2002 Journal of Software,,. :(1) (simultaneous perturbation stochastic approximation, SPSA)

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1 /2002/3(0) Journal of Software Vol3, No0, (, ) ttp://wwwfzueducn : X, Spall Crition (SPSA),,,,, : ; X ;L ; ; : TP39 : A, :() ;(2),,,,,,,,,,,, X, X [], [2] [3],, Jacobi, Spall Crition (imultaneou perturbation tocatic approximation, SPSA) [4], (tocatic perturbation gradient approximation, SPGA),, ( ), f(x,y) H (, n(x,y) g(x,y), g(x,y)=h[f(x,y)]n(x,y) () () H[f(x,y)] g(x,y) D(x,y), g = Df n, : ; : : 973 (G ); (F0003) : (963 ),,,,,, ; (94 ),,,,

2 : 2045, g = Df n p = Ξf ε [2],p,f,ε,,,,, g = Df n p = Ξf ε p p, R x x,, p = up x 0 x / x u D, D R, L(u), L ( u) ( ), ( ) (Ω,F,P) n, δ = û 0 a 2 e D / L = Rf n (2), L SPGA L, 2 2 ( ) (potfiltering) SPGA, Llacer, ( ) ( ),,, ( ), Snyder [5] 2 SPGA L(u) ( ), () u, [6], L(u), G(u), G (u) = (u) L u SPSA [4] a, a, Spall, ˆ ), (3) u ( ' ), (3), δ = a /, 0 < δ, =, 2, uˆ ˆ ˆ = u a H (, u ), = 0,, SPSA, [7], f(z) z, >0 ( S f )( z) = f ( z ) S, δ ( δ f)f(z)=f(z/2) f(z /2) / 2, δ δ ( S ) ( S ) (4) D / 2 =,z Taylor D f ( z ) = e f ( z), (4) D f ( z ) = ( S f )( z), S = e (4), e D / 2 =2in(D/2), D D = 2in S ( δ / 2) (5) P 2m ( δ ) (5) 2m Taylor, 2m f, 2m m 2 (2m ) f ( z) = P2 m ( δ ) ( ) ( m!) f ( ξ ) /(2m )!, ξ [z m,zm], (5) 0, L(u), >0, v, z f(z)=l(uzv), δ v, L 2m, u ( 2 m v, T 2, f 0) = v ( L( u) / u) = P ( δ ) L( u) O( m ), P δ ) f(z) 2m 2m( v, Taylor v =, 0 c u ± ( i / 2) c i =,, m, <, δ (3) δ Llacer, J, Velerov, E, Nunez, J Te concept of cauality (feaibility) in tomograpic image recontruction In: Proceeding of te NATO Adv Study Int, Povoa, Portugal, Sept 988

3 2046 Journal of Software 2002,3(0) ± ( i / 2) 2) L( u ± ( i / 2) c ) ε ( ε ± ( i / ), m=, u SPSA i=,,m, u ± ( i / 2) c D0 (u) u, (u)=l(u)ε u, H (, u) = P m ( δ, 2 c ) ( u) /( c ) (6), (3), (3) H(,u) SPSA SPSA ћ=min(4m, 2)>0 u=u, L Heian λ, aλ>ћ/2, 4 :() L(u) u D K< D,L(u) u (2) () σ ( F n, F n ) [6] T L (3) m E,,, ) iid m p l <, i = i ( ') = ( (4) int D, u int D, c>0 D(c), D(c) D ξ t, (t,,ξ) D, x 0 C y aλ 0,λ>0, (t,,ξ) ξ C 0( / t) ( ) 4 : D, L 2m, (4), ( (m)c/2) D, (3) (6) H, (7) : uˆ u = O aλ aλ<ћ/2, u u = O ( ), aλ= ћ/2, ε>0, ˆ u = O ( ˆ ( û / 2) m /( 2m ) uˆ u = O ( aλ ε ) uˆ,(3) û = uˆ a H (, uˆ ), 2 p (7) ˆ u ˆ wen ˆ = u u D 0, (9) ˆ u ˆ wen ˆ = u0 u D 0 (0) 4m= 2, ћ=min(4m, 2) =/(4m2),ћ=2m/(2m) m, /2 3 0 u ), (8) SPGA,, SPGA Hoffman,, (3), a=05,δ=,=,2, Collie Collie, Collie, 35, SPGA (b) (c) (a) Original degraded image (b) Recontruction of invere filtering (c) Recontruction of SPGA algoritm (a) (b) (c) SPGA Fig Recontruction of te degraded Collie image Collie

: 2047,, 52 52 45, SPGA 2(b) 2(c) (a) Original degraded image (b) Recontruction of invere filtering (c) Recontruction of SPGA algoritm (a) (b) (c) SPGA Fig2 Recontruction of te military airport

4 : 2047,, , SPGA 2(b) 2(c) (a) Original degraded image (b) Recontruction of invere filtering (c) Recontruction of SPGA algoritm (a) (b) (c) SPGA Fig2 Recontruction of te military airport aerial poto , SPGA 3(b) 3(c) (a) Original degraded image (b) Recontruction of invere filtering (c) Recontruction of SPGA algoritm (a) (b) (c) SPGA Fig3 Recontruction of te model image 3 3,,SPGA, Potoop, Hoffman, 3, 4 SPGA,,SPGA Sepp-Logan, Sepp-Logan,, ( 4),, 4, Hoffman, Fig4 45: UCLA(Univerity of California, Lo Angele) 4 ECAT-III X [8], 3000,, ( [9]): () (random coincidence) Hoffman, 65%,,,SPGA Poion

5 2048 Journal of Software 2002,3(0) (2) H-value H 90% confidence level Hitogram teting function,, VL [0] [], R,Llacer Velerov, λ, R ji λ, LE(maximum lieliood etimate),,, λ=65% Iteration No, 90%, Fig5 5 Hoffman, 5 Hoffman,, H, Hoffman (a) SPGA 25 6(b) 45, 6(c) 0 6(d) 300 6(e) 6(d),,, σ=055 4 (a) (b) (c) (d) (e) Fig 6 6 SPGA Hoffman,,SPGA LE FAPE (te fat maximum a poteriori wit entropy), SPGA, SPGA, Heian /2, /2,, SPGA,,, ( 4 ),, UCLA ECAT-III X Hoffman ( )!

6 : 2049 Reference: [] Hielcer, AH, Kloe, AD, Hanon, K, Gradient-Baed iterative image recontruction ceme for time-reolved optical tomograpy IEEE Tranaction on edical Imaging, 999,8(3):262~27 [2] Hanon, K Object detection and amplitude etimation baed on maximum a poteriori recontruction SPIE, edical Imaging IV on Image Formation, 990,I-23(4):64~75 [3] Pan, Xiao-cuan Conitency condition and linear recontruction metod in diffraction tomograpy IEEE Tranaction on edical Imaging, 2000,9():5~54 [4] Spall, JC, Crition, JA odel-free control of nonlinear tocatic ytem wit dicrete-time meaurement IEEE Tranaction on Automatic Control, 998,43(9):98~20 [5] Snyder, DL, iller, I, Toma, LJ, et al Noie and edge artifact in maximum-lieliold recontruction for emiion tomograpy IEEE Tranaction on edical Imaging, 987,I-6(3):228~238 [6] Gerenc r, L Convergence rate of moment in tocatic approximation wit imultaneou perturbation gradient approximation and reetting IEEE Tranaction on Automatic Control, 999,44(5):894~905 [7] Fox, L Two-Point Boundary Problem in Ordinary Differential Equation Oxford: Clarendon, 957 [8] Hoffman, EJ, Ricci, AR, van der Stee, LA, et al ECAT-III Baic deign conideration IEEE Tranaction on Nuclear Science, 983,30():729~733 [9] Nunez, J, Llacer, J A fat Bayeian recontruction algoritm for emiion tomograpy wit entropy prior converging to feaible image IEEE Tranaction on edical Imaging, 990,9(2):59~7 [0] Velerov, E, Llacer, J Stopping rule for te LE algoritm baed on tatitical ypotei teting IEEE Tranaction on edical Imaging, 987,6(4):33~39 [] Llacer, J, Velerov, E Feaible image and practical topping rule in iterative image recontruction IEEE Tranaction on edical Imaging, 989,8(2):86~93 [2] Gerenc r, L On a cla of mixing procee Stocatic, 989,26():65~9 [3] Gerenc r, L Rate of convergence of recurive etimator SIA Journal on Control and Optimization, 992,30(5):200~227 : δ ( J ε ) : (Ω,F,P ),, ( ( ε ( ) ( )) σ ( F r, F r r δ ( J ε ) = δ ( J ε ε r ) r ) L, [2] δ ( J /(2m) (, )) m>, ( E ) δ ( J (, )) ε 2m p C / 2 Ω, (,dp), δ ( J C δ ( J ) C (ii) δ ( J ) ε ( / 2 ) ε ( / 2 ) ) = δ ( J ) e ρ ( J ) = O (e ) = O ( ε / 2 e [6], : up δh C ( t) = O ( σ t qσ τ ( σ ), σ q σ t ( a / r) δh, ε ),,C ) (i) ) (ii) ( r) dr 0,, ( J C ) = O ( ) δ ( J ), σ u, (iii) δ J ( r, y( r, σ, u)) = ( r ( ) T r C 2 q ( a / r) Cr dr C ( ) I) G( y( r, σ, u)) (iii) (iii),r σ, u, (Ω,F,P ) L 2 [3] 3 ( ) ( /, 2m δ J = O ) δ ( J ), δ ( J 2 2 q q C, 2m 2m 2m 2m 2m δ ( J ) ( a / r) Ccr dr ac( c / r )dr C, 2m / 2 ) = O ( ) ћ=min(4m, 2), = ( ),2m δ O > 0 [6] : (iv)

7 2050 Journal of Software 2002,3(0) An Image Retoration and Recontruction Algoritm Baed on Stocatic Perturbation Gradient Approximation LIU Cuan-cai, FU Qing-xiang (Department of Computer Science and Tecnology, Fuzou Univerity, Fuzou , Cina) ttp://wwwfzueducn Abtract: In order to retore degenerative image, wic are go ort of priori nowledge about original image, and explore new way of x-ray tomograpic image recontruction, te experience of Spall and Crition imultaneou perturbation tocatic approximation (SPSA) metod i drawn on, and ti algoritm i extended to te ig order and multivariate cae, ten a new gradient approximation algoritm wit tocatic perturbation i preented Ti algoritm doe not need eiter a priori nowledge or a poteriori probability, and a convergence wit excellent tability Comparative experiment ow tat ti algoritm converge to viually good image wit excellent tability for retoration and recontruction of image Key word: image recontruction; x-ray tomograpy; L-mixing procee; image retoration; tocatic perturbation gradient Received November 8, 2000; accepted arc 9, 200 Supported by te National Grand Fundamental Reearc 973 Program of Cina under Grant NoG ; te Natural Science Foundation of Fujian Province of Cina under Grant NoF OA : OA () OA ;(2) ;(3) OA ;(4) OA OA ;(5) / ;(6) OA ;(7) / ;(8) OA ;(9) :(025) yangming@eueducn

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