Constraining strategies for Kaczmarz-like algorithms

Transcription

1 Constraining strategies for Kaczmarz-like algorithms Ovidius University, Constanta, Romania Faculty of Mathematics and Computer Science Talk on April 7, 2008 University of Erlangen-Nürnberg

2 The scanning procedure in geotomography A : m n; big, sparse normalization A ij [0, 2] A : rank deficient, i.e. N(A) 0 nonzero vectors from N(A) have negative components x ex = (x1 ex,..., x n ex ) T R n exact image [a, b] [0, 1] (color/greys scale) x ex i

3 The least squares formulation - 1 ( ) Ax b = min!; LSS(A; b) = { P N(A) (x) + x LS, x R n} x ex = P N(A) (x ex ) + x LS x LS = x ex P N(A) (x ex ) / [0, 1] n Shadows appear in the reconstructed image x LS!

4 The least squares formulation - 2 Figure: Original (x ex ) Figure: Kaczmarz (x LS )

5 The classical Kaczmarz (K) algorithm Algorithm (K): for k = 0, 1,... do x k+1 = (f 1 f m )(ω; b, x k ) not = K(ω; b; x k ) where f i (ω; b; x) = x ω x,a i b i A A i 2 i Th. (Elsner, Koltracht, Num. Math., 1991) For x 0 R n, ω (0, 2), lim k x k = P N(A) (x 0 )+x LS +G P N(A T )(b) }{{} not = δ Note If b R(A), P N(A T )(b) = 0, thus δ = 0 and lim x k LSS(A; b) = S(A; b) k Moreover, for x 0 = 0, lim k x k = x LS

6 The constraining function C : R n R n (generally nonlinear) Cx Cy x y (I) Cx Cy = x y Cx Cy = x y (II) y Im(C) y = Cy (III) Examples. (i) an orthogonal projection on a closed convex subset C R n (e.g. C = [a 1, b 1 ] [a n, b n ]) x i, x i [0, 1] (ii) (Cx) i = 0, x i < 0, Im(C) = [0, 1] n 1, x i > 1 Note. (I) = nonexpansive; (I) + (II) = strictly nonexpansive;

7 The Constrained Kaczmarz (CK) algorithm Th. (Koltracht + Lancaster, IMA J. Num. An., 1990) x 0, ω (0, 2), if (I), (II), (III) hold, then lim k x k = x and it satisfies (i) x Im(C) (ii) x δ LSS(A; b) (iii) consistent case (δ = 0) x LSS(A; b) = S(A; b) Algorithm (CK) x k+1 = C[K(ω; b; x k )], k 0(x 0 R n ) Notes. (i) The bigger is δ, the bigger will be d(x, LSS(A; b)). (ii) It is supposed that Im(C) {z R n, z δ LSS(A; b)}

8 The Kaczmarz Extended (KE) algorithm Algorithm (KE) x 0 R n ; y 0 = b; k = 0, 1,... do y k+1 = (ϕ 1 ϕ n )(y k ) b k+1 = b y k+1 x k+1 = (f 1 f m )(b k+1 ; x k ) ϕ i (y) = y y,aj A j A j 2 f i (β; x) = x x,a i β i A i 2 A i Th. (C. Popa, Int. J. Comp. Math., 1995) For x 0 R n, the KE converges to a solution of (*) (also in the inconsistent case).

9 The Constrained Kaczmarz Extended (CKE) algorithm Algorithm (CKE) Algorithm (KE) x 0 R n ; y 0 = b; for k = 0, 1,... do y k+1 = (ϕ 1 ϕ n )(y k ) b k+1 = b y k+1 x k+1 = C[K(b k+1 ; x k )] x 0 R n ; y 0 = b; k = 0, 1,... do y k+1 = (ϕ 1 ϕ n )(y k ) b k+1 = b y k+1 x k+1 = K(b k+1 ; x k ) Th. (C. Popa, submitted 2007) If Im(C) LSS(A; b), and (I) and (III) hold (without (II)!!!), then x 0 R n lim k x k = x Im(C) LSS(A; b)

10 Experiments x ex : pixels (similar with Lancaster et. al, LAA, 1990) Figure: x ex Figure: x LS alg. (K) Figure: x LS alg. (KE)

11 Experiments Perturbation Let v N(A T ) and pert(ɛ) = ɛ b 100 v v ; b def = A x ex b pert (ɛ) = b+pert(ɛ) Ax b pert (ɛ) = min! x pert LS (ɛ) = x LS Note The bigger is ɛ, the bigger will be δ = G P N(A T )(b pert ) ; thus for CK the reconstruction will be not so good as for CKE (for which δ doesn t appear anymore!)

12 Experiments (CK) reconstruction Figure: ɛ = 0 Figure: ɛ = 10 Figure: ɛ = 30 Figure: ɛ = 60

13 Experiments (CKE) reconstruction Figure: ɛ = 0 Figure: ɛ = 10 Figure: ɛ = 60

14 Experiments A = T 1 T 2. T 300, b = τ 1 τ 2. τ 300 Figure: Fan beam scanning provided by Prof. Dr. J. Hornegger and Dr. M. Prümmer Ax b = min!; A : ; resolution of the image: a highly inconsistent problem: Ax LS b = P N(A T ) (b) 629.2, by comparing it with the norm of the right hand side b = (thus, a very big δ!) no information about the exact solution a special choice for the constraining intervals and for the representation of the computed solution as an image (see next slide)

15 Experiments constraining intervals: several tests with KE and CKE, for different values of [a, b] z k i ; then we kept the values a = 0, b = 0.15, for which we got the best reconstruction representation of z k as an = 4096 pixels image: compute mzk = min{z k i, i = 1,..., 4096}, Mzk = max{z k i, i = 1,..., 4096} scale the components of z k to the interval [0, 255] by zi k zi k mzk = 255, i = 1,..., 4096 Mzk mzk allocate to each integer from 0 to 255 a grey level ((grey(i), i = 0,..., 255)) put in the i-th pixel on the representing picture, the grey level grey(j i ) for which z k i j i = min{ z k i j, j = 0,..., 255}

16 Experiments Figure: (CK) reconstruction Figure: (KE) reconstruction Figure: (CKE) reconstruction

17 Final comments (CKE) doesn t need assumption (II) (strict nonexpansivity) and gives better results than (CK) for inconsistent problems extend the results to (KERP) - (KE) with Relaxation Parameters, i.e. get a (CKERP) method the simultaneous ART (e.g. Cimmino - like and Jacobi - like algorithms) seem to be compatible with a constraining procedure (work in progress...!) the CGLS algorithm - is no more compatible with a constraining procedure! the idea of constraining in iterative methods can be useful outside the CT image reconstruction context (e.g. projecting on special convex sets)

18 Thanks for your attention!