Sparse tomography. Samuli Siltanen. Department of Mathematics and Statistics University of Helsinki, Finland

Transcription

1 Sparse tomography Samuli Siltanen Department of Mathematics and Statistics University of Helsinki, Finland Minisymposium: Fourier analytic methods in tomographic image reconstruction Applied Inverse Problems Conference Daejeon, Korea, July 1, 2013

2 Finland

3 This is a joint work with Keijo Hämäläinen, University of Helsinki Aki Kallonen, University of Helsinki Ville Kolehmainen, University of Eastern Finland Matti Lassas, University of Helsinki Esa Niemi, University of Helsinki Kati Niinimäki, University of Eastern Finland Eero Saksman, University of Helsinki

4 Outline Sparse sampling and tomography Total variation regularization Discretization-invariance Besov space regularization Parameter choice: the S-curve method

5 Outline Sparse sampling and tomography Total variation regularization Discretization-invariance Besov space regularization Parameter choice: the S-curve method

6 Let us study a simple two-dimensional example of tomographic imaging

7 Tomography is based on measuring densities of matter using X-ray attenuation data Detector X-ray source (=4+4+5)

8 A projection image is produced by parallel X-rays and several detector pixels (here three pixels) Detector (=4+4+5) 8 (=1+3+4) 3 (=1+0+2)

9 For tomographic imaging it is essential to record projection images from different directions

10 The length of X-rays traveling inside each pixel is important, thus here the square roots

11 The direct problem of tomography is to find the projection images from known tissue

12 The inverse problem of tomography is to reconstruct the interior from X-ray data ?????????

13 We write the reconstruction problem in matrix form and assume Gaussian noise m 3 m 2 m 1 f 1 f 4 f 7 f 2 f 5 f 8 f 3 f 6 f 9 f 2 f 3 m 1 f 4 m 2 f = f 5, m = m 3 f 6 m 4, f 7 m 5 f 8 m 6 f 9 Our measurement model is m = Af + ε with independently distributed Gaussian noise (white noise) with standard deviation σ > 0. m 4 m 5 m 6 f 1 m = Af

14 Outline Sparse sampling and tomography Total variation regularization Discretization-invariance Besov space regularization Parameter choice: the S-curve method

15 We reconstruct the unknown pixel values using total variation regularization and non-negativity We consider the minimizer of the anisotropic TV functional Af m α { L H f 1 + L V f 1 } where L H and L H are horizontal and vertical first-order difference matrices. [Rudin, Osher and Fatemi 1992] Primal-dual algorithms Chambolle, Chan, Chen, Esser, Golub, Mulet, Nesterov, Zhang Thresholding Candès, Chambolle, Chaux, Combettes, Daubechies, Defrise, DeMol, Donoho, Pesquet, Starck, Teschke, Vese, Wajs Bregman iteration Cai, Burger, Darbon, Dong, Goldfarb, Mao, Osher, Shen, Xu, Yin, Zhang Splitting approaches Chan, Esser, Fornasier, Goldstein, Langer, Osher, Schönlieb, Setzer, Wajs Nonlocal TV Bertozzi, Bresson, Burger, Chan, Lou, Osher, Zhang

16 We use quadratic programming (QP) to compute the non-negative minimizer of the TV functional The minimizer of the functional { } arg min Af m α L H f 1 + α L V f 1 f R n + can be transformed into the standard form { 1 2 zt Qz + c T z arg min z R 5n }, z 0, Ez = b, where Q is symmetric and E implements equality constraints. Large-scale primal-dual interior point QP method was developed in Kolehmainen, Lassas, Niinimäki & S (2012) and Hämäläinen, Kallonen, Kolehmainen, Lassas, Niinimäki & S (2013).

17 Reduction to arg min z R 5n { 1 2 zt Qz + c T z } Denote horizontal and vertical differences by where u ± H, u ± V arg min f R n + L H f = u + H u H and L V f = u + V u V, 0. TV minimization is now { f T A T Af 2f T A T m + α1 T (u + H + u H + u + V + u V ) }, where 1 R n is vector of all ones. Further, we denote 1 f A T A u + σ 2 H z = u H u +, Q = V , c = u V A T m α1 α1 α1 α1.

18 We collected X-ray projection data of a walnut from 1200 directions The data was collected by Keijo Hämäläinen and Aki Kallonen at University of Helsinki.

19 This is the reconstruction using all 1200 projections and filtered back-projection

20 When only few projection angles are available, TV regularization performs better than FBP 23 angles 15 angles 10 angles FBP TV These images were computed by Kati Niinimäki.

21 Outline Sparse sampling and tomography Total variation regularization Discretization-invariance Besov space regularization Parameter choice: the S-curve method

22 X-ray tomography: Continuum model

23 X-ray tomography: Practical measurement model m R k k = 4

24 X-ray tomography: Computational model m = Af m R k f R n k = 4 n = 48

25 The numbers k and n are independent m = Af m R k f R n k = 8 n = 48

26 The numbers k and n are independent m = Af m R k f R n k = 8 n = 156

27 The numbers k and n are independent m = Af m R k f R n k = 8 n = 440

28 The numbers k and n are independent m = Af m R k f R n k = 16 n = 440

29 The numbers k and n are independent m = Af m R k f R n k = 24 n = 440

30 Bayesian inversion using total variation prior does not have a well-defined continuous limit Theorem (Lassas and S 2004) Total variation prior is not discretization-invariant. Sketch of proof: Apply a variant of the central limit theorem to the independent, identically distributed random consecutive differences. New numerical experiments are reported in Kolehmainen, Lassas, Niinimäki and S (2012) and Lucka (2012).

31 Bayesian inversion using Besov space priors has a well-defined continuous limit Theorem (Lassas, Saksman and S 2009) Besov space priors are discretization-invariant. Sketch of proof: Construction of well-defined Bayesian inversion theory in infinite-dimensional Besov spaces that allow wavelet bases. Discretizations are achieved by truncating the wavelet expansion. Numerical experiments are reported in Kolehmainen, Lassas, Niinimäki and S (2012). Deterministic Besov space regularization was first introduced in Daubechies, Defrise and De Mol (2004); this applies to Bayesian MAP estimates.

32 Outline Sparse sampling and tomography Total variation regularization Discretization-invariance Besov space regularization Parameter choice: the S-curve method

33 The wavelet transform divides an image into three types of details at different scales

34 This is how the wavelet decomposition is defined for two-dimensional periodic images We can represent periodic functions by wavelet expansion f (x, y) = 2 J 0 1 k 1 =0 2 J 0 1 k 2 =0 J 1 c J0 k φ J0, k (x, y)+ 3 2 j 1 2 j 1 j=j 0 l=1 k 1 =0 k 2 =0 w j kl ψ l j, (x, y), k where k = (k 1, k 2 ) and the coefficients c J0 k and w j kl are defined by c J0 k = f, φ J0 k = f (x, y)φ j k (x, y)dxdy, T 2 w j kl = f, ψ l j = f (x, y)ψ k T l 2 j (x, y)dxdy. k

35 Besov space norm can be defined in terms of the wavelet coefficients A function f belongs to B s pq(t 2 ) if and only if the following expression is finite: 2 J 0 1 k 1 =0 2 J 0 1 k 2 =0 c J0 k p + 1 p j=j 0 2 jq(s+1 2 p ) We focus on the case p = 1 = q and s = 1: 3 2 j 1 2 j 1 l=1 k 1 =0 k 2 =0 w j kl p q p 1 q f B 1 11 (T 2 ) = k c J0 k + j,l, k w j kl.

36 We promote sparsity by minimizing a sum of l 2 and l 1 norms In the case of Besov space penalty we minimize f MAP = arg min f R n + { 1 2σ 2 Af m 2 l 2 + α f B 1 11 (T 2 ) where ψ j are the wavelet basis functions. The above kind of minimization task has been studied in Chambolle, DeVore, Lee & Lucier 1998, Daubechies, Defrise & De Mol 2004, Candès, Romberg & Tao 2006, Klann, Maaß & Ramlau 2006, Grasmair, Haltmeier & Scherzer 2008, Kolehmainen, Lassas, Niinimäki & S }, We use constrained quadratic programming for the minimization.

37 History of multiresolution tomography 1994 Olson and DeStefano 1994 Sahiner and Yagle 1995 Delaney and Bresler 1996 Berenstein and Walnut 1996 Bhatia, Karl and Willsky 1997 Rashid-Farrokhi, Liu, Berenstein and Walnut 1997 Zhao, Welland and Wang 1998 Louis, Maaß and Rieder 1999 Candès and Donoho 1999 Madych 2000 Smith and Adhami 2000 Bonnet, Peyrin, Turjman and Prost 2002 Das and Sastry 2002 Frese, Bouman and Sauer 2004 Soleski and Walter 2004 Zhong, Ning and Conover 2006 Sastry and Das 2006 Soleski and Walter 2006 Lee and Lucier 2006 Rantala, Vänskä, Järvenpää, Kalke, Lassas, Moberg and S 2007 Niinimäki, S and Kolehmainen 2008 Soussen and Idier 2009 Vänskä, Lassas and S 2011 Klann, Ramlau and Reichel 2011 Terzija and McCann 2011 Frikel

38 Besov regularized sparse-data reconstructions compared to filtered back-projection

39 Outline Sparse sampling and tomography Total variation regularization Discretization-invariance Besov space regularization Parameter choice: the S-curve method

40 We took photographs of walnuts cut in half These photos are used for estimating the expected number of nonzero wavelet coefficients in a two-dimensional tomographic reconstruction. Special thanks go to Esa Niemi for his careful job in sawing the walnuts.

41 The S-curve method determines a regularization parameter value giving the right sparsity level Number of nonzero wavelet coefficients Regularization parameter α Kolehmainen, Lassas, Niinimäki & S 2012

42 Something needs to be done to deal with the erratic behaviour of the raw S-curve

43 Reconstruction from 30 projections using interpolated S-curve α = 0.019

44 Interpolated S-curve method for TV These unpublished images were computed by Kati Niinimäki.

45 All Matlab codes freely available on a website! Part I: Linear Inverse Problems 1 Introduction 2 Naïve reconstructions and inverse crimes 3 Ill-Posedness in Inverse Problems 4 Truncated singular value decomposition 5 Tikhonov regularization 6 Total variation regularization 7 Besov space regularization using wavelets 8 Discretization-invariance 9 Practical X-ray tomography with limited data 10 Projects Part II: Nonlinear Inverse Problems 11 Nonlinear inversion 12 Electrical impedance tomography 13 Simulation of noisy EIT data 14 Complex geometrical optics solutions 15 A regularized D-bar method for direct EIT 16 Other direct solution methods for EIT 17 Projects

46 Inverse Days, Dec 11 13, 2013, Inari, Finland Organizers: Maarten de Hoop Matti Lassas Markku Lehtinen Lassi Roininen S. S. Gunther Uhlmann

47 Thank you for your attention! Preprints available at