Sparse Approximation by Wavelet Frames and Applications Bin Dong Department of Mathematics The University of Arizona 2012 International Workshop on Signal Processing , Optimization, and Control June 30- July 3, 2012 USTC, Hefei, Anhui, China

Outlines Wavelet Frame Based Models for Linear Inverse Problems (Image Restoration) Applications in CT Reconstruction 1-norm based models Connections to variational model 0-norm based model Comparisons: 1-norm v.s. 0-norm Quick Intro of Conventional CT Reconstruction CT Reconstruction with Radon Domain Inpainting

Tight Frames in Orthonormal basis Riesz basis Tight frame: Mercedes-Benz frame Expansions: Unique Not unique

Tight Frames General tight frame systems Tight wavelet frames Construction of tight frame: unitary extension principles [Ron and Shen, 1997] They are redundant systems satisfying Parseval’s identity Or equivalently where and

Tight Frames Example: Fast transforms Lecture notes: [Dong and Shen, MRA-Based Wavelet Frames and Applications, IAS Lecture Notes Series,2011] Decomposition Reconstruction Perfect Reconstruction Redundancy Lecture notes: self-contained theoretical review of MRA-based wavelet frames, as well as recent applications in image restorations and analysis.

Image Restoration Model Image Restoration Problems Challenges: large-scale & ill-posed Denoising, when is identity operator Deblurring, when is some blurring operator Inpainting, when is some restriction operator CT/MR Imaging, when is partial Radon/Fourier transform

Frame Based Models Image restoration model: Balanced model for image restoration [Chan, Chan, Shen and Shen, 2003], [Cai, Chan and Shen, 2008] When , we have synthesis based model [Daubechies, Defrise and De Mol, 2004; Daubechies, Teschke and Vese, 2007] When , we have analysis based model [Stark, Elad and Donoho, 2005; Cai, Osher and Shen, 2009] Synthesis: sharper edges but more artifact. Analysis: less artifact but edges are a bit blurry. Balance in between. Which one to choose depends on applications Key: It works because images are piecewise smooth functions and have sparse representation under tight frame transforms Resembles Variational Models

Connections: Wavelet Transform and Differential Operators Nonlinear diffusion and iterative wavelet and wavelet frame shrinkage 2nd-order diffusion and Haar wavelet: [Mrazek, Weickert and Steidl, 2003&2005] High-order diffusion and tight wavelet frames in 1D: [Jiang, 2011] Difference operators in wavelet frame transform: True for general wavelet frames with various vanishing moments [Weickert et al., 2006; Shen and Xu, 2011] Filters Transform Approximation

Connections: Analysis Based Model and Variational Model [Cai, Dong, Osher and Shen, Journal of the AMS, 2012]: The connections give us Leads to new applications of wavelet frames: Converges For any differential operator when proper parameter is chosen. Geometric interpretations of the wavelet frame transform (WFT) WFT provides flexible and good discretization for differential operators Different discretizations affect reconstruction results Good regularization should contain differential operators with varied orders (e.g., total generalized variation [Bredies, Kunisch, and Pock, 2010]) Synthesis: sharper edges but more artifact. Analysis: less artifact but edges are a bit blurry. Balance in between. Which one to choose depends on applications Key: It works because images are piecewise smooth functions and have sparse representation under tight frame transforms Image segmentation: [Dong, Chien and Shen, 2010] Surface reconstruction from point clouds: [Dong and Shen, 2011] Standard Discretization Piecewise Linear WFT

Frame Based Models: 0-Norm Nonconvex analysis based model [Zhang, Dong and Lu, 2011] Motivations: Related work: Restricted isometry property (RIP) is not satisfied for many applications Penalizing “norm” of frame coefficients better balances sparsity and smoothnes Synthesis: sharper edges but more artifact. Analysis: less artifact but edges are a bit blurry. Balance in between. Which one to choose depends on applications Key: It works because images are piecewise smooth functions and have sparse representation under tight frame transforms “norm” with : [Blumensath and Davies, 2008&2009] quasi-norm with : [Chartrand, 2007&2008]

Fast Algorithm: 0-Norm Penalty decomposition (PD) method [Lu and Zhang, 2010] Algorithm: Change of variables Quadratic penalty There are other algorithms also, but split Bregman was proved to be efficient for frame based models.

Fast Algorithm: 0-Norm Step 1: Subproblem 1a): quadratic Subproblem 1b): hard-thresholding Convergence Analysis [Zhang, Dong and Lu, 2011] : There are other algorithms also, but split Bregman was proved to be efficient for frame based models.

Numerical Results Comparisons (Deblurring) Most important: relationship between minimizers. PFBS/FPC: [Combettes and Wajs, 2006] /[Hale, Yin and Zhang, 2010] Balanced Split Bregman: [Goldstein and Osher, 2008] & [Cai, Osher and Shen, 2009] Analysis 0-Norm PD Method: [Zhang, Dong and Lu, 2011]

Numerical Results Comparisons Portrait Couple Balanced Analysis Most important: relationship between minimizers. Couple Balanced Analysis

Faster Algorithm: 0-Norm Start with some fast optimization method for nonsmooth and convex optimizations: doubly augmented Lagrangian (DAL) method [Rockafellar, 1976]. Given the problem: The DAL method: where There are other algorithms also, but split Bregman was proved to be efficient for frame based models. We solve the joint optimization problem of the DAL method using an inexact alternative optimization scheme

Faster Algorithm: 0-Norm Start with some fast optimization method for nonsmooth and convex optimizations: doubly augmented Lagrangian (DAL) method [Rockafellar, 1976]. The inexact DAL method: Given the problem: The DAL method: where There are other algorithms also, but split Bregman was proved to be efficient for frame based models. Hard thresholding

Faster Algorithm: 0-Norm However, the inexact DAL method does not seem to converge!! Nonetheless, the sequence oscillates and is bounded. The mean doubly augmented Lagrangian method (MDAL) [Dong and Zhang, 2011] solve the convergence issue by using arithmetic means of the solution sequence as outputs instead: MDAL: There are other algorithms also, but split Bregman was proved to be efficient for frame based models.

Comparisons: Deblurring Comparisons of best PSNR values v.s. various noise level There are other algorithms also, but split Bregman was proved to be efficient for frame based models.

Comparisons: Deblurring Comparisons of computation time v.s. various noise level There are other algorithms also, but split Bregman was proved to be efficient for frame based models.

Comparisons: Deblurring What makes “lena” so special? Decay of the magnitudes of the wavelet frame coefficients is very fast, which is what 0-norm prefers. Similar observation was made earlier by [Wang and Yin, 2010]. 1-norm 0-norm: PD 0-norm: MDAL There are other algorithms also, but split Bregman was proved to be efficient for frame based models.

Applications in CT Reconstruction With the Center for Advanced Radiotherapy and Technology (CART), UCSD Applications in CT Reconstruction

3D Cone Beam CT Cone Beam CT

3D Cone Beam CT Discrete = Animation created by Dr. Xun Jia

Cone Beam CT Image Reconstruction Goal: solve Difficulties: Related work: Unknown Image Projected Image In order to reduce dose, the system is highly underdetermined. Hence the solution is not unique. Projected image is noisy. Total Variation (TV): [Sidkey, Kao and Pan 2006], [Sidkey and Pan, 2008], [Cho et al. 2009], [Jia et al. 2010]; EM-TV: [Yan et al. 2011]; [Chen et al. 2011]; Wavelet Frames: [Jia, Dong, Lou and Jiang, 2011]; Dynamical CT/4D CT: [Chen, Tang and Leng, 2008], [Jia et al. 2010], [Tian et al., 2011]; [Gao et al. 2011];

CT Image Reconstruction with Radon Domain Inpainting Idea: start with Benefits: Instead of solving We find both and such that: is close to but with better quality Prior knowledge of them should be used Safely increase imaging dose Utilizing prior knowledge we have for both CT images and the projected images

CT Image Reconstruction with Radon Domain Inpainting Model [Dong, Li and Shen, 2011] Algorithm: alternative optimization & split Bregman. where p=1, anisotropic p=2, isotropic

CT Image Reconstruction with Radon Domain Inpainting Algorithm [Dong, Li and Shen, 2011]: block coordinate descend method [Tseng, 2001] Convergence Analysis Problem: Algorithm: Note: If each subproblem is solved exactly, then the convergence analysis was given by [Tseng, 2001], even for nonconvex problems.

CT Image Reconstruction with Radon Domain Inpainting Results: N denoting number of projections N=15 N=20

CT Image Reconstruction with Radon Domain Inpainting Results: N denoting number of projections N=15 N=20 W/O Inpainting With Inpainting

Thank You Collaborators: Mathematics Stanley Osher, UCLA Zuowei Shen, NUS Jia Li, NUS Jianfeng Cai, University of Iowa Yifei Lou, UCLA/UCSD Yong Zhang, Simon Fraser University, Canada Zhaosong Lu, Simon Fraser University, Canada Medical School Steve B. Jiang, Radiation Oncology, UCSD Xun Jia, Radiation Oncology, UCSD Aichi Chien, Radiology, UCLA