1 / 30

Sparse Approximation by Wavelet Frames and Applications

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.

calla
Télécharger la présentation

Sparse Approximation by Wavelet Frames and Applications

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. 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

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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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]

  11. Fast Algorithm: 0-Norm • Penalty decomposition (PD) method [Lu and Zhang, 2010] • Algorithm: Change of variables Quadratic penalty

  12. Fast Algorithm: 0-Norm • Step 1: • Subproblem 1a): quadratic • Subproblem 1b): hard-thresholding • Convergence Analysis [Zhang, Dong and Lu, 2011] :

  13. 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

  14. Numerical Results • Comparisons Portrait Couple Balanced Analysis

  15. 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

  16. 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

  17. 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:

  18. Comparisons: Deblurring • Comparisons of best PSNR values v.s. various noise level

  19. Comparisons: Deblurring • Comparisons of computation timev.s. various noise level

  20. 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

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

  22. 3D Cone Beam CT Cone Beam CT

  23. 3D Cone Beam CT Discrete = • Animation created by Dr. XunJia

  24. 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];

  25. 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

  26. 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

  27. 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.

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

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

  30. 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

More Related