Sparse modeling and the resolution of inverse problems in biomedical imaging

Transcription

1 Sparse modeling and the resolution of inverse problems in biomedical imaging Michael Unser Biomedical Imaging Group EPFL, Lausanne, Switzerland Plenary talk, IEEE Int. Symp. Biomedical Imaging (ISBI 5), 6-9 April, 205, New York, USA IEEE International Symposium on Biomedical Imaging: Macro to Nano 7-0 July 2002 Washington, DC, USA ISBI INTERNATIONAL SYMPOSIUM ON BIOMEDICAL IMAGING Logo design: Annette Unser 2

2 Variational formulation of image reconstruction Linear forward model H noise g = Hs + n s Integral operator Ill-posed inverse problem: recover s from noisy measurements g Reconstruction as an optimization problem s =argmin g Hs 2 2 }{{} + λr(s) }{{} data consistency regularization n 3 Classical reconstruction = linear algorithm Quadratic regularization (Tikhonov) R(s) = Ls 2 s =argmin g Hs 2 2 }{{} + λr(s) }{{} data consistency regularization Formal linear solution: s =(H T H + λl T L) H T g = R λ g L = C /2 s : Whitening filter Statistical formulation under Gaussian hypothesis Wiener (LMMSE) solution = Gauss MMSE = Gauss MAP s MAP =argmin s σ 2 g Hs 2 2 }{{} Data Log likelihood + C /2 s s 2 2 }{{} Gaussian prior likelihood Signal covariance: C s = E{s s T } 4

3 Current trend: non-linear algorithms (l optimization) s =argmin y Hs 2 2 }{{} + λr(s) }{{} data consistency regularization Wavelet-domain regularization Wavelet expansion: s = Wv (typically, sparse) Wavelet-domain sparsity-constraint: R(s) = v l with v = W s (Nowak et al., Daubechies et al. 2004) l regularization (Total variation=tv) (Rudin-Osher, 992) R(s) = Ls l with L: gradient Compressed sensing/sampling (Candes-Romberg-Tao; Donoho, 2006) 5 Key research questions (for biomedical imaging) Formulation of ill-posed reconstruction problem Statistical modeling (beyond Gaussian) supporting non-linear reconstruction schemes (including CS) Sparse stochastic processes 2 Efficient implementation for large-scale imaging problem ADMM = smart chaining of simple modules 3 Future trends and open issues 6

4 OUTLINE Variational formulation of inverse problems Statistical modeling Introduction to sparse stochastic processes Generalized innovation model Statistical characterization of signal Algorithm design Reconstruction of biomedical images Discretization of inverse problem Generic MAP estimator (iterative reconstruction algorithm) Applications Deconvolution microscopy Computed tomography Cryo-electron tomography Differential phase-contrast tomography 7 An introduction to sparse stochastic processes EDEE Course 8

5 Random spline: archetype of sparse signal non-uniform spline of degree Ds(t) = n a n δ(t t n )=w(t) Random weights {a n } i.i.d. and random knots {t n } (Poisson with rate λ) Anti-derivative operators Shift-invariant solution: D ϕ(t) =( + ϕ)(t) = Scale-invariant solution: D 0 ϕ(t) = t 0 ϕ(τ)dτ t ϕ(τ)dτ 9 Compound Poisson process Stochastic differential equation Ds(t) =w(t) with boundary condition s(0) = 0 Innovation: w(t) = n a n δ(t t n ) Formal solution s(t) =D w(t) = n = n a n D {δ( t n )}(t) a n + (t t n ) 0

6 Lévy processes: all admissible brands of innovations Generalized innovations : white Lévy noise with E{w(t)w(t )} = σ 2 wδ(t t ) Ds = w (perfect decoupling!) (Wiener 923) (Paul Lévy circa 930) Generalized innovation model Theoretical framework: Gelfand s theory of generalized stochastic processes Generic test function ϕ Splays the role of index variable innovation process X = ϕ, w L White noise 3 w Solution of SDE (general operator) 2 Whitening operator L s =L w sparse stochastic process Y = ϕ, s = ϕ, L w = L ϕ, w 4 Approximate decoupling Proper definition of continuous-domain white noise (Unser et al, IEEE-IT 204) Regularization operator vs. wavelet analysis Main feature: inherent sparsity (few significant coefficients) 2

7 Infinite divisibility and Lévy exponents Definition: A random variable X with generic pdf p id (x) is infinitely divisible (id) iff., for any N Z +, there exist i.i.d. random variables X,...,X N such that X = d X + +X N. Rectangular test function i.i.d. X = w, rect =, =, + +, Proposition The random variable X = w, rect where w is a generalized innovation process is infinitely divisible. It is uniquely characterized by its Lévy exponent f(ω) =logˆp id (ω). p id (ω) =e f(ω) = R Bottom line: There is a one-to-one correspondence between Lévy exponents and infinitely divisible distributions and, by extension, innovation processes. n p id (x)e jωx dx n 3 Probability laws of innovations are infinite divisible X = w, ϕ =, lim n, = lim n, + +, Statistical description of white Lévy noise w (innovation) Characterized by canonical (p-admissible) Lévy exponent f(ω) Generic observation: X = ϕ, w with ϕ L p (R d ) X is infinitely divisible with (modified) Lévy exponent f ϕ (ω) =log p X (ω) = f ( ωϕ(x) ) dx R d 4

8 Sparser Probability laws of sparse processes are id Analysis: go back to innovation process: w =Ls Generic random observation: X = ϕ,w with ϕ S(R d ) or ϕ L p (R d ) (by extension) {}}{ Linear functional: Y = ψ,s = ψ, L w = L ψ,w If φ =L ψ L p (R d ) then Y = ψ, s = φ,w is infinitely divisible with Lévy exponent f φ (ω) = R d f ( ωφ(x) ) dx p Y (y) =F {e fφ(ω) }(y) = R e f φ(ω) jωy dω 2π = explicit form of pdf Unser and Tafti An Introduction to Sparse Stochastic Processes 5 Examples of infinitely divisible laws p id (x) (a) Gaussian p Gauss (x) = x2 e 2σ 2 2πσ (b) Laplace p Laplace (x) = λ 2 e λ x (c) Compound Poisson p Poisson (x) =F {e λ(ˆp A(ω) ) } (d) Cauchy (stable) p Cauchy (x) = π (x 2 +) Characteristic function: p id (ω) = R p id (x)e jωx dx =e f(ω) 6

9 Sparser Examples of id noise distributions p id (x) Observations: X n = w, rect( n) (a) Gaussian f(ω) = σ2 0 2 ω (b) Laplace ( f(ω) =log ) +ω (c) Compound Poisson f(ω) =λ R (ejxω )p(x)dx (d) Cauchy (stable) f(ω) = s 0 ω Complete mathematical characterization: ( Pw (ϕ) =exp f ( ϕ(x) ) ) dx R d 7 Generation of self-similar processes: s =L w L F (jω) H+ 2 L : fractional integrator fbm; H = 0.50 Poisson; H = 0.50 fbm; H = 0.75 fbm; H =.25 fbm; H =.50 H=.5 H=.75 H=.25 H=.5 Poisson; H = 0.75 Poisson; H =.25 Poisson; H =.50 Gaussian Sparse (generalized Poisson) Fractional Brownian motion (Mandelbrot, 968) (U.-Tafti, IEEE-SP 200) 8

10 Aesthetic sparse signal: the Mondrian process L=D x D y F (jω x )(jω y ) λ =30 9 Scale- and rotation-invariant processes Stochastic partial differential equation : ( Δ) H+ 2 s(x) =w(x) Gaussian H=.5 H=.75 H=.25 H=.75 Sparse (generalized Poisson) (U.-Tafti, IEEE-SP 200) 20

11 Powers of ten: from astronomy to biology 2 RECONSTRUCTION OF BIOMEDICAL IMAGES Discretization of reconstruction problem Signal reconstruction algorithm (MAP) Examples of image reconstruction Deconvolution microscopy X-ray tomography Cryo-electron tomography Phase contrast tomography 22

12 Discretization of reconstruction problem Spline-like reconstruction model: s(r) = k Ω s[k]β k (r) s =(s[k]) k Ω Innovation model Ls = w s = L w Discretization u = Ls (matrix notation) p U is part of infinitely divisible family Physical model: image formation and acquisition y m = s (x)η m (x)dx + n[m] = s,η m + n[m], (m =,...,M) R d y = y 0 + n = Hs + n n: i.i.d. noise with pdf p N [H] m,k = η m,β k = η m (r)β k (r)dr: R d (M K) system matrix 23 Posterior probability distribution p S Y (s y) = p Y S(y s)p S (s) p Y (y) = p ( ) N y Hs ps (s) p Y (y) (Bayes rule) = Z p N (y Hs)p S (s) u = Ls p S (s) p U (Ls) k Ω p ( ) U [Ls]k Additive white Gaussian noise scenario (AWGN) ) y Hs 2 ( ) p S Y (s y) exp ( 2σ 2 p U [Ls]k k Ω... and then take the log and maximize... 24

13 Sparser General form of MAP estimator s MAP =argmin ( 2 y Hs ) σ2 n Φ U ([Ls] n ) Gaussian: p U (x) = 2πσ0 e x2 /(2σ 2 0 ) Φ U (x) = 2σ 2 0 x 2 + C Laplace: p U (x) = λ 2 e λ x Φ U (x) =λ x + C 2 Student: p U (x) = B ( ) r, 2 ( ) r+ 2 x 2 + Φ U (x) = ( r + 2) log( + x 2 )+C 3 Potential: Φ U (x) = log p U (x) Proximal operator: pointwise denoiser 3 5 σ 2 Φ U (u) linear attenuation soft-threshold l 2 minimization l minimization shrinkage function l p relaxation for p 0 26

14 Maximum a posteriori (MAP) estimation Constrained optimization formulation Auxiliary innovation variable: u = Ls L A (s, u, α) = 2 g Hs σ2 n Φ U ([u] n )+α T (Ls u)+ μ 2 Ls u Alternating direction method of multipliers (ADMM) L A (s, u, α) = 2 g Hs σ2 n Φ U ([u] n )+α T (Ls u)+ μ 2 Ls u 2 2 Sequential minimization s k+ arg min s R N L A (s, u k, α k ) α k+ = α k + μ ( Ls k+ u k) Linear inverse problem: Nonlinear denoising: s k+ = ( H T H + μl T L ) ( H T y + z k+) ( ) u k+ = prox ΦU Ls k+ + μ αk+ ; σ2 μ with z k+ = L T ( μu k α k) 3 2 Proximal operator taylored to stochastic model prox ΦU (y; λ) =argmin u 2 y u 2 + λφ U (u)

15 Deconvolution of fluorescence micrographs Physical model of a diffraction-limited microscope g(x, y, z) =(h 3D s)(x, y, z) 3-D point spread function (PSF) h 3D (x, y, z) =I 0 p λ ( x M, y M, z M 2 ) 2 p λ (x, y, z) = P (ω,ω 2 )exp R 2 ( j2πz ω2 + ω2 2 ) ( 2λf0 2 exp j2π xω + yω 2 λf 0 ) dω dω 2 Optical parameters λ: wavelength (emission) M: magnification factor f 0 : focal length P (ω,ω 2 )= ω <R0 : pupil function NA = n sin θ = R 0 /f 0 : numerical aperture 29 Deconvolution: numerical set-up Discretization ω 0 π and representation in (separable) sinc basis {sinc(x k)} k Z d Analysis functions: η m (x, y, z) =h 3D (x m,y m 2,z m 3 ) [H] m,k = η m, sinc( k) = h 3D ( m), sinc( k) = ( sinc h 3D ) (m k) =h3d (m k). H and L: convolution matrices diagonalized by discrete Fourier transform Linear step of ADMM algorithm implemented using the FFT s k+ = ( H T H + μl T L ) ( H T y + z k+) with z k+ = L T ( μu k α k) 30

16 2D deconvolution experiment Astrocytes cells bovine pulmonary artery cells human embryonic stem cells L : gradient Deconvolution results in db Optimized parameters Gaussian Estimator Laplace Estimator Student s Estimator Astrocytes cells Pulmonary cells Stem cells D deconvolution with sparsity constraints Maximum intensity projections of image stacks; Leica DM 5500 widefield epifluorescence microscope with a 63 oil-immersion objective; C. Elegans embryo labeled with Hoechst, Alexa488, Alexa568; (Vonesch-U. IEEE Trans. Im. Proc. 2009) 32

17 Computed tomography (straight rays) Projection geometry: x = tθ + rθ with θ =(cosθ, sin θ) Radon transform (line integrals) R θ {s(x)}(t) = s(tθ + rθ )dr R = s(x)δ(t x, θ )dx R 2 R θ {s}(t) y r θ x t sinogram (b) ( ) θ Equivalent analysis functions: η m (x) =δ ( t m x, θ m ) 33 Computed tomography reconstruction results (a) (b) Figure 0.6 Images used in X-ray tomographic reconstruction experiments. (a) The Shepp-Logan (SL) phantom. (b) Cross section of a human lung. Table 0.4 Reconstruction results of X-ray computed tomography using different estimators. Directions Estimation performance (SNR in db) Gaussian Laplace Student s SL Phantom SL Phantom Lung Lung L: discrete gradient 34

18 EM: Single particle analysis Cryo-electron micrograph Bovine papillomavirus C.-O. Sorzano 35 Image reconstruction (real data) Standard Fourier-based reconstruction High-resolution Fourier-based reconstruction High-resolution reconstruction with sparsity 36

19 Differential phase-contrast tomography x 2 θ Paul Scherrer Institute (PSI), Villigen x g CCD x g (y, θ) θ x interference pattern Intensity (Pfeiffer, Nature 2006) Mathematical model g(t, θ) = t R θ{f}(t) g = Hs [H] (i,j),k = t P θ j β k (t j ) 37 Experimental results Rat brain reconstruction with 72 projections ADMM-PCG (TV) FBP (Pfeiffer et al. Nucl. Inst. Meth. Phys. Res. 2007) 2 3 L: discrete gradient, Φ(u) =λ u Collaboration: Prof. Marco Stampanoni, TOMCAT PSI / ETHZ 38

20 Reducing the numbers of views Rat brain reconstruction with 8 projections ADMM-PCG g-fbp SSIM =.96 SSIM =.5 SSIM =.95 SSIM =.60 SSIM =.49 SSIM =.5 SSIM =.89 SSIM =.43 Collaboration: Prof. Marco Stampanoni, TOMCAT PSI / ETHZ (Nichian et al. Optics Express 203) 39 Performance evaluation Goldstandard: high-quality iterative reconstruction with 72 views (a) (b) Reduction of acquisition time by a factor 0 (or more)? 40

21 CAN WE GO BEYOND MAP ESTIMATION? A detailed investigation of simpler denoising problem Test case: Lévy processes Gaussian : MAP = Wiener = MMSE Poisson: MAP = zero. Can we compute the best = MMSE estimator? Yes, by using belief propagation (Kamilov et al., IEEE-SP 203) 2. Can we compute it with an iterative MAP-type algorithm? Yes (with the help of Haar wavelets) by optimizing the thresholding function 4 Pointwise MMSE estimators for AWGN Minimum-mean-square-error (MMSE) estimator from y = x + n x MMSE (y) =E{X Y = y} = x p X Y (x y)dx R AWGN probability model = p Y X (y x) =g σ (y x) and p Y = g σ p X Stein s formula for AWGN g σ : Gaussian pdf (zero mean with variance σ 2 ) x MMSE (y) =y σ 2 Φ Y (y) where Φ Y (y) = d dy log p Y (y) = p Y (y) p Y (y) 3 2 MMSE LMMSE 0 MAP = soft threshold Laplacian 42

22 Iterative wavelet-based denoising: MAP MMSE Consistent Cycle Spinning (CCS) (Kamilov, IEEE-SPL 202) Algorithm3: CCSdenoisingsolves { Problem (.4) where A is a tight-frame matrix y τ M Φ(As)} where A is a tight frame CCS denoising: Solves min s 2 s input: y,s 0 R N,τ,μ R + set: k = 0,λ 0 = 0,u = Ay; repeat z k+ ( = prox ( Φ +μ u + μas k + λ k) ; s k+ = A ( z k+ μ λk) λ k+ = λ k μ ( z k+ As k+) k = k + until stopping criterion return s = s k τ +μ ) or z k+ = v MMSE ( z k+ ) σ ; 2 +μ CCS constraint: z = As with z 2 = M s 2 (enforces energy convervation) Variation on a theme: substitute MAP shrinkage by MMSE shrinkage (Kazerouni, IEEE-SPL 203) Iterative MMSE denoising 43 Comparison of wavelet denoising strategies Key empirical finding: CCS MMSE denoising yields optimal solution!!!! Unser and Tafti An Introduction to Sparse Stochastic Processes Chap. 44

23 CONCLUSION Unifying continuous-domain stochastic model Backward compatibility with classical Gaussian theory Operator-based formulation: Lévy-driven SDEs or SPDEs Gaussian vs. sparse (generalized Poisson, student, SαS) Regularization Sparsification via operator-like behavior (whitening) Specific family of id potential functions (typ., non-convex) Conceptual framework for sparse signal recovery Principled approach for the development of novel algorithms Challenge: algorithms for solving large-scale problems in imaging: Cryo-electron tomography, diffraction tomography, dynamic MRI (3D + time), etc... Beyond MAP reconstruction: MMSE with learning (self-tuning) 45 Conclusion (Cont d) The continuous-domain theory of sparse stochastic processes is compatible with both 20th century SP = linear and Fourier-based algorithms, and 2st century SP = non-linear, sparsitypromoting, wavelet-based algorithms... but there are still many open questions... 46

24 References Theory of sparse stochastic processes M. Unser and P. Tafti, An Introduction to Sparse Stochastic Processes, Cambridge University Press, 204; preprint, available at M. Unser, P.D. Tafti, Stochastic models for sparse and piecewise-smooth signals, IEEE Trans. Signal Processing, vol. 59, no. 3, pp , March 20. M. Unser, P. Tafti, and Q. Sun, A unified formulation of Gaussian vs. sparse stochastic processes Part I: Continuous-domain theory, IEEE Trans. Information Theory, vol. 60, no. 3, pp , March 204. Algorithms and imaging applications E. Bostan, U.S. Kamilov, M. Nilchian, M. Unser, Sparse Stochastic Processes and Discretization of Linear Inverse Problems, IEEE Trans. Image Processing, vol. 22, no. 7, pp , 203. C. Vonesch, M. Unser, A Fast Multilevel Algorithm for Wavelet-Regularized Image Restoration, IEEE Trans. Image Processing, vol. 8, no. 3, pp , March M. Nilchian, C. Vonesch, S. Lefkimmiatis, P. Modregger, M. Stampanoni, M. Unser, Constrained Regularized Reconstruction of X-Ray-DPCI Tomograms with Weighted-Norm, Optics Express, vol. 2, no. 26, pp , 203. A. Entezari, M. Nilchian, M. Unser, A Box Spline Calculus for the Discretization of Computed Tomography Reconstruction Problems, IEEE Trans. Medical Imaging, vol. 3, no. 8, pp , Acknowledgments Many thanks to Dr. Pouya Tafti Prof. Qiyu Sun Prof. Arash Amini Dr. Cédric Vonesch Dr. Matthieu Guerquin-Kern Emrah Bostan Ulugbek Kamilov Masih Nilchian Members of EPFL s Biomedical Imaging Group Preprints and demos: 48

25 ebook: web preprint An Introduction to Sparse Stochastic Processes Michael Unser and Pouya D. Tafti Cover design: Annette Unser 49