Gradient-based Sparse Approximation for Computed Tomography Elham - PowerPoint PPT Presentation

Gradient-based Sparse Approximation for Computed Tomography Elham Sakhaee , Manuel Arreola 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 p f tomographic sinogram system matrix data intensity § Least-squares solution: image k Au � p k 2 ˆ f = arg min 2 u ∈ R N § Regularize the solution: ˆ k Au � p k 2 f = arg min 2 + λ R ( u ) u ∈ R N § R(u) can be sparsity promoting regularizer 4

Related Work (Sparsity) § 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] § TV minimization: - Very promising for biomedical images - ASD-POCS [Pan & Sidky 2009] 5

Gradient Domain Sparsity § TV-based reconstruction: ˆ k Au � p k 2 f = arg min 2 + λ ( k D x u k 1 + k D y u k 1 ) u ∈ R N Seek a solution with sparse gradient magnitude 6

Gradient Components are Sparser § Gradient Magnitude (TV image): § Horizontal and vertical partial derivatives: Vertical Derivative Horizontal Derivative 7

Method: Recovering Partial Derivatives § Horizontal derivative: k Au x � p x k 2 ˆ 2 + λ k u x k 1 f x = arg min u x ∈ R N § Vertical derivative: k Au y � p y k 2 ˆ 2 + λ k u y k 1 f y = arg min u y ∈ R N [ˆ f x , ˆ § May result in a non-integrable vector field f y ] T 8

Method: Curl-free Constraint § For a vector field to be gradient field, it must be curl-free (zero curl): curl( r f ) = D x f y � D y f x = 0 § Adds a prior knowledge to the ill-posed problem 9

Method: incorporating the curl constraint § Recover the gradient components simultaneously § Consider integrability constraint at recovery stage ˆ f x , ˆ k Au x � p x k 2 2 + k Au y � p y k 2 f y = arg min 2 + u x , u y ∈ R N λ ( k u x k 1 + k u y k 1 ) + µ k D x u y � D y u x k 2 2 10

Method: LASSO Formulation § Define:     A 0 p x p 0 = G = 0 A p y  ,    µ D y − µ D x 0 § Reformulate as minimization: ` 1 f y ] T = arg min [ˆ f x , ˆ k Gv � p 0 k 2 2 + λ k v k 1 v 2 R 2 N 11

Method: Final Image Reconstruction [ˆ f x , ˆ § Given the gradient vector field f y ] T § Recover the final image by Poisson Equation [Perez et al., 2003] : r 2 ˆ f = D x ˆ f x + D y ˆ f y 12

Method: derivation of X-ray measurements § Q: Given , find: p = P θ ⊥ ( f ) and p x = P θ ⊥ ( f x ) p y = P θ ⊥ ( f y ) p y = P θ ⊥ ( f y ) ) p = P θ ⊥ ( f ) f ( x ⊥ P θ = p x § A: Projection-slice theorem: s = S θ ( F{ f } ) = F{P θ ⊥ ( f ) } 13 13

Method: derivation of X-ray measurements § Fourier transform properties: ω y = sin( θ ) ω F{ f x } = j ω x F{ f } ω F{ f y } = j ω y F{ f } θ ω x = cos( θ ) ω § From projection-slice theorem: S θ ( F{ f x } )( ω ) = S θ ( j ω x F{ f } )( ω ) = cos( θ ) j ω s ( ω ) S θ ( F{ f y } )( ω ) = S θ ( j ω y F{ f } )( ω ) = sin( θ ) j ω s ( ω ) 14

Method: derivation of X-ray measurements § Intuitively: p x = P θ ⊥ ( f x ) = cos( θ ) D P θ ⊥ ( f ) p y = P θ ⊥ ( f y ) = sin( θ ) D P θ ⊥ ( f )

Results: 15 projection views ( 4% of full range ) FBP, SNR: 13.03 dB Ground Truth TV minimization Proposed SNR: 26.45 dB SNR: 30.15 dB 16

Results: 15 projection views ( 4% of full range ) FBP, SNR: 2.61 dB Ground Truth TV minimization Proposed SNR: 23.08 dB SNR: 23.59 dB 17

Results: 15 projection views ( 4% of full range ) TV minimization Proposed 18

Results: 10 projection views ( 2.7% of full range ) TV minimization Separate Recovery Proposed SNR: 23.50 dB SNR: 23.42 dB SNR: 27.02 dB 19

Results: Accuracy Comparison § Accuracy vs. number of projection angles for Catphan dataset: 35 SNR (db) 30 SGF(proposed) 25 Separate Recovery TV minimization 20 8 12 15 20 24 30 36 45 projection angles 20

Results: Noisy Data (15 projection angles) TV minimization Separate Recovery Proposed SNR: 14.15 dB SNR: 16.78 dB SNR: 23.58 dB 21

Summary § We propose: - Leveraging higher sparsity of individual gradient components - Enforcing curl-free constraint at recovery stage - Leveraging interdependency of partial derivatives § Provided a recipe for deriving of X-ray measurements corresponding to derivative images 22

Future Work § Application to 3D CT reconstruction § Robustness against other types of noise § Analytical derivation of X-ray measurements corresponding to derivative images using box-splines 23

References § 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 § Sakhaee, E., Entezari, A. : Learning splines for sparse tomographic reconstruction . Advances in visual computing, (Proc. of ISVC) Springer Lecture Notes, pp1-14, 2014. § Muller, J.L. and Siltanen, S., Linear and Nonlinear Inverse Problems with Practical Applications . Society for Industrial and Applied Mathematics, USA, 2012. § 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) § Patel, V.M., Maleh, R., Gilbert, A. C., and Chellappa R., Gradient-based image recovery methods from incom- plete fourier measurements. Image Processing, IEEE Transactions on, vol. 21, no. 1, pp. 94–105, 2012 . § Perez, P., Gangnet, M., and Blake, A., Poisson Image Editing , ACM transactions on Graphics, Vol 22, no. 3, pp313-318, 2003. 24

Acknowledgements § This research was supported in part by the ONR grant N00014-14-1- 0762 and the NSF grant CCF/ CIF-1018149. § We thank the imaging physicists at Shands hospital for providing the catphan phantom scan. 25

Thank you … Questions? 26

Related Work (MRI) § Derive partial Fourier measurements corresponding to [Patel et al. 2012] : - Horizontal partial derivative: F f x = (1 − e − 2 π i ω x /N ) F f - Vertical partial derivative: F f y = (1 − e − 2 π i ω y /N ) F f § Recover each component separately § Fit an integrable field to the recovered non-integrable field. § Reconstruct the image 27

Results: Noisy Data (27 projection angles) TV minimization Separate Recovery Proposed SNR: 13.66 dB SNR: 14.17 dB SNR: 15.85 dB 28

Objective § Leverage higher sparsity of partial derivatives, to reduce required measurements. § Given sinogram data, recover the gradient components simultaneously. § Enforce integrability constraint at recovery stage, as opposed to post-processing. 29