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This workshop outlines wavelet frame based models for linear inverse problems, including CT reconstruction and image restoration. It covers 1-norm and 0-norm based models, connections to variational models, and comparisons between the two. The construction of tight frames, their examples, and applications in image restoration are discussed. Various models for image restoration are explored, such as balanced, synthesis-based, and analysis-based models. The workshop also delves into the connections between wavelet transforms and differential operators, highlighting applications like nonlinear diffusion and iterative wavelet shrinkage. Additionally, it discusses standard discretization methods, fast algorithms, and numerical results in the context of the 0-norm model.
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Sparse Approximation by Wavelet Frames and Applications Bin Dong Department of Mathematics The University of Arizona 2012 International Workshop on Signal Processing , Optimization, and ControlJune 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 Unique Not unique • Orthonormal basis • Riesz basis • Tight frame: Mercedes-Benz frame • Expansions:
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
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] 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]) • 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 • “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
Fast Algorithm: 0-Norm • Step 1: • Subproblem 1a): quadratic • Subproblem 1b): hard-thresholding • Convergence Analysis [Zhang, Dong and Lu, 2011] :
Numerical Results • Comparisons (Deblurring) PFBS/FPC: [Combettes and Wajs, 2006] /[Hale, Yin and Zhang, 2010] Balanced Split Bregman: [Goldstein and Osher, 2008] & [Cai, Osher and Shen, 2009] Analysis PD Method: [Zhang, Dong and Lu, 2011] 0-Norm
Numerical Results • Comparisons Portrait 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 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 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:
Comparisons: Deblurring • Comparisons of best PSNR values v.s. various noise level
Comparisons: Deblurring • Comparisons of computation timev.s. various noise level
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
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. XunJia
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 • ZuoweiShen, NUS • Jia Li, NUS • JianfengCai, 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 • XunJia, Radiation Oncology, UCSD • Aichi Chien, Radiology, UCLA