Spline-based Sparse Tomographic Reconstruction with Besov Priors - PowerPoint PPT Presentation

Spline-based Sparse Tomographic Reconstruction with Besov Priors Elham Sakhaee and and Alireza Entezari University of Florida esakhaee@cise.ufl.edu

Tomographic Reconstruction § Recover the image given X-ray measurements X-ray detector Sinogram X-ray source 2

Motivation § X-ray Exposure Reduction Images courtesy of Pan et.al [1] Limited-Angle Few-View Half-Detector § ill-posed problem A b x 3

Sparse CT A b x tomographic sinogram system matrix data intensity § Least-squares solution: image || Ax − b || 2 x = min ˆ 2 x § Regularize the solution: || Ax − b || 2 ˆ x = min 2 + λ R ( x ) x § R(x) can be sparsity promoting regularizer 4

Related Work (Sparsity) § TV minimization: - Very promising for piece-wise constant images - ASD-POCS [Pan & Sidky 2009] § X-let sparsity: - Wavelet [Rantala 2006] - Curvelet [Hyder & Sukanesh, 2011] § Adaptive sparsity via dictionary learning - K-SVD [Liao & Sapiro 2008, Sakhaee & Entezari 2014] § Besov space priors: - Bayesian inversion [Siltanen et al. 2012] 5

A b Bayesian Inversion x § Find posterior distribution P(x|b), given: - Likelihood distribution - Prior distribution - Bayes formula: P ( x | b ) ∝ P ( b | x ) P ( x ) - Zero-mean Gaussian noise: P ( b | x ) = P ✏ ( Ax − b ) ∼ exp ( − 1 2 σ 2 || Ax − b || 2 2 ) - x is maximum a posteriori estimate: x MAP = argmax P ( x | b ) x 6

Discretization Invariance [Siltanen, 2012] Image courtesy of www.siltanen- research.net/MarseilleDiscInv.pdf 7

Discretization Invariance [Siltanen, 2012] Image courtesy of www.siltanen- research.net/MarseilleDiscInv.pdf 8

Discretization Invariance [Siltanen, 2012] Image courtesy of www.siltanen- research.net/MarseilleDiscInv.pdf 9

Discretization Invariance [Siltanen, 2012] § Posterior estimate must converge as n & k tend to infinity § Otherwise: - Adding number of measurements is not helpful - A higher resolution image does not converge to the true image § TV is not discretization invariant 10

Besov Space Priors § Bayesian inversion using Besov priors is discretization invariant [Lassas et. al. 2008] § Besov space B s p,q : the space of functions with certain level of smoothness § B 1 1,1 : space of functions with (up to) first (weak) derivatives in L 1 11

Common Pixel Representation Finite grid Continuous object vs. reconstruction Image courtesy of C.G. Koay, https://science.nichd.nih.gov 12

Expansion Sets § Alternatives for pixel-basis - Blob functions [Lewitt 1990] - Kaiser-Bessel functions - Higher-order box-splines • Tensor-product linear B-spline • Tensor-product cubic B-spline • Zwart-Powell function N X f ( x ) = c n ϕ ( x − x n ) n =1 13

Spline Formulation § Besov Prior: P ( c ) = C exp( − λ || c || B 1 1 , 1 ) § Norm in B 1 1,1 : equivalent to weighted Haar wavelet coefficients 2 i − 1 N − 1 X X 2 i/ 2 | w i,j | || c || B 1 1 , 1 = | w 0 | + i =0 j =0 14

Spline Formulation § Posterior distribution of spline coefficients: P ( c | b ) = C exp( − 1 2 || Hc − b || 2 2 − λ || c || B 1 1 , 1 ) Spline System Matrix Spline Coefficients § MAP estimate for higher order splines: 1 2 || Hc − p || 2 2 + λ || c || B 1 c MAP = arg min 1 , 1 c 15

Results: pixel-basis vs. Cubic box-spline § 45 projection views (12.5% of full range): Cubic FBP Pixel-basis (fourth-order box-spline) (first-order box-spline) SNR: 16.67 dB SNR: 11.92 dB SNR: 14.64 dB 16

Results: The impact of Besov Prior § 60 projection views (16.6% of full range): With Without Besov Prior Besov Prior SNR: 19.07 dB SNR: 17.85 dB 17

Results: TV minimization vs. Besov priors § 120 projection views (33.3% of full range): Cubic box-spline TV minimization Ground Truth w/ Besov prior SNR: 15.50 dB SNR: 19.94 dB 18

Results: Resilience to reduction of views 90 views 45 views 60 views SNR: 14.67 dB SNR: 14.67 dB SNR: 14.67 dB 19

Results: Accuracy Comparison § Accuracy of tensor-product Box, Linear and Cubic & non-separable Zwart-Powel 22 20 SNR (dB) 18 16 pixel − basis Linear 14 Zwart − Powell Cubic 12 30 45 60 90 number of projection angles 20

Summary § Recovering sparse spline coefficients with Besov space priors: § Advantages: - Higher approximation order - Edge-preserving prior - Discretization invariance 21

Future Work § Mixed spline representations § Analysis of approximation error as a function of grid resolution 22

References § Pan, X., Sidky, E.Y., Vannier, M.: Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction? Inverse Problems 25 (2009) § Rantala, M., Vanska, S., Jarvenpaa, S., Kalke, M., Lassas, M., Moberg, J., & Siltanen, S. (2006) . Wavelet-based reconstruction for limited-angle X-ray tomography . Medical Imaging, IEEE Transactions on, 25(2), 210-217. § Hyder, S. Ali, and R. Sukanesh. "An efficient algorithm for denoising MR and CT images using digital curvelet transform." Software Tools and Algorithms for Biological Systems. Springer New York, 2011. 471-480. § Liao, H., Sapiro, G.: Sparse representations for limited data tomography . In Biomedical Imaging: From Nano to Macro, 2008. ISBI 2008. 5th IEEE International Symposium on. (2008) 1375–1378 § Muller, J.L. and Siltanen, S., Linear and Nonlinear Inverse Problems with Practical Applications . Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2012. § Kolehmainen, V., Lassas, M., Niinimaki, K., Siltanen, S.: Sparsity-promoting bayesian inversion . Inverse Problems 28 (2012) § Saksman M.L., Siltanen S., Discretization-invariant bayesian inversion and Besov space priors , arXiv preprint (2009) 23

Thank you … Questions? 24