Union of Learned Sparsifying Transforms Based Low-Dose 3D CT Image Reconstruction

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Union of Learned Sparsifying Transforms Based Low-Dose 3D CT Image Reconstruction Xuehang Zheng 1, Saiprasad Ravishankar 2, Yong Long 1, Jeff Fessler 2 1 University of Michigan - Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai, China 2 Department of Electrical Engineering and Computer Science, University of Michigan, MI, USA June 21, 2017

Introduction Outline 1 Introduction 2 Problem Formulations 3 Optimization Algorithm 4 Experimental Results 5 Conclusion and Future Work Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 2 / 25

Introduction Dictionary Learning-Based LDCT Reconstruction Challenges in Low-Dose CT (LDCT): significantly reduce patient radiation exposure maintain high image quality Apply the Prior Information Learned from Big Datasets of Normal-Dose CT Images into LDCT Reconstruction. Training Phase = Prior = Reconstruction Phase Dictionary Learning-Based Approaches 1 : have shown promising results for LDCT typically use an overcomplete dictionary NP-Hard sparse coding 1 [Xu et al., IEEE T-MI, 2012] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 3 / 25

Introduction Dictionary Learning-Based LDCT Reconstruction Challenges in Low-Dose CT (LDCT): significantly reduce patient radiation exposure maintain high image quality Apply the Prior Information Learned from Big Datasets of Normal-Dose CT Images into LDCT Reconstruction. Training Phase = Prior = Reconstruction Phase Dictionary Learning-Based Approaches 1 : have shown promising results for LDCT typically use an overcomplete dictionary NP-Hard sparse coding 1 [Xu et al., IEEE T-MI, 2012] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 3 / 25

Introduction Dictionary Learning-Based LDCT Reconstruction Challenges in Low-Dose CT (LDCT): significantly reduce patient radiation exposure maintain high image quality Apply the Prior Information Learned from Big Datasets of Normal-Dose CT Images into LDCT Reconstruction. Training Phase = Prior = Reconstruction Phase Dictionary Learning-Based Approaches 1 : have shown promising results for LDCT typically use an overcomplete dictionary NP-Hard sparse coding 1 [Xu et al., IEEE T-MI, 2012] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 3 / 25

Introduction Union of Learned TRAnsforms (ULTRA) Sparsifying Transform Learning 2 : a generalized analysis operator Learning A Union of Transforms 3 : one for each class of features (group of patches) Closed-form solutions for sparse coding (and clustering): Computational cost: O(l 2 N) vs O(l 3 N) for Dictionary 2 [Ravishankar & Bresler, IEEE T-SP, 2015] 3 [Wen et al., Int J Comput Vis, 2015] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 4 / 25

Introduction Union of Learned TRAnsforms (ULTRA) Sparsifying Transform Learning 2 : a generalized analysis operator Learning A Union of Transforms 3 : one for each class of features (group of patches) Closed-form solutions for sparse coding (and clustering): Computational cost: O(l 2 N) vs O(l 3 N) for Dictionary 2 [Ravishankar & Bresler, IEEE T-SP, 2015] 3 [Wen et al., Int J Comput Vis, 2015] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 4 / 25

Introduction Union of Learned TRAnsforms (ULTRA) Sparsifying Transform Learning 2 : a generalized analysis operator Learning A Union of Transforms 3 : one for each class of features (group of patches) Closed-form solutions for sparse coding (and clustering): Computational cost: O(l 2 N) vs O(l 3 N) for Dictionary 2 [Ravishankar & Bresler, IEEE T-SP, 2015] 3 [Wen et al., Int J Comput Vis, 2015] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 4 / 25

Problem Formulations Outline 1 Introduction 2 Problem Formulations 3 Optimization Algorithm 4 Experimental Results 5 Conclusion and Future Work Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 5 / 25

Problem Formulations Learning A Union of Transforms min K {Ω k,z i,c k } k=1 i C k Sparsification Error Sparsity Penalty { {}}{{}}{ } Ω k X i Z i 2 2 + η 2 Z i 0 + K λ k Q(Ω k ) k=1 (P0) {Ω k } K k=1 : union of square transforms. Z i : sparse code of the training signal X i. Q(Ω k ) Ω k 2 F log det Ω k : controls the properties of Ω k 4. C k : the set of indices of signals matched to the kth class. An efficient alternating algorithm is used for (P0). 4 [Ravishankar & Bresler, IEEE T-SP, 2015] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 6 / 25

Problem Formulations Image Reconstruction Penalized Weighted-Least Squares (PWLS): 1 min x 0 2 y Ax 2 W + βr(x) (P1) y: noisy sinogram (measurement) A: system matrix x: unknown image (volume) W: diagonal weighting matrix R(x): regularizer β: regularization parameter Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 7 / 25

Problem Formulations Image Reconstruction: PWLS-ULTRA R(x) min 1 min x 0 2 y Ax 2 W + βr(x) (P1) {z j,c k } k=1 K { j C k Ω k P j x z j 2 2 + γ 2 z j 0 } (1) Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 8 / 25

Optimization Algorithm Outline 1 Introduction 2 Problem Formulations 3 Optimization Algorithm 4 Experimental Results 5 Conclusion and Future Work Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 9 / 25

Optimization Algorithm Image Update Step 1 min x 0 2 y Ax 2 W + R 2 (x) {}}{ K β Ω k P j x z j 2 2 (2) k=1 j C k We solve it using the relaxed linearized augmented Lagrangian method with ordered-subsets (relaxed OS-LALM) 5 : s (k+1) = ρ(d A x (k) h (k) ) + (1 ρ)g (k) x (k+1) = [x (k) (ρd A + D R2 ) 1 (s (k+1) + R 2 (x (k) ))] C ζ (k+1) MA T mw m (A m x (k+1) y m ) g (k+1) = ρ ρ + 1 (αζ(k+1) + (1 α)g (k) ) + 1 ρ + 1 g(k) h (k+1) = α(d A x (k+1) ζ (k+1) ) + (1 α)h (k) (3) 5 [Nien & Fessler, IEEE T-MI, 2016] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 10 / 25

Optimization Algorithm Image Update Step 1 min x 0 2 y Ax 2 W + R 2 (x) {}}{ K β Ω k P j x z j 2 2 (2) k=1 j C k We solve it using the relaxed linearized augmented Lagrangian method with ordered-subsets (relaxed OS-LALM) 5 : s (k+1) = ρ(d A x (k) h (k) ) + (1 ρ)g (k) x (k+1) = [x (k) (ρd A + D R2 ) 1 (s (k+1) + R 2 (x (k) ))] C ζ (k+1) MA T mw m (A m x (k+1) y m ) g (k+1) = ρ ρ + 1 (αζ(k+1) + (1 α)g (k) ) + 1 ρ + 1 g(k) h (k+1) = α(d A x (k+1) ζ (k+1) ) + (1 α)h (k) (3) 5 [Nien & Fessler, IEEE T-MI, 2016] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 10 / 25

Optimization Algorithm Sparse Coding and Clustering Step min {z j },{C k } k=1 K { } Ω k P j x z j 2 2 + γ 2 z j 0 j C k (4) Hard-thresholding operator H γ ( ): sets entries with mag. < γ to 0. For each patch, the optimal cluster assignment: ˆk j = arg min Ω k P j x H γ (Ω k P j x) 2 2 + γ 2 H γ (Ω k P j x) 0. (5) 1 k K The optimal sparse code: ẑ j = H γ (Ωˆkj P j x). Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 11 / 25

Optimization Algorithm Sparse Coding and Clustering Step min {z j },{C k } k=1 K { } Ω k P j x z j 2 2 + γ 2 z j 0 j C k (4) Hard-thresholding operator H γ ( ): sets entries with mag. < γ to 0. For each patch, the optimal cluster assignment: ˆk j = arg min Ω k P j x H γ (Ω k P j x) 2 2 + γ 2 H γ (Ω k P j x) 0. (5) 1 k K The optimal sparse code: ẑ j = H γ (Ωˆkj P j x). Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 11 / 25

Optimization Algorithm Sparse Coding and Clustering Step min {z j },{C k } k=1 K { } Ω k P j x z j 2 2 + γ 2 z j 0 j C k (4) Hard-thresholding operator H γ ( ): sets entries with mag. < γ to 0. For each patch, the optimal cluster assignment: ˆk j = arg min Ω k P j x H γ (Ω k P j x) 2 2 + γ 2 H γ (Ω k P j x) 0. (5) 1 k K The optimal sparse code: ẑ j = H γ (Ωˆkj P j x). Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 11 / 25

Experimental Results Outline 1 Introduction 2 Problem Formulations 3 Optimization Algorithm 4 Experimental Results 5 Conclusion and Future Work Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 12 / 25

Experimental Results 3D Axial Cone-beam CT with XCAT phantom 6 Training: 512 512 54 XCAT image volume with patch size 8 8 8 and patch stride 2 2 2 ( 1.5 10 6 patches). 6 [Segars et al., MP, 2008] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 13 / 25

Experimental Results 3D Axial Cone-beam CT with XCAT phantom 7 Testing: Sinogram size 888 64 984; Volume size 420 420 96 (air cropped); x = y = 0.977 and z = 0.625 mm; Patch size 8 8 8 with stride 2 2 2 ( 2 10 6 patches). Reconstruction Methods: FDK with a Hanning window. PWLS-EP with Lange3 Edge-Preserving regularizer. PWLS-ST based on a learned single Square Transform (K = 1). PWLS-ULTRA based on a Union of Learned TRAnsforms. 7 [Segars et al., MP, 2008] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 14 / 25

Experimental Results RMSE & SSIM Comparison Table: RMSE (HU) & SSIM of reconstructions for two incident photon intensities. Intensity FDK EP ST (K = 1) ULTRA (K = 15) 1 10 4 67.8 33.7 31.9 31.5 0.536 0.917 0.976 0.979 5 10 3 89.0 39.9 37.4 37.2 0.463 0.894 0.967 0.969 ULTRA scheme further improves the reconstruction than ST. Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 15 / 25

Experimental Results Performance Across Slices 50 45 PWLS-EP PWLS-ULTRA 60 55 PWLS-EP PWLS-ULTRA RMSE (HU) 40 35 RMSE (HU) 50 45 40 30 35 25 10 20 30 40 50 60 Slice Index 30 10 20 30 40 50 60 Slice Index Figure: RMSE of axial slices for 1 10 4 (left) and 5 10 3 (right). ULTRA provides improvement for most of axial slices. Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 16 / 25

Experimental Results FDK PWLS-EP PWLS-ULTRA FDK PWLS-EP PWLS-ULTRA Figure: Photon intensity: 1 10 4 (top row) and 5 10 3 (bottom row). Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 17 / 25

Experimental Results PWLS-EP PWLS-ULTRA Figure: Photon intensity: 1 10 4. Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 18 / 25

Experimental Results An Example of Clustering Result Class 1 Class 2 Class 3 Class 4 Class 5 Figure: Pixel clustering results (top) for the central axial slice of PWLS-ULTRA (K = 5) for 1 10 4, and a slice of the corresponding 3D transforms (bottom). Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 19 / 25

Experimental Results 3D Reconstructions of a Helical Chest Scan Sinogram size 888 64 3611; Pitch 1.0 (about 3.7 rotations with rotation time 0.4s); Volume size 420 420 222, x = y = 1.167 and z = 0.625 mm; Patch size 8 8 8 with stride 3 3 3 ( 1.5 10 6 patches); Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 20 / 25

Experimental Results 3D Reconstructions of a Helical Chest Scan FDK PWLS-EP PWLS-ULTRA Use the transforms learned from XCAT phantom! (K = 5) Might not need closely matched dataset for training. 7 FDK Reconstruction is provided by GE Healthcare. Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 21 / 25

Conclusion and Future Work Outline 1 Introduction 2 Problem Formulations 3 Optimization Algorithm 4 Experimental Results 5 Conclusion and Future Work Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 22 / 25

Conclusion and Future Work Conclusion We proposed PWLS-ST and PWLS-ULTRA for 3D LDCT imaging, which combine PWLS reconstruction with regularization based on learned sparsifying transforms. Both PWLS-ST and PWLS-ULTRA significantly improve the reconstruction quality compared to PWLS-EP. The ULTRA scheme with a richer union of transforms model provides better reconstruction of various features such as bones, specific soft tissues, and edges, compared to a single transform model. Future Work Convergence guarantees and automating the parameter selection. New transform learning-based LDCT reconstruction methods, such as involving rotationally invariant transforms, or online transform learning 8, etc. 8 [Ravishankar et al., IEEE J-STSP, 2015] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 23 / 25

Conclusion and Future Work Conclusion We proposed PWLS-ST and PWLS-ULTRA for 3D LDCT imaging, which combine PWLS reconstruction with regularization based on learned sparsifying transforms. Both PWLS-ST and PWLS-ULTRA significantly improve the reconstruction quality compared to PWLS-EP. The ULTRA scheme with a richer union of transforms model provides better reconstruction of various features such as bones, specific soft tissues, and edges, compared to a single transform model. Future Work Convergence guarantees and automating the parameter selection. New transform learning-based LDCT reconstruction methods, such as involving rotationally invariant transforms, or online transform learning 8, etc. 8 [Ravishankar et al., IEEE J-STSP, 2015] Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 23 / 25

Thanks for your attention! Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 24 / 25

While the total runtime for the 200 iterations (using a machine with two 2.80 GHz 10-core Intel Xeon E5-2680 processors) was 110 minutes for PWLS-DL, it was only 56 minutes for PWLS-ST and 60 minutes for PWLS-ULTRA(K = 15). Xuehang Zheng (UM-SJTU JI) PWLS-ULTRA 25 / 25

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