ""( ( ", % ")% ! # . 3" ! ":

U (~x t) = u(~x) exp(i!0t): (2.193)

o (2.193) ) I "( "(. " " u(~x): u + k2(~x)u = 0 (2.194)

k2 = !02=c2 - " ". 3" # "( " G : "X 1 Am(~x) # (2.195) u(~x) = m exp(ik0 (~x)) m=0 k0

k0 - " ". " jk ; k0j " ) "J # " > # . "(, * ! # > " "( " " ) ! # .

(2.195) (2.194) ") " ! ! k0: " k2

#

~ )2 = 0 ; (r (2.196) 2 k0 ~ A0 r ~ = 0 A0 + 2r ~ A1 r ~ = i A0: A1 + 2r G "! , # "( !

! # (2.196), # .

2.5. # $ .

69

A :

~ )2 = k2(~x)=k02 n2(~x) (r (2.197) )% . # (2.218), ) >#"J # # ) . 0. n(~x), ! ! " #, >**. "" . E , " G ** ."(! # ! ! ) ! , # " "( G ) "(! G " ! ** ."(! #. > G ) >#". 3" ! ! ( # " #) ! ** ."(! #: dxi = pi dpi = 1 dn2 d = n: ds n ds 2n dxi ds # : ~ r ~ ~ = r = jr~ j n G (2.198) :

(2.198)

(2.199)

d~r = ~ (2.200) ds dn~ = r~n (2.201) ds d = n (2.202) ds

~x = (x1 x2 x3). ~ - #, (2.200) " , :

70

2.

ds2 = dxi dxi : (2.203) ," "(, ds - >" " # ~r = ~r(s), ~ - "(# ># # (! ).

" ~ (2.200) (2.201), ! : d Xn(~x) d~r ] = r ~ n(~x) (2.204) ds ds # *. n = n(~x) " # # ! (" #), ! %! ( !), "( # "(# "" "( ~ . E (2.202) "( ! ~r = ~r(s), " >#" (*): Zs

(~r(s)) = n(~r(s))ds + 0 s0

(2.205)

0 - "( >#". , !( = const "( ! % " " * ". A ( "0" ):

~Ar ~= 0 A + 2r (2.206) " " A "( " ( " ! " " > , > , (2.206) " "). " , G (2.206) :

r~ (A2n~ ) = 0

(2.207) " , ")% # # # . # ) "(, ") ) * 1 = const 2 = const, *

2.5. # $ .

71

1 = const -" "% d1 , # ! * 2 = const, ")% ) !( " )% * -") "% d2 . E " : Z

<

r~ (A2n~ )d = 0:

(2.208)

) " " *" I- , "( ~ # ! "( ~ , ! , :

n1A21d1 = n2A22d2 : (2.209) W " # "() ( n0A20d0, % G (2.209) # ( ")# ".

" "( ") ( ) : "( ", " # !! 1, 2, "( )%! *, )% ( ( ! ! , # " , # ! # 1 2, " ! ! , ! . E "( " (: - T12(;(~xi)) ~xi 2 1 % , G ~xi " ;(~xi) !() 2: T12 =

ZM2 ds M1

c(~r(s)) M1 2 1 M2 2 2J

(2.210)

- " A1 A2J - " * " * ( " > ! ( , " ! M ).

72

2.

' , ! )% )) M , " ""( ( c(~x) " # () "( "" n(~x). E " (2.210) " # % ' "# " #! # . # . % , " > . < , " , " # , ) % ) ' . 3" ""). > M "(! ", M , " " # , , " " #! #, , " ), *# "(, " " " " "(. ,( c0 " (2.210) ": ! 1 (2.211) T~12 = c(s) ; c ds 0 0 "( ") " ~x M J H - " " . W " ( ;1 1, " ) ' , " % ( ")# ! G . , " % ' " ! #. % " " # ( G "( " , , , . ) " " *. c(~x) " n(~x), . ! ! # . " " " ! ! "(! ! #, , ZH 1

2.5. # $ .

73

% " " X16].

2.5.2. 7 (+ '" . ' M , " # " # #, # ~v = ~v(~x t).

" )%) : # " # ~v " "( # ># ". 3 " - " " # ! . , " , ! # " # "(G! G! " ! , "! ! ! !. 3" G G # ! *! "(! " >" ! ", # ! % . " ) ! ", ! ! " !0 )% # " " !: ! = c22c!;0vv2 (2.212) ) ( % # J " c *" (2.212) - > ( )% # " (" )% "(), v - . )% # . " ". " >** 3" , " ! " % " " . 3" " "(# .) " * , )% "( " " ~x, # ! J > " #, " ~v (! " "(). ( ! ! % ! "(# .

74

2.

" , # " # # ", ! (

" ! ! " # )% ". 3" * " " ( ( " ! " " ! s " . E # " ", " )% # *. : Z

(t) = expXi(!0 + !)t]ds ;

(2.213)

s - " , - " - ! = !(s). % > (, # s = s(! ; !0) # ) : Z (t) = exp(i!t)s0! d! ;

(2.214)

- G! ># *" ** . !.

" "() *.) " " , ! , (t) " # ( () ") 0( # (!t)s0! # !. <# 0( # > *. (t), , # # . , # *.) (!t)s0! , " , "(( *"# (2.212), "( " v " . , . " (, , " ) " , ( s(!) ( *. - # " "(-*. 3, - " # .

" (, " , "( ( s = s(; v ), v " " # *" (2.212) " - %

" - *" :

2.5. # $ .

! 2!c 0 v kv "# " , :

75

(2.215)

j~vj = v << c:

(2.216) 3" " # *. " "(# : Z Z (S 0 ) = v ds = ~v d~s ;

(2.217)

;

)%# % .. " ". , " (S 0 ), # !( , > " "(! " . , ># . "() "" "(! " ; ; += , %! (. .2.11). ( , ") ) > ", " . @ " ~v d~s, ! . # ": Z

@<

Z Z 0 0 ~v d~s = (S )j += ; (S )j + ~v d~s + ~v d~s: AB

" " *" ,: Z @<

~v d~s =

Z <

r~ ~v dS~ =

! @v 1 @x1 ; @x2 dx1 dx2

Z @v2

<

CD

(2.218)

(2.219)

dS~ - # >" "% ! - "# " .

(2.219) *" (2.218) " " ) ) ( " , "( "

76

2.

'. 2.11 " (x1 x2) " (s ) "( * : Z @v2

! Z @v2 @v1 ! @v 1 ; @x dx1 dx2 = @x ; @x ds (2.220) @x 1 2 1 2 < ;

") + , " , : @ (S 0 ) = Z @v2 ; @v1 !ds: (2.221) @ @x @x 1 2 ; 0" (2.221) " # " . ! . " " (, " ! .# " ( !( , " . " - > " *" % .. "(. ") , .

2.5. # $ .

77

" " # # *. S 0 ( "( ) .

3. 3.1. + 0# ' <<= () *>> M ( .# , ! " # " . , "" ! . ' % , "( . , ) * , % | '0 . 3" " '0 "(), ", ". ># * " ( ( % ! ! " ! % ( G ". % # " " ! "( , ! #, ")% . # ,

># () # *.

3.1.1. . < ! " G! . " # " # *. 78

3.1. # %

79

4 )%! >" ! " . =" " % " " >" | "(! >" ! . # me 9:11 10;28 . "( >" e; ;4:803 10;10 : CGSE ;1:602 10;19 ". @ #. < # >" #" J " "( 1839 "(G >" .

" " "( >" e+, ")# " >" . 1836 "(G >" . # ) " # # . | ". N" ) # Z J > " >" # " . "# , , Ze+ J ! "! " Ze+ >" , . . . " #" . " ! "! ( ( >" " !( ! ", %( " "( " . "( # . N" # ) # N J N + Z = A ". 3" -" % "() ( Z XA " XAZ, X | " >" . < , " " X = OJ " " , )% " 16 8 ( .! ), : 8O16 O16 8 . W Z ). 4 Z " N ) , : 1) Z = 1, N = 0, 1 >" : J 2) Z = 1, N = 1, 1 >" : # #J 3) Z = 1, N = 2, 1 >" : #J 4) Z = 2, N = 2, 2 >" : "#. 4, )% ", "# Z , ) .

80

3. %

3.1.2. 6 3 ' () ) "( % % > " % , >" ! . /" . ' ( #, ( % ! ! >" , #, ) " >" , " ! .! " "! >" ! . ( # G ). ~" ( " " "( ! " 2He4, ! -., (G 2 . ( .! >" ), " (G 4 .. ," "(, - ". " %( 2 " " ! >" .

- , )% " . "( >" ( -.), " , " 1J > - ) ; - . ," "(, ". " " ! >" . " ) - % , )% " e+ ( +- ) >" # ! (~-!), ! >" # (K , L, M . .) ". "( ! , ! -. ~" # " -, % # >" ! , > " . @ )% >" ) , | .

" ! > . ! " >" ! ., , # e # ~e, )%! -

3.1. # %

81

.

". ! , "() " . - " . ". 3.1. - - :

;- +-

E-* + -$ . ,/* $ .

Z

A

Z ;2 Z 1

A;4 A

Z +1 Z ;1 Z ;1 Z ; 0:5Z

A A A A ; 0:5A

Z ;1

A;1

Z ;2

A;2

-! # $ (p) ' (n): n p + (e; + ~e ) p n + (e+ + e ) p + e; n + e , $ $ $

$ /*

! !

!

"( # . - " " )% .: Mo99 ! mTc99 + e; + ~e (3.1)

mTc99 >" ! .# , # "( - ( ! mTc99 ! Tc99 + ). 3" " - "( ". >" >" () e+, )% + - (. ". 3.1).

+ - : Ga68 ! Zn68 + e+ + ~e: (3.2) 4 ". >" - > # 511 =, " )%! "! " !.

82

3. %

" # ! # " G % " Q(t) % " Q0 "(# t = 0. E , Q(t) = Q0 exp(;t) (3.3)

= const > 0J # "() " T1=2: T1=2 = (ln2)= (3.4)

(3.4) (3.3) Q(t) = Q0 exp(;t ln 2=T1=2) (3.5) @ . ) % ! , ! # >" . "

. ," ( ! .# : A + a ! B + b " A(a b)B (3.6)

A | " ! , B | # ( ), a | ! . ., b | .- . @ . ) ! | , > , "(, , ". " (, " > )%! .) . " | G 6 I=.

! .#: Mo98(n )Mo99 S32(n p)P32 (3.7) Te130(n )Te131 ! I131 (3.8) (< >" : Mo | " , S | , P | **, Te | "", I | ).

3.1.3. (+ 3 , . 76 E " )% '0 :

3.1. # %

83

) ## ! " . #J ) " ( ) J ) " # ! . . 3" .# "() % , " ! . ."(! #! | ! " . " % , ")% ( " ! >** , # " " " 10;3 10;11 . < ## ! | > "% # ! (3.7). "( " # # >" " ! " (G ), , , "( ! , " . % , . = ! " " ! : ) " Mo99 > - 740 =, " | 66.2 J ) " . (3.8) > - 364 =, " | 8.05 . ' . " | > . , "( " (, " . .) " "% # ) " ! >" " "(! #. # .: 266 99 133 U235 92 + n ! U92 ! Mo42 + Sn50 + 4n

(3.9)

"G Mo99 42 . " ), , > ! " , % , % ( " ( " ! " .. . (3.9) ) 4 " "(! #-

84

3. %

, ) ( #( , ! " J . " ! )% ". # " ! G " '0 | > . (H, D+, He3+ 2 ) "(! . A " # " " ." >" " .

! ! .#:

p + Zn68 ! Ga67 + 2n

(3.10)

+ O16 ! F18 + p + n (3.11) 3 " | "( .# | " ! ! , " "( # # "(# ().

" " .# ) " )% : 1) " " ( ! # " , # " %# . )J 2) " ( " -J 3) > - " ( # " " ( , )% " # "(# .

" # >** # " TE , # , # , * # " T1=2 , # , ! TB '0 . TB "() " '0 ") TE *" 1 = 1 + 1 (3.12) T T T E

B

1=2

3.1. # %

85

% 90% " # # . "( ! .# Tc99 m . 0 # " ! . " 6.02 , Tc99 - > # 140 =. "% " " "" -" 4.6 , " " " . , # . ) In111, Ga67, Tl201, Kr81.

I123,

3" - ) " )% #. 1) I" . . A# >" * " # , " ) "(G Z . I ! "(G " "( >" " . - ! ., )% >" " , !G "( . 2) ,."". . A ., # ". " . E ")*# >, )%# > ) - " . " > # *>" # "(, ")%# "( ( -, ( ! > ). 3) " . 3) ( > - | ."".

. " ! | , )% # , , )% !" . # G)% ( ! 1 =.

86

3. %

3.2. ) 3.2.1. . 6. 6 .# , ># * "() )%! | - = . ! >" >! " "" | #, ")% "( - " ! " #. #G# "" " # ." " !G "%)% ", .J " " " (G ( "( "( # "" "( (. 3.1). <

'. 3.1 > ."".# *>" # "(. 3 # " ( ( " - "( ", * | "( ." , " "". ~" ( "% -, ( ."(# " '0 "( " (." ), ! % ( "". f (x y z ) "( '0

3.2.

87

(x y z ) ) ! .! "J " ( " (" -) EAB , ! "", ZB

EAB p = A f Xx y z ] d (3.13)

- , "( ". A " " *. f ) | | " x y z " AB , . . ") *. # . ~ , *" (3.13) "% - " AB . 3" (G G # "() ! , " ! # # . . 3" " # (<< >>), " # " ), %) ! - "! " !, , , . 3.2. ) , ) |

, ! # , ". *. ) , " ! . W , " ( %( G , ! , " " # ! # () # *, # ( M " , "( "" "( # "", . 3.1. , " # , ( M " "(! "" "(! #. # "" | *)%, %() ! . "( " '0 "# " (" ), "# "". 3) # . ) "(# * # ("( "" "( <<>> . ), " G " # *, # * #. , # *)%# "" " # -

88

3. %

'. 3.2 # .# "(G! #, ! ) # | * . 3" " "( G)% # ." ! # "() , ! (G) " ) *"(# . # ( "( , "(, "( . . . "() "(( *"(# ", " *"(# " *. "" . 3 > # ! - | 140 511 =.

3.2.

89

M *)%# "" #, )% ! , " # . 3.2, ") "( * " # " . . < ! " ( ! , " ( , "( !. ! . N >! " " ## , "(- "( ")% # #. " " , "" )%! "". < , # " * Cleon " 10 "", ! " | . .

3.2.2. - <"(G .# "" - , 1969 = . E ( * " # "% ,

" "( . E - " " ) !" )%! "! "" "(G# "% | "! " 50 , # , "" | NaI(Tl). = "" ") - " # " "# " 415 , " "( ! G" ! 0=A. >**. " # "% - > # 150 = 2.22 cm;1, " "" "%# 10 90% "% ). "" 230 , # . "" !G " , " >**. "" , ! ( ) > ) 0=A %() . E "" * " , -

90

3. %

! . " " . , ". "! !! "() "" " | !". < "" > * , * ( , - ). "" "" "( ! " . , %! " ! "". "J "" "" 400 , " # | 6000 30000, # | 3.4 1.8 , "% " ( # ) | 1.4 0.3 . A " " # (G ! *# "% " ) G)% # , " # | " # "(. G)% # () ) "( Rc, "( M # ". (, " "" "" "(

Rc d(L + z )=L

d | , L | "% , z | - "". E"() "" ! % ( ) , ")% ( G "( (G " , "" ! % , ")% "( " !( "(G) "% (. 3" . " # " "(# * ) "" "" "( , " ". ,% *.# - , ")% ."".! G , *., *. . .J - ) % #G! .# ! # .

3.2.

91

3.2.3. 0+ . ' ( #G) ) "( ! G . *., " , % ( .# - * - '0 ), )% ;, " "# ! (""). " , )% # " # ") " - = , "

! , " ! "# " , "" "(# ")% .

"( " '0 M ( *. # f (x0 y0 z0)J " '0 " " " z = a, :

f (x0 y0 z0) = (x0 y0)(z0 ; a) (3.14)

a - " " " # , () "(-*. 3, (x0 y0) - " '0 " z = a - *.. % " -, ! ~x = (x2 y2 0), g(x2 y2). 6 " (, " " " z = z1 " *)% #, # *. # h(x1 y1). # *. h(x1 y1) " . # *)% #J , " !"( "" " (x1 y1): h(x1 y1) = (x ; x1)(y ; y1)J (3.15)

" # " 0 " . "(# # # r1: h(x y) = 1 "#$ sinX(x2 + y2)=r12] > 0 h(x y) = 0 "#$ sinX(x2 + y2)=r12] < 0: (3.16)

92

3. %

' * " ; " J " # ! ( ; ."( "% ! * , ) " " . , ! # "( ; , ( " )% " # ; (x2 y2 0): Z f (x0 y0 z0)

cos()h(x1 y1)dx0dy0dz0 (3.17) 2 4

r 0 <

2 R3 - "(, '0 , cos() = z0=j~r2 ; ~r0j ; " " " ## ; Oz .

( " " "), cos() 0 r0 z0J *" (3.14), # , : Z 1 g(x2 y2) = 4 a2 (x0 y0)h(xa ya)dx0dy0 (3.18) D

:

g(x2 y2) =

xa = x2 ; z1(x2 ; x0)=a ya = y2 ; z1(y2 ; y0)=a (3.19) A (3.18) - > "(# 0 "( I- "( " (x0 y0) '0 " D " z = aJ ( ) G % 0( "( . 3" " "( !: (3.20) x = za1 (1 ; za1 );1x0 y = za1 (1 ; za1 );1y0 : x0 = ;( za ; 1)x dx0 = dx 1

(3.21)

3.2.

93

y0 = ;( za ; 1)y dy0 = dy 1

(3.18) :

+Z 1 Z1 1 X(;( za ; 1)x ;( za ; 1)y ] (3.22) g(x2 y2) = 4 a2 1 1 ;1 ;1 hX(1 ; za1 )(x2 ; x) (1 ; za1 )(y2 ; y )]dxdy

(3.22) , *. (x y) ") " D, > " ( (.

(3.23) X(;( za ; 1)x ;( za ; 1)y ] ~(~) ~ = (x y ) 1 1 hX(1 ; za1 )(x2 ; x) (1 ; za1 )(y2 ; y )] ~h(x~2 ; ~) x~2 = (x2 y2)

(3.22) : 1 ~ h~ (3.24) g(x2 y2) = 4 a 2 " - ## " (2.40). E"( , ! G : ~(~!) = 4 a2g(~! )Xh~ (~! )];1 ~! = (u v) (3.25) " " 0( . *. ~(~! ) ~h(~! ) hJ " ) : ZZ 1 ~(~!) = 21 ;1 (; a ;z z1 x ; a ;z z1 y ) exp(i~! ~)d~: 1 1

: x = ; a ;z z1 x y = ; a ;z z1 y 1 1

(3.26) (3.27)

94

3. %

" , : " :

~(~!) = ( a ;z1 z )2(; a ;z1 z ~! ) 1 1

(3.28)

a !2 ! a h~ (~!) = =h ; !~

(3.29) a ; z1 a ; z1

" (3.28)-(3.29) *" (3.25), ! , :

(; a ;z1 z ~!) = ( a ;z z1 )24 a2( a ;a z1 )2g(~! )Xh(; a ;a z ~! )];1 (3.30) 1 1 1 " (3.30) 0( , ! G " # . *.) *"( h(x y) .) , " )% # *. Xh( a;az1 ~! )];1 " "( ", " G # *. g(x y) > ".

3.3. 0# %& # ' 3.3.1. 6 + 6 6 < () ("#) ) ) , ! )%) " " ! M " . J .# ( ) ) ), %) "( "% " J "( ! # -*" - () " ( % # . "(# # ) " f (x y z ) '0 , , " # "% h, "" "( . # # ) " '0 , " ># .

3.3. # %

95

, *)% "" ") ( "( ! 3.2. 3" ! ! "(( "" "" "( ( % ", "# , | > " ! # -*" -. W , ! # " % 1963 " = , . . 10 " () *. >! ! " " " " " # f (x y z ), "( " # . ~" - "" "" "( ", % ", "# , "( *.), ) " "(! ! .

;, ! "# !() " " # "# ! M *)% >" "(( " . "% " " ( #, ( "% ; " M *, * )% # " " ; " !. *)% # .# ) !". 6" " ) "( "" " ) !" " *)% # # " 0 ", "( " ; . # " ) G)%) (, > ! "( " " *.. < " M , " ! # " '0 " " '0 " " . . > ,

96

3. %

( - " "% , #G " ! >**. " # "% = (~r) = (x y z ). , " # ># () # * ") " " f (~r) = = f (x y z ) " * >**. " # "% (~r) M " -. < .) *"# # " #| " ) "( " f " '0 , " , >** "% ", " >**. " # "% -" ( , %() # ) | # " # "# # # . ' . ! " G , " ". 2. ' G "# | " ! " # f | ( # "( . , " G " #! " ! #.

"(! | ! " # '0 "! !, "" "(! ", " % *)% "" " # = ,

" G # *. = ( " , % ! "() () " , ! *., . ( " <<.*>> *. | %() "( " ! # . () # ># * " = ) ! >! G . # , " > | " " '0 .

- >) % ") *) >) (% 0-

3.3. # %

97

=) ) >) *) (% =).

3.3.2. A

?

12 ? 3 '" . *# ># (% - 0=) ;, ( ! M , ") . , ! . Tc99 m J >! .! " " "( -, " ! "# . E % ! ! , ! . 3.2. ' #G# "#, ! # " " # ." " "(G# ", , - " # ". ", " " , > . 2.3.1 | " Oxy # " O * " " % "" "( O # (%)% #) (. . 2.4). E .# " , " '0 ! "( - ! '0 "(, , % # " , , ! . . , ! % '0 (

! ! . E , "( " " , % , M , ")% "(, "% ;.

# " " ( - *" (3.18), ( " " " " # Ox, % O %)% # O , ."( " " f (x( ) y( )) f Xx y] #

98

3. %

" . " # .. 3.2.3 .91 ( > ."( "% >**. "% = (x y) " *)% #. (G ; " ! ."( " : " Z 4

A = exp ; Xx y]d 0

#

(3.31)

~" ( ! (. .. 3.2.3) , " ! "() f Xx y] X2 3] % ", ") , q( ) ( " % # . "() ") ! " #) : Z 3 f Xx y]

" Z 4

#

q( ) = 4 (R ; )2 exp ; Xx y]d 0 d (3.32)

2

" " # 0=J % " > "( " " f (x y), >**. " # "% (x y). < ) # " "( *. f (x y), " *.) (x y) #, , # # . < ( "#, >**. " # "% (x y) = 0 = constJ " 0 = 0 ( ! , ! # # . "(, " " 0 > 0 ( % ! " , ! " G # # . < % ' . E (3.32) = 0 = const " , : Z 3 f Xx y]

"

#

q( ) = 4 (R ; )2 exp ;0 (4 ; ) d

2

(3.33)

3.3. # %

99

" exp(;04), % " , % " , " "( # # " , " (, ( " , " ! . A " ) ) ( (3.33) exp(0 4) " " "(, # ! () (, . .: 1 1 = const (3.34) 4 (R ; )2 4 R} 2

R} - # # J (3.34) " , " " # "(G ! ! " " .

"( " f (x y) = 0 " " , " 2 3 *" (3.32) ( ;1 +1 . , :

p( ) = q( ) exp(04)4 R} 2 (3.35) ): Z1 p( ) = f Xx y] exp(0 )d (3.36) ;1 "( *. f (x y). E " # > > ."( ' .

3.3.3. 8 6 3 + '" 4( .

3" G (3.36) X?], ")%! # % , " G # .

100

3. %

' 0( . , , " # (3.36) " # " OxyJ , ..2.2.2, " , : ZZ 1

p( ) = ;1 f (x0 y0) expX0 (x0 cos + y0 sin )] X ; (;x0 sin + y0 cos )]dx0dy0

(3.37)

, , () - "(-*. 3.

" # # (3.37) 0( # : Z1 1 p ( ) = p p( ) exp(;i )d (3.38) 2 ;1 Z1 (Z Z 1 1 =p f (x y ) expX0 (x0 cos + y0 sin )] 2 ;1 ;1 0 0 ) X( ; (;x0 sin + y0 cos )]dx0 dy0 exp(;i )d

"( " "(*., ! : 1 ZZ 1 p f (x y ) expf;iXx0 (; sin + i0 cos ) (3.39) p ( ) = 2 ;1 0 0 p + y0( cos + i0 sin )]gdx0dy0 = 2 f (; sin + i0 cos cos + i0 sin )

f (u v) - 0( *. f (x y).

" 0( # *. 0( " '0 , " (, " # " . "( .

3.3. # %

101

, " G (2.21) " # . "( (3.39) "( ( *.) f (x y) * )%) *.)

f = f (; sin + i0 cos cos + i0 sin ) (3.40) 0( % # ", "( 0 6= 0 " # *" (3.39) " " # *. # ! " ! !.

3.3.4. 0 '- '" 4(.

"(, % ! } = }( ) } = }( ), " # *. f ((} }) (} }) *. # % ! ! , " (, " " *. f (x y) ( 0( . = (X?]):

= (}2 + 20)1=2 = } + iarcsh(0=}) (3.41)

arcsh() *., " . 3" "( *" " *. f (3.39): q

; sin + i0 cos = ; }2 + 20 sin(} + iarcsh(0=})) + i0 cos(} + iarcsh(0=}))

(3.42)

3. %

102

q cos + i0 sin = }2 + 20 cos(} + iarcsh(0=})) + i0 sin(} + iarcsh(0=}))

(3.43)

E"( >" *" ( *. " *. % , ! , :

; sin + i0 cos = ;} sin }

(3.44)

cos + i0 sin = }cos} (3.45) *" . ! ! . "( (*" (3.39)) : Z q 1 2 2 } p ( } + 0 + iarcsh(0=})) = p f (x0 y0) expXf;i} (3.46) 2

} } (;x0 sin + y0 cos )]dx0 dy0 = f (}u v})

} cos(} + =2) v} = } cos } = rho } sin(} + =2) (3.47) u} = ;} sin } = rho - " " ! 0( , # " =2. % *" (3.46) # : Z

q

f (x y) = p1 p( }2 + 20 } + iarcsh(0=})) expXi(}ux + v}y)]du}dv} (3.48) 2

3.3. # %

103

3.3.5. 0 3 3.

: # *.) H = H ( ), " # G % > ."( ' * # .:

:

Z2 1 G ( (x y) )d f (x y) = p 2 0

G( ) =

Z1

p( )H( ; 0)d0

(3.49) (3.50)

;1 *. p( ) - > ."( ' (3.36) *. f (x y) - > ".

" (3.36) *" (3.50), "( ># - " (3.49), ! , : # ) 1 Z2( Z1 " f (x y) = 2

f (x0 y0) exp(0 ) H ( ; 0)d0 d 0 ;1

(3.51)

E"( " (3.37), (3.51) " )% : Z2( Z1 "Z Z 1 1 f (x y) = 2

f (x0 y0) expX0(x0 cos ;1 0 ;1

#

(3.52)

)

+ y0 sin )]X( ; (;x0 sin + y0 cos )]dx0 dy0 H ( ; 0)d0 d

"( " "(*., " , :

104

3. % ( 1 Z2

ZZ 1

(3.53) f (x y) = ;1 f (x0 y0) 2 expX0((x ; x0) cos 0 ) + (y ; y0) sin )]H X;(x ; x0) sin + (y ; y0) cos ]d dx0 dy0 ) , " "(-*., " )% : 1 Z2 expX;0 (x cos + y sin )]H (;x sin + y cos )d = (x)(y) (3.54) 2 0

*.) H( ) -0( : ZZ 1 (3.55) H ( ) = (2 1)1=2 ;1 H() exp(i )d

(3.55) *" (3.54) "( *" (2.61) : 1 Z1 Z2fexpXi(x cos + y sin )]gdd = 1 Z1 Z2H ()(3.56) (2 )2 0 0 (2 )3=2 ;1 0 expXi(;x sin + y cos ) ; 0(x cos + y sin )]d]d ") (r ) " Oxy " , *":

p2

I0( b

; a2 =

1 Z2 exp(ia cos x + b sin x)dx 2 0

(3.57)

I0(x) = J0(ix) (3.58)

I0 - *. 6 ", J0 - *. *. 6 ", " , :

3.4. %

105

1 Z1J (r)d = p2 Z H ()I (rq2 ; 2)d + Z H ()J (rq2 ; 2)d 0 0 o 2 0 0 2 0 0 (3.59) ~" "(: 8< jj < 0 H() = : p2 j0j " " jj > 0 2 (3.59), (3.49), (3.50) " . 3" " H( ) ( ! %) .), , (2.2.7).

3.4. 0 # ># (% - =) " " ")% # <<>" {>> "() .) Ga68, F18, C11. , G# "( # ., ! ( "# 0.28 1.35 , " "( ". - > # " 511 =, " )%! "! " !. 4 ". #J " # # "! . # # . < "( "(. .! , # ! " - " " . 4 >" " ."".# "" BGO " .

( L | " , " . #, )% # , ( = 0 = const | >**. " # "% . " ( () ! # ( ".# ) ."(

106

3. %

) exp(;0( ; 1)) exp(;0 (4 ; )) = exp(;0(4 ; 1)) (3.60) = exp(;0L)

| , # G" "..

" (, , : Z 4

q( ) = f Xx y] exp(;0L)d

1

(3.61)

*" (3.61) , *" (3.32). E) ( p( ) - " "( . # " # " exp(L). W )% ", % " " -, ") # ! ( .) - > "( .. 3.3.2, " "( " "(! ! # - " - " . . %( " " . ) # % " .

" . " " '0 , , # # " G ( ") , " " >**. " # "% # . = "( " " % = ) 0=. 3 = " "" , " > , " "(( G)% (. = " '0 , )%! J '0 ") % " # >" ! ..

4. - - ! 4.1. 2 "% 32$ 4.1.1. B :* "(. " (@') ") "( # ! , % ! " , # # ! % # >" # ". , >** @' "G( # # # !, ! % .

! . , "( % G # ! >" , " G "(G " ! " "(G! " # (<<"" >>) >" ! .. , " ) >" . @ ! ( ! ! ! >" ) " ) " ( " ) L~ . % >" " ) ", ! "( m ~ p ( " " ). E *": 107

108

4. --

p~m = L~ (4.1) >**. ."( # G . 3" ( ) ; e = 2m (4.2) p

e; | >" # ( >" ), mp | . , )% "(

" (). 6 ( >** ( ! . " .. (??)) "( " # ! - "(. 6 ( # "( %)% ") ". " ( " L~ ! #)%! %)% " ". *"# (??) ",

#)%! "( ~pm B~ 0 , B~ 0 - . # . ," "(, " " ( " )% : dL~ = ~p B~ (4.3) m 0 dt A *" (4.1), ! ): d~pm = ~p B~ (4.4) m 0 dt " " ) ") " # # # " " , " . # #. E"( " ) " ), ") , "( , % # O . O1 ~pm, " ; B~ 0 , " > " " ) Oz # # #

4.1.

109

, p~m ( ! " - " , "( # , G( . ). A " ( ( ) . ! " # :

!~ 0 = ; B~ 0 " . N

(4.5)

!0 = j~!0j = jB~ 0j (4.6) "# #. 3" , !( % " " "(! " , "( #G " (4.6) ) ( ( "-". ~ " " # ") , ."( . G " B~ 0 . , B~ 0, "( # . !0. , " B~ 0 ( , ""( " " # #.

) (4.4) .) ") " #: N 1X (i) = M ~ p~(i) p ~ (4.7) m m 2 0 N i=1

0 - -" "(, % "(G " N ! " # - , " M~ " ( # " - # *. # . 0. M~ (~x) () % % " "( ! , .) % " G ". , * # " "( ( - # () ~x " ! # , ! " *"(# -

110

4. --

# - "( #G - . ' "( . ( ! #) ) (4.4) " )%: dM~ = M~ B~ (4.8) 0 dt

" 6"!.

", -" ( > ) M~ " B~ 0 # "J , " , % , "( M~ . " "( , B~ 0 " "( Oz # "# . ," "(, " M~ B~ 0 , M~ " t = 0 :

M~ jt=0 = M~ (0) = Mx0~i + My0~j + Mz0~k (4.9)

(~i ~j ~k ) | # # . ~" t > 0 " #) G ", # . B~ 0, M~ "( G G " ! "! ! ** ."(! #: dMx = B M (4.10) 0 y dt dMy = ;B M 0 x dt dMz = 0 dt "( " (4.9). ' G ># : Mx(t) = Mx0 cos !0t ; My0 sin !0t

(4.11)

4.1.

111

My (t) = ;Mx0 sin !0t + My0 cos !0 t Mz (t) = Mz0 . . M~ ) ) !( () Oz . E > , M~ t " ) # . B~ , > . , # ". #, ! ! T1 T2. T2 ! . (G Mx My " -# " # ".. T1 ~ j | " " ") Mz ; M0, M0 = jM , G "(G " . " T1 - G # " "(# ".J ! ! "! T2 6 T1.

" " . ". > ."(, *. (4.8) " )% : M~ = dM~ B~ ; Mx~i + My~j ; (Mz ; M0)~k (4.12) 0 dt T2 T1

" , * " . ".! . , 6"! " . #. W G " > "( " (4.9) G : Mx(t) = exp(;t=T2 )(Mx0 cos !0 t ; My0 sin !0 t) (4.13) My (t) = exp(;t=T2 )(;Mx0 sin !0 t + My0 cos !0t) Mz (t) = Mz0 exp(;t=T1 ) + M0(1 ; exp(;t=T1 )) ) " M~ " Oz , )% % Oz Mx(t) My (t) " ", > Mz (t) ) M0 > ."( J 6"! (4.12) "( , ! "! " > G (4.13).

112

4. --

~" " ) ":

Mxy = Mx + iMy *" (4.13) ( :

(4.14)

Mxy (t) = Mxy0 exp(i!0t ; t=T2 ) (4.15)

Mxy0 = Mx0 + iMy0 A exp(i). ") > .. , "( " M~ M~ 0 = (0 0 M0) " % >** , # ( " )% ..

4.1.2. A ? "

( . " ( ! . " . ) % " . N # , %# ", ) "). 3" , "( ! " . ! , , ( . " )-" ( . , " %)% # , "( ( () "- "J , " " >" " , )% . ), ) *" (4.13). 3" . . "() " )% # . " " @' ) ."( >" " . (

" " " # > " "" " Ox, , ( B~ b (t) "

B~ b(t) = 2B1(t) cos !t~i

(4.16) " )% :

(4.16)

4.1.

B~ b(t) = B1(t)X(cos !t)~i + (sin !t)~j ] + B1(t)X(cos !t)~i ; (sin !t)~j ]

113

(4.17)

( "( 2 (4.16) " " Bb " "

.) <, " (4.16) ! " # ) "", ! ! | ", " J ". " # # " " ) ". # # . "! " # # | ! "# # | , " )% | > ") >** " % - >** @'.

", >** @' " J B~~ b(t) = B1 (t)X(cos !t)~i + (sin !t)~j ] (4.18) >** " J "( #G " >** " (. <# , "( 6"!, # >** " (4.18) M~ (t). W , ,

" "(( "( G 2 , ". 40 . ," "(,

" , G)%! 40 , . # ( "(( (4.8):

dM~ = M~ B~ dt

") " (: B~ (t) = B1(t)(cos !t)~i + B1 (t)(sin !t)~j + B0~k

(4.19) (4.20)

114

4. --

E"( (4.19) " G ", G (4.19) !:

dMx = M B ; M B (t) sin !t (4.21) y 0 z 1 dt dMy = M B (t) cos !t ; M B z 1 x 0 dt dMz = M B (t) sin !t ; M B (t) cos !t x 1 y 1 dt

"(# B1 (t) G # (4.21) ( "(, " , B1 (t) = B1 = const (4.22) G ( .

" " "(, ! = !0 = ;B0 , # *. u v %() u = Mx cos !0t + My sin !0t (4.23) v = ;Mx sin !0t + My cos !0t Mz = Mz E *" (4.23) , " u . M~ ( ~i0 # , %)% # " Oz "# () !0, " v . M~ ( ~j 0 # # (~k0 = ~k ).

# (4.23) (4.21) " )% # # #: du = 0 (4.24) dt dv = B M 1 z dt dMz = ;B v 1 dt

4.1.

115

" )% "( ":

ujt=0 = 0 vjt=0 = 0 Mz jt=0 = Mz0 (4.25)

Mz0 | M~ . ' G (4.24) "( " (4.25) : u(t) = 0 (4.26) 0 v(t) = Mz sin !1t Mz (t) = Mz0 cos !1 t !1 = ;B1 , # (~i0 ~j 0 ~k0 = ~k) M~ (t) % " ~i0 , ( " (~j 0 ~k0 ), "# () !1 . A " = !1 (4.27)

, ! ( # , "( ) , " " =2 "

. ") > .., " , B1 = B1 (t) | "( # " "( , " " "( *" Z = ; o B1(t) dt (4.28) = "( " % #.

4.1.3. 4 0- , ?*.

~" " M~ # " B1 ")(, " B~ 0, "( #G ( *" (4.13). " (4.13) ( G! # ( ) # " | ", * . 4.1. =

116

4. --

'. 5.1 " " # ., % | ,,,E. 4" " ( , G " ,,,E " )% ") *.) " % # ". ,,,E "( ". T2, ! ! # % . ' z - *.) G # ". T1. < " G "() ."( " "( " . , "(, )% "

=2 "() ) ! "(, " )%! TR. E ,,,E, " "( ( # ) "(-

4.1.

117

'. 5.2 . 3 " "(( "( " "(() % ( " Oxy) | " ( . " ) Oz ). 3 # | -" J ( " M~ " , " TI * )%# "( " =2. =** >! " - ! 6"!. ,( ") , . * , "")%# . M~ "( Oxy " % # "( ( *" (5.27)) << ( (>>. = " * -"( ("( *) (TE =2) " # ( =2)-"(. ' " "( , G" *)% "(, ( > % # * | *#. ,* # TE =2 > # ,,,E, # >! (. . 5.2).

118

4. --

<< '- *)>> ( . '. W. , A. 4#!**) " )% "( " >!. ~" ( (<<">>), . ) ! G # (, , " ! ( !"(, " "( "( "( . , ( ( . . % " "(, " "(( " >! ( .

4.2. + 32$- #

4.2.1. E"3 + , " + ' - " , +6

~" " B~ , % "( . ( ), , , *"# (4.5) " (4.6) " . ( . = ( " M , " ( B~ .

#G# # , ) ") ( ! # (G) | > " # ( # " . "( >**. (x y z ) ) " B , " , " # % , ) , )% G | G. E, ( B~ (~r) = B~ (x y z ) = (B0 + Gxx + Gy y + Gz z )~k (B0 + G~ ~r)~k (4.29) (x y z ) " ! ! = ;B = ; (B0 + G~ ~r) (4.30)

4.2. -

119

I # "# # " "(), "# G~ . " , Gx = Gy = 0, " (4.29) " ( " , " Oz . ~", " # )% # z - # " M ( " # )%) y- , # "# # , "" "(# Ox. < ., " ( B~ G " " # )% " ! ! # ( " "( ") B~0 "( Oz ), M ( " " . ~" " )% "(, "( > "( # , ! , " ! | ". I " ( " " "( ( # )%# ), "( ,,,E . ") > .. , # "( , , " , " # "# ( " ), # "# "#, # "% (." " , G " . " # ). = ") G)%) ( .

) . "% ", " Oz . (

B~ (t) = B1(t) cos !t~i + B1(t) sin !t~j + (B0 + h)~k (4.31)

h = Gz z Gz | " "( Oz . " (4.31) 6"! (4.8) ( .), (4.21), # B0 ! B = B0 + h.

(4.23) " , ! = !0 = ;B0 ,

120

4. --

du = hv ;! v (4.32) h dt dv = ;hu + B M ! u + B (t)M 1 z h 1 z dt dMz = ;B v 1 dt ' "() ) . , " % B1 " "(, " dMz 0 0 M (4.33) z M = const dt " ) *.) (t) = u(t) + iv(t) (4.34) " (4.43)) *" (4.34), ! , dc = i! c + iB (t)M 0 (4.35) h 1 dt

"( " : ujt=0 = 0 " " )% G :

vjt=0 = 0 Zt

(4.36)

exp(;i!h s)B1(s) ds (4.37)

" h = Gz , "( B1(t) # (0 ), # " % " Oxy: Z jc( z )j = M 0 j 0 exp(iGzs)B1(s + 2 ) dsj (4.38) ~" "(, )% B1 "( *. " # I: B1 (s + 2 ) = expX;(saG)=8] (4.39)

(t) = iM 0 exp(i!

h t) 0

4.2. -

121

! # . " , 95% "% # (4.38) ! " jz j 6 a. ," "(, "% " 2aJ " a " ! .

4.2.2. + '' ...%. 1 ( 6, " 6,

,,,E % " # G# ( #), . ( ! ) M~ (t) | G =3,, ."( . ~" ( ("# "), " ,,,E " "() % # J ( ,,,E " ". | > . .. 5.2.4. A" ", " # " M "( *( > " -" # ". )% . = " " ". ~ ". " , ( "(G , > % " "( . ~" , " "( ! , #( " ,,E. ' "()%# " "( " )% ! : Z d V (t) = ; dt < M~ (t ~r) Q~ c(~r) d~r (4.40)

| "(, . )% , Q~ c (~r) | ! "( G # ~r .

122

4. --

! "! !G " " " )% "( :

Q~ c (~r) = a~i + b~j a = const b = const (4.41)

(4.41) *" (4.40) " " =3, : Z (4.42) V (t) = ; dtd <(aMx + bMy ) d~r " " a + ib = k0 exp(i0) = k0 (cos 0 + i sin 0) (4.43) Mxy = Mx + iMy = A(~r) expXi( + !0 t) ; t=T2] !0 = !0 + !h = ;B0 ; G~ ~r = arctg(My =Mx) G (4.42) " )% : Z V (t) = ; dtd < ReXA(~r) exp(;t=T2 )k0 expXi(!0 t + + 0)] d~r

(4.44) (4.45)

(4.46)

, >** # # G " | " k0 * " 0. " (4.46) .) " % # c Re ~r ** . t: Z V (t) = ; <XA(~r) exp(;t=T2 (~r))(; T 1(~r) )k0 cosX(!0 + !h )t + + 0] (4.47) 2 0 +A(~rZ) exp(;t=T2 (~r))X;k sin((!0 + !h )t + + 0)(!0 + !h)] d~r = k0 !0 < A(~r) exp(;t=T2(~r))X(1 + !h =!0) sin((!0 + !h)t + + 0) +(!0T2(~r));1 cos((!0 + !h)t + + 0)] d~r E"( ( > " )% *:

4.2. -

123

1) " !h =!0 " ) 1 (" " G~ ~r)J 2) " (!0T2);1 " ) 1.

( (4.47) " , ! "J ": Z V (t) = k0 !0 < A(~r) exp(;t=T2 (~r)) sin((!0 + !h)t + + 0 ) d~r

(4.48)

:

k k0!0

: Z

0 ; =2

V (t) = k < A(~r) exp(;t=T2 (~r)) cos((!0 + !h)t + + ) d~r

(4.49) (4.50)

> " "( #G # . ,! # ". ,,,E . 5.3. ," ! # . >" . < ( ! * " , ( # " * 0( " | " " M (~r), , "(, ."( " " )% ). # *# " )% . # "( " # " ", ! ! *.) c cos !t, c = const, ! # " !0. 4" . " , " ", * G ) " " 90. "( " " ( !h t + + ):

124

4. --

'. 5.3 1 1 cos !t cos(!0t + ) = cosX(! + !0)t + ] + cosX(!0 ; !)t + ] 2 2 " :

(4.51)

(4.52) cos !t cos(!0t + + ) = 1 cosX(! + !0)t + + ] 2 2 2 1

+ cosX(!0 ; !)t + + ] 2 2

" " *"( ! , )%# ")% J ! *"( " Z SI(t) = k2 < A(~r) exp(;t=T2(~r)) cosX(!0 ; !)t + ] d~r (4.53) Z SII(t) = ; k2 < A(~r) exp(;t=T2 (~r)) sinX(!0 ; !)t + ] d~r (4.54)

4.2. -

125

M " " " ) *.) Z

S (t) = SI ; iSII = k < A(~r) exp(;t=T2 (~r)) expXi(!0 ; ! + !h(~r))t + i((~r) + )] d~r

(4.55)

,

Mxy = Mx + iMy = A(~r) expXi( + !0 + !h )t ; t=T2] #

Z

S (t) = K < Mxy (t ~r) exp(;i!t) d~r (4.56)

K | " , ) " %! G . " # "( # 1, " (. 3"( #G " S (t) , * Mxy (t ~r) ) *.) ! " ( " " ,,E.

4.2.3. B * , (3 ' "#, "( # %)% "( M M " =2, t = 0 ( " )

M~ jt=0 = ;~iM0(~r) (4.57) (, "( ( M0 ."( " (~r): M0(~r) = (~r) (4.58)

| ."(, ) , G , "( # 1.

126

4. --

t = 0 # %)% "( , "( G, % G~ , "(# # ~r "( *" :

Mxy (t ~r) = Mxy (0 ~r) expXi!0 (~r)t ; t=T2 (~r)] M )% # (4.15) (4.30)J !0 :

(4.59)

!0 = !0 ; G~ ~r (4.60)

" (4.58){(4.60) *" (4.56), # , Z

S (t) = < Mxy (0 ~r) expXi!0 t ; i G~ ~rt ; i!t ; t=T2] d~r (4.61)

", "# !0 = = ;B0 , ("( *" (4.58)): Z

S (t) = < (~r) exp(;t=T2 (~r)) exp(;i G~ ~rt) d~r (4.62)

" *" , # # G# # * " ,,E " G~ " # 0( " % , G # *. # expX;t=T2 (~r)]J 0( (<<>>) : ~ = Gt ~ (4.63) ," "(, " G~ "(G# " , > "( # *. (~r) expX;t=T2 (~r)] " %() 0( # G ) "( (~r) expX;t=T2 (~r)]: ~" " exp(;t=T2 ) " "( # 1, (t=T2) 1, . ! " " (~r).

4.2. -

127

'. 5.4 , # "(, *. 0( - # * , " " :

Gx(jk) (t) = G cos j cos k 0 6 t 6 (4.64) Gy(jk) (t) = G cos j sin k 0 6 t 6 Gz(jk) (t) = G sin j 06t6 j = (j ; 1=2) =n j = 1 2 : : : n k = k2 =m k = 0 1 : : : m ; 1 C! # " "( . 5.4. ~% # , # # " " # 0( ! " , . 5.5. W ( >

128

4. --

'. 5.5 " " " =2 ) Oy Oz . " > > # "( " Ox. ', "( (4.56){(4.62), # 0( - " . 3" # "( *" (4.60) :

Mxy ( ~r) = ;iM0(~r) expXi(!0 ; (Gy y + Gz z ))] (4.65)

" Gx . ,,,E ": Mxy (t ~r) = M ( ~r) expXi(!0 ; Gxx)(t ; ) ; t=T2 (~r)] = ;iM0(~r) expXi(; )(Gy y + Gz z ) + i!0t ; Gx x(t ; ) ; t=T2 (~r)]

(4.66)

4.2. -

129

(4.66) *" (4.55) (" ! = !0 ) Z

S (t) = < (~r) expX;t=T2 (~r)] expX;i (Gy y + Gz z + Gxx(t ; ))] d~r (4.67) ( ;i - " ) , ! "( G ). (4.67) " # 0( # ", G # *. # exp(;t=T2 ), XGx (t ; ) gy Gz ]. "( t , * " 0( #, "# " Oyz . %() Gy Gz " ># #, " # 0( .

4.2.4. (3 + " , ' ,,,E " ", " Oz (. .. 5.2.1). 3" 0( " "( " "(( "(, " ) . 5.6. " , "% )% ", *" (4.62) > . ,,,E, > -# ". # , Gz = 0, (

S (t) =

ZZ @

(x y z = z0 = const) expX;i (Gx x + Gy y)t] dx dy

(4.68)

"", , 0( *. " z = z0 = const. " G , " # "(, Gx, Gy , ! "# 0( *. (x y z = z0). #-" % 0( , " G " # . ' "( . " ,,E " >! , " ! "( ("),

130

4. --

'. 5.6 " J % ! "( . ' " ! , " * # "( # Oz Oy, " # "( " =2. "( , "" "(# Ox. 3" * >! ! # ( *)%# "( " # " Oxy, # "# , # " ) >! , " ! "( , "" "( Ox. '" > " # *)% "(. , " >! " )% # *"#:

4.2. - Z

S (t y0 z0) = M0 exp(;ty =T2) (x y0 y0) exp(;iGx xt) dx Ox

131

(4.69)

")% # # 0( " " J " ty ( | > # Gy . ") > " , " "( @'- ", *. G % )% # % " "(# !, ")% # "(# .

4.2.5. %" , T1 + , T2 3 E"( " "(( "( % -" "( TD . " # *" (5.13) "( " ,,,E # =2-"( :

Mz (TD ) = M0(1 ; exp(;TD =T1)) E > , (:

(4.70)

f (TD ) = ln 1 ; MzM(TD ) = ; T1 TD (4.71) 0 1 " " ## >**. (;1=T1 ). ) # " TD , ( * (4.71), "( "(# (- G #) ". T1. ~" "(( " "(( "( << " >> ( " . G ), " "

132

4. --

T1 "(# ". *" (4.13) " " )%) (: Mz (TD ) = M0X1 ; 2 exp(;TD =T1)] (4.72) ) ) ! > . 3" " T2 # (-#) ". *" (4.13), ! " , " ! " T2 ( )%) ,,,E. 3" (G G # ) # ,,,E, ) ! ". " " , " (G " ( " " T2 ! * ** . " "(( >** >!. " *" " " T2 "( # ,,,E =2 "(.

5. # 5.1. : / . # * ("( " ."( *") ! # # M " >" (, " ?? " " ^ - . *" (??. G # - > " ! M , )% ) "( , " >" ." , )% M " .", " # M " >" # . G " # " . M " ! ." " n = I~ ~ !(,

)%# " >" .". , # * ") , , " >" ." ! M )% " !(, # " >" # M . # % # " G , "( >" ( % " ! ! , " " "(

*. ! !. ~" "( ># . ) # , " (, 133

134

5.

)% " " % # "(# " # . , , # # #, "" G( ") . ~" ( # . # , ( (

M "( % . = " # "(# *. # * )% "( ! ! > , ")%! " >" ." !. 0 ( ( , # " M " ! " ) ." " >" !(. # # "

4. ."( , "" ( ( " .. ># , # "(, " ! % > " " ! G #. "( G # 3!" , ")% # () " . *. # *, " "( , " ! > . ~ , "( > , > ( G ) " > , "( ! ") "G # G ). , "( # " ( ! > ), > " ( ( " ), " " % " ) >" # # "# *. - >" # .

5.2. .

135

5.2. 2 "%. 5.2.1. + , "+. ' " ) - %) "( " M >" ) . (, ! " # N . N( ! " ( " , . . ! >" ." ( " "), ." "(! " "( " ! ( "). 3" ") " ! ." ( , )% N ! >" . "( # >" "), # * ( G "( " , (N ; 1)2 > M , "( " ! # (N ; 1)2. "( . *. " ## " " )% .

( ~ @ = (1 2 ::: N )T - -" . ! ." ( "" - . ),

I~@ = (I1 I2 ::: IN )T (5.1) - ! . N - ." J i# ( " ( )% " ) >" ) ! !, i- , ( 1. > ~ (i) @ , )%# (i) I~@ . "( " " ##, # * " . XA](~ ), )% # .", . .: I~@(i) = XA]@~ (i) (5.2) @ i = 1 2 ::: N: " " G ")

136

5.

>" ") > , (5.2) (N ; 1)2 ! #. #, ! (N ; 1)2 > M " # # "( .", ! G) (G! - > . W( >" Aij . XA] " # " " ## " . M . ~ , >

M J ( # % ! *. , ! ( )N " ! > ( )N ." ~ (i) ! I~@(i) @ i=1 I=1

# G ) " ## " # # (5.2). G # .

# ( "(# ( #) , , # . ", " " ! , . . " ..

5.2.2. 1 ( .

, " ! .# # * " ) .*, , - !, ") , ( in vivo >" ." - .", ." , " ! . "( "( ! #, ( # *.. -!, ( " " # ") ) , " ,

!) - " "!,

5.2. .

137

* " # " "G# , *. !, " , " " ", !. , . ! "G# , ( 1 (G " . # "( " ( " # ", .# - , # # " X19] "G( " ! ( # "() " - " % # .

- " *" G " ) "G# - " " M J , G # - *. # !, % , " " ." = (~x) " " )% ! ! (. (??)):

div(grad) = 0 > ) ") ":

= @(~x) ~x 2 @

~ ) ~

= j@ (r @

(5.3) (5.4)

(5.5) # * ") , # " >" # " . ># " *. @ j@ . W , * ( " ! *.# " , "( ." (5.3) .

"( (5.3) " (5.5) *. ! , "(# # * " " ##.

138

5.

,*" " ! ! >**. ! # * " , , #, > ! ( ! % " ".. ( " # "., # ) . " G ! .

5.2.3. 7"3. ' " ! (5.3) "# (5.4) - (5.5) ! .. 3" ># . " ) - ! #, ! >" ! >" , " ! >" ! " ( * "( G " " # (5.3). "( * "( G "( " # >**. , (5.3) ! "( ( ) >**. J > "(, , ( " )," " , ! G ! " (5.3) "(( .# " ! "! % " ). 6" M ) ") ), " " # "G " ! >" " - " ! - ! >" . ' ". ! " # , > "( # . ( " "(! " #) ! #. ' ") " "(, "" "( . ") ( G hx hy ) "() , " # # i = 1 2 :::Nx j = 1 2 ::: Ny . , . ! # G" " " (. " -

5.2. .

139

X20]) " )% # " ! #:

; 21h2 (ij + ij;1 )ij;1 ; 2h1 2 (ij + i;1j )i;1j " 1

y

x

(5.6)

# 1 + 2 (ij;1 + 2ij + ij+1) + 2 (i;1j + 2ij + i+1j ) ij 2hy 2hx ; 2h1 2 (ij + i+1j )i+1j ; 21h2 (ij + ij+1)ij+1 = bij x y

ij ij - ! *.# (x y) (x y) " (xi = ihx yj ; jhy . (5.6), )% # 3!" , " ij ) ) #J " " (i ; 1 j ) (i j ; 1) (i +1 j ) (i j +1) ., )% " " # (5.6) )

" ) ) ( bij . -". ! ~ ~ e ~J ~ e - > # ." ! "! , ~ " ~ e , )%! ", ! ." . ( XA(~ )] - . # (5.6), # * > ( " )% :

XA(~)]~ = ~b

(5.7)

~b - # . 3" " G "( . XY (~)], )% ." ~ e ! "! ! - # " : I~@ = XY (~)]~ e (5.8) # # ! I~@ # " # ! I~@C . 3 * G)% # # ! # ( ,

140

5.

)% *" ( , "(, ( " G " ! # .

5.3. " 5 . 5.3.1. 0 @ . # " G . " ") . - " # (5.2), (5.7) !, *"( "(# * , " > ! - " ! , . G . . 1) # G " ") , ( "( " " ~ = ~(0) " " , " G( " #) # (5.7) # ." ~ (0) " " J , " G , # "( " " ~(0) ( N G # ~ (0)i (5.7), )%! N > , "i" - > , )% i- (. G ). 2) E"( G (5.8), " ! :

I~@(0)i = XY (~(0))]~ (0)i (5.9) e W , >! " ! "( "# ." ~ e, ")%# # G (5.7) ~

" ! ! "# J > .) "G " ~ ( " comp. 3) " I~(0)i " " : (0)i I~(0)i = I~@i ; I~@C 4) 3" " %!

~ = ~ ; ~(0)

(5.10) (5.11)

5.3. # '.

141

" .) # XA(~)] XY (~)] % ~. W , > > ! # " . XA(~ e)] XY (~ e)] *": XA(~)]~ e = XA(~ e)]~ XY (~)]~ e = XY (~ e)]~ (5.12) E") ." ! ! ! %() G (5.7), " -" . I~(0) , " ! *" (5.10) ! ) > i, " )% # # " " ~(0) : XP ]~(0) = I~(0) (5.13) . XP ] # (N ; 1)2 Ne " . : (0) (0) ;1 ~ (0)i )]) XP ](i) = XY (~ (0)i e )] ; XY (~ )]comp(XA(~ )] XA( e

(5.14)

' G # (5.13) % " (G! : ~(0) = (XP ]T XP ]);1XP ]I~(0) (5.15) " % " . # *" :

~(1) = ~(0) + ~(0) (5.16) # - G , # " "(G # ! . , > ". (G! " ( - " "( . (XP ]T XP ]) (G G hx hy ( ").

142

5.

#G# ". ") " "( >" . (XP ]T XP ]) " ", # " ! > .

" . *" (5.16) ! ! " # J . . " " ! # . W . ~" ( . XA] XY ] ~ " " ##, . * . XA(~ e)] XY (~ e)] *" (5.12) ! ( " . XA(~)] XY (~)]. = "( M - <(). 3

$ % X1] The Physics of Medical Imaging. Medical Scince Series Ed. by S.Webb. BristolJ Philadelphia: Adam Hilger IOP Publ. 1988. 633 p. = 0 ". # . . . A>. .: , 1991. .2. 408 . X2] I "(* E.., I .E., " E.@. E "( # " #. , "% *.", .5. .: 0 , 1962, 656 . X3] :* .4. E "( ! " #. "<", <, 1993. 229 . X4] < G)%# "( . ,. . ..") . .: G , 1995. 487 . X5] < 0. () # *. .: , 1990, 279 . X6] 4.,. " "( *. .:I4 E, 1996. 136. X7] ' E. " *.# ! " "( ! #. - .: - " ,. ' . .: , 1983. X8] "( . . 0 . . .-"". .:, 1989. 567 . X9] Tretiak O.J., Metz C. The exponential Radon transform. SIAM J. Appl. Math., 1980, v.39 pp. 341-354. 143

144

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