Tikhonov Regularization in Image Reconstruction with Kaczmarz Extended Algorithm

Transcription

1 Tikhonov Regularization in Image Reconstruction with Kaczmarz Extended Algorithm Paper supported by the PNCDI INFOSOC Grant 131/2004 Andrei Băutu 1 Elena Băutu 2 Constantin Popa 2 1 Mircea cel Bătrân Naval Academy Constantza, Romania 2 Ovidius University Constantza, Romania ASIM 2005 Conference, September 12-15, 2005 Erlangen, Germany

2 The LS formulation Ax = b, A : m n, b R(A) IR m ART (Kaczmarz): x 0 IR n, lim k x k = x(x 0 ) S(A; b) real world applications: b b = b + δb / R(A) Ax b = min!; LSS(A; b); x LS ART: still exists lim k x k = x(x 0 ), but (Popa/Zdunek, 2004) x 0 IR n, d ( x(x 0 ), LSS(A; b) ) = c > 0.

3 The KE algorithm KE: x 0 IR n, y 0 = b, for k = 0, 1, 2,... y k+1 = Φ(α; y k ), b k+1 = b y k+1, x k+1 = F (ω; b k+1 ; x k ) x 0 IR n, α, ω (0, 2) lim k x k = x(x 0 ) LSS(A; b) (Popa, 1998) R1: If b = b + δb, δb = δb A + δb A R(A) N(At ), then δb A is completely eliminated during the KE iterations. The remaining problem: δb A.

4 Perturbations Ax = b x LS = A + b Ax b = min!, thus x LS = A + b = A + ( b + δb A + δb A ) = A+ ( b + δb A ) x LS x LS = A + δb A The small singular values of A (ill-conditioning) determine a (possible) big value for A + δb A, although δb A is small.

5 Tikhonov regularization Ax b 2 + γ 2 Rx, x = min! ( ) R : n n, positive semidefinite and symmetric For appropriate construction of R and values of γ we get Problem: 1 How to construct R? 2 How to solve ( ) with KE? x LS (γ) x LS = O ( δb A )

6 Construction of the regularization matrix R w h, if j H i D i V i D i w (R) H i P i H ij = v, if j V i i w d, if j D i D i V i D i 0, otherwise (R) ii = j i (R) ij + ɛ H i the set of horizontally neighbour pixels of pixel P i V i the set of vertically neighbour pixels of pixel P i D i the set of diagonally neighbour pixels of pixel P i Note: 1 ɛ > 0 R: SPD matrix (RKE-1 algorithm) 2 ɛ = 0 R: symmetric and positive semidefinite (RKE-2 algorithm)

7 First regularized KE version (RKE-1) R = SPD R = LL t (Cholesky) [ A ( ) Âx ˆb = min!, Â = γl t ] [ b, ˆb = 0 ] ( 1) RKE-1 = Just apply KE to ( 1) R2: x 0 IR n, α, ω (0, 2), γ IR, lim k x k = x(x 0 ; γ); for x 0 = 0, x(0, γ) = x LS (γ).

8 Second reguralized KE version (RKE-2) R = symmetric and positive semidefinite ( ) Ax b 2 W + γ2 Rx, x = min! ( ) 1 W = diag a 1 2,..., 1 a m 2 RKE-2: x 0 IR n, y 0 = b; for k = 0, 1, 2,... y k+1 = Φ(α; y k ), b k+1 = b y k+1, x k+1 = F (ω; b k+1 ; x k ) γ 2 Rx k Note: Unfortunately, no convergence proof (yet).

9 Numerical experiments parameters of the reconstruction procedure Parameter KE RKE-1 RKE-2 α ω γ ɛ initialization 1 zero initialization: x 0 i = 0 2 Herman initialization: x 0 i = m i=1 b i m n i=1 j=1 (A) ij

10 Numerical experiments (2) reconstruction error e xex (x) = x x ex Algorithm Initialization Reconstruction error Test 1 Test 2 KE Zero Herman RKE-1 Zero Herman RKE-2 Zero Herman

11 Test 1 reconstruction

12 Test 2 reconstruction

13 Test 1 reconstructions errors x 0 i = 0 x 0 i = x

14 Test 2 reconstructions errors x 0 i = 0 x 0 i = x

15 Tikhonov Regularization in Image Reconstruction with Kaczmarz Extended Algorithm Thank you!