Sparse Approximation by Wavelet FdAlitiFrames and Applications
35
Sparse Approximation by Wavelet Sparse Approximation by Wavelet F dA li ti F dA li ti Frames andApplications Frames andApplications Bin Dong Department of Mathematics University of Arizona Compressive Sensing Workshop: Leveraging Sparsity at UCLA & Beyond. March 6-8, 2012
Transcript of Sparse Approximation by Wavelet FdAlitiFrames and Applications
Sparse�Approximation�by�Wavelet�Sparse�Approximation�by�Wavelet�F d A li tiF d A li tiFrames�and�ApplicationsFrames�and�Applications
Bin�DongDepartment�of�MathematicspUniversity�of�Arizona
Compressive�Sensing�Workshop:�Leveraging�Sparsity�at�UCLA�&�Beyond.March�6-8,�2012
OutlinesOutlinesOutlinesOutlinesI. Wavelet�Frame�Based�Models�for�Linear�a e e a e ased ode s o ea
Inverse�Problems�(Image�Restoration) 1-norm�based�models
Connections�to�variational�model
0-norm�based�model
II. Applications�in�CT�Reconstruction
Comparisons:�1-norm�v.s.�0-norm
II. Applications�in�CT�Reconstruction Quick�Intro�of�Conventional�CT�Reconstruction
CT�Reconstruction�with�Radon�Domain�Inpaintingp g
Tight Frames inTight Frames inTight�Frames�in�Tight�Frames�in� Orthonormal basis Orthonormal basis
Riesz basis Riesz basis
Tight�frame:�Mercedes-Benz�frame
E i Expansions:Unique
N t iNot�unique
Tight FramesTight FramesTight�FramesTight�Frames General tight frame systems General�tight�frame�systems�����
• They�are�redundant�systems�satisfying�Parseval’s�identity
• Or�equivalently
Tight�wavelet�frames
Construction of tight frame: unitary extension principles
where���������������������������������������and�
Construction�of�tight�frame:�unitary�extension�principles�[Ron and Shen, 1997]
Tight FramesTight FramesTight�FramesTight�Frames Example:
Fast�transforms
Decompositionp
Reconstruction
Perfect�Reconstruction
Redundancy
Lecture�notes:�[Dong and Shen, MRA-Based Wavelet Frames and Applications IAS Lecture Notes Series 2011]
Redundancy
Applications, IAS Lecture Notes Series,2011]
Image Restoration ModelImage Restoration ModelImage�Restoration�ModelImage�Restoration�Model Image Restoration Problems Image�Restoration�Problems
D i i h i id tit t• Denoising,�when������is�identity�operator
• Deblurring,�when������is�some�blurring�operator
Inpainting when is some restriction operator• Inpainting,�when������is�some�restriction�operator
• CT/MR�Imaging,�when������is�partial�Radon/Fourier
transform
Challenges:�large-scale�&�ill-posed
transform�
Frame Based ModelsFrame Based ModelsFrame�Based�ModelsFrame�Based�Models Image restoration model: Image�restoration�model:
Balanced�model�for�image�restoration�[Chan, Chan, Shenand Shen, 2003], [Cai, Chan and Shen, 2008]
When�������������,�we�have�synthesis�based�model�y[Daubechies, Defrise and De Mol, 2004; Daubechies, Teschke and Vese, 2007]
When , we have analysis based model [Stark, When��������������,�we�have�analysis�based�model�[Stark, Elad and Donoho, 2005; Cai, Osher and Shen, 2009]
Resembles�Variational ModelsVariational�Models
Connections:�Wavelet�Transform�and�Connections:�Wavelet�Transform�and�Differential�OperatorsDifferential�Operators Nonlinear diffusion and iterative wavelet and wavelet Nonlinear�diffusion�and�iterative�wavelet�and�wavelet�
frame�shrinkageo 2nd-order�diffusion�and�Haar wavelet:�[Mrazek,
W i k t d St idl 2003&2005]Weickert and Steidl, 2003&2005]o High-order�diffusion�and�tight�wavelet�frames�in�1D:�
[Jiang, 2011] Difference�operators�in�wavelet�frame�transform:
Filters
Transform
Approximation
True for general wavelet frames with various vanishing
Approximation
True�for�general�wavelet�frames�with�various�vanishing�moments�[Weickert et al., 2006; Shen and Xu, 2011]
Connections:�Analysis�Based�Connections:�Analysis�Based�Model�and�Variational�ModelModel�and�Variational�Model Connection [Cai, Dong, Osher and Shen, 2011]: Connection�[Cai, Dong, Osher and Shen, 2011]:
Gamma-Converges
For�any differential�operator�when�proper�parameter�is�chosen.
Merits of Frame Based Model Merits�of�Frame�Based�Model• Rich�in�geometric�information
• Provides�good�and�flexible�discretizaiton for�differential�operators
Leads�to�new�applications�of�wavelet�frames:
g p
Image�segmentation:�[Dong, Chien and Shen, 2010] Surface reconstruction from point clouds: [Dong and Shen 2011] Surface�reconstruction�from�point�clouds:�[Dong and Shen, 2011]
Connections:�Analysis�Based�Connections:�Analysis�Based�Model�and�Variational�ModelModel�and�Variational�Model Connection [Cai, Dong, Osher and Shen, 2011]: Connection�[Cai, Dong, Osher and Shen, 2011]:
Gamma-Converges
For�any differential�operator�when�proper�parameter�is�chosen.
Merits of Frame Based Model Merits�of�Frame�Based�Model• Rich�in�geometric�information
• Provides�good�and�flexible�discretizaiton for�differential�operators
Leads�to�new�applications�of�wavelet�frames:
g p
Image�segmentation:�[Dong, Chien and Shen, 2010] Surface reconstruction from point clouds: [Dong and Shen 2011] Surface�reconstruction�from�point�clouds:�[Dong and Shen, 2011]
Frame Based SegmentationFrame Based SegmentationFrame�Based�SegmentationFrame�Based�Segmentation General variational model [Chan and Vese General�variational�model�[Chan and Vese,
1999&2000], [Chan, Esedoglu and Nikolova, 2004]:
Image�Seg:
Points�Seg: [Ye, Bresson, Goldsteind O h 2010
Frame�based�model: and Osher, 2010]
Results:�Image�segmentation�(2D)
Original Observed Results Zoom-In
[Dong, Chien and Shen, 2010]
Frame Based SegmentationFrame Based SegmentationFrame�Based�SegmentationFrame�Based�Segmentation General variational model [Chan and Vese General�variational�model�[Chan and Vese,
1999&2000], [Chan, Esedoglu and Nikolova, 2004]:
Image�Seg:
Points�Seg: [Ye, Bresson, Goldsteind O h 2010
Frame�based�model: and Osher, 2010]
Results:�Image�segmentation�(3D)
[Dong, Chien and Shen, 2010]
Frame Based SegmentationFrame Based SegmentationFrame�Based�SegmentationFrame�Based�Segmentation General variational model [Chan and Vese General�variational�model�[Chan and Vese,
1999&2000], [Chan, Esedoglu and Nikolova, 2004]:
Image�Seg:
Points�Seg: [Ye, Bresson, Goldsteind O h 2010
Frame�based�model: and Osher, 2010]
Results:�Surface�Reconstruction�(2D)
[Dong and Shen, 2010]
Frame Based SegmentationFrame Based SegmentationFrame�Based�SegmentationFrame�Based�Segmentation General variational model [Chan and Vese General�variational�model�[Chan and Vese,
1999&2000], [Chan, Esedoglu and Nikolova, 2004]:
Image�Seg:
Points�Seg: [Ye, Bresson, Goldsteind O h 2010
Frame�based�model: and Osher, 2010]
Results:�Surface�Reconstruction�(3D)
[Dong and Shen, 2010]
Frame Based Models: 0Frame Based Models: 0 NormNormFrame�Based�Models:�0Frame�Based�Models:�0--NormNorm Nonconvex analysis based model [Zhang, Dong and Lu, Nonconvex�analysis�based�model�[Zhang, Dong and Lu,
2011]
Motivations:• Restricted isometry property (RIP) is not• Restricted�isometry property�(RIP)�is�not
satisfied�for�many�applications�
• Penalizing�����“norm”�of�frame�coefficients
Related�work:
g
better�balances�sparsity�and�smoothnes
• “norm”�with��������������:�[Blumensath and Davies,
2008&2009];�blind�inpainting:�[Yan, 2011]
• quasi-norm�with������������������:�[Chartrand, 2007&2008]
Fast Algorithm: 0Fast Algorithm: 0 NormNormFast�Algorithm:�0Fast�Algorithm:�0--NormNorm Penalty�decomposition�(PD)�method�[Lu and Zhang, 2010]
Change�of�variables
Quadratic penalty
Algorithm:�
Quadratic�penalty
Fast Algorithm: 0Fast Algorithm: 0 NormNormFast�Algorithm:�0Fast�Algorithm:�0--NormNorm Step�1:
Subproblem 1a): quadratic Subproblem�1a):�quadratic
S b bl 1b) h d h h ldi Subproblem�1b):�hard-thresholding
Convergence�Analysis�[Zhang, Dong and Lu, 2011] :
Numerical ResultsNumerical ResultsNumerical�ResultsNumerical�Results Comparisons (Deblurring) Comparisons�(Deblurring)
Balanced PFBS/FPC:�[Combettes and�Wajs,�2006]�/[Hale,�Yin�and�Zhang,�2010]
Split Bregman: [Goldstein and OsherAnalysis
0-Norm
Split Bregman:�[Goldstein�and�Osher,�2008]�&�[Cai,�Osher and�Shen,�2009]
PD Method:�[Zhang,�Dong�and�Lu,�2011]
Numerical ResultsNumerical ResultsNumerical�ResultsNumerical�Results Comparisons Comparisons
Portrait
CoupleCouple
Balanced Analysis
Faster Algorithm: 0Faster Algorithm: 0 NormNormFaster�Algorithm:�0Faster�Algorithm:�0--NormNorm Start�with�some�fast�optimization�method�for�nonsmooth and�
convex optimizations: doubly augmented Lagrangian (DAL)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: 0Faster Algorithm: 0 NormNormFaster�Algorithm:�0Faster�Algorithm:�0--NormNorm Start�with�some�fast�optimization�method�for�nonsmooth and�
convex optimizations: doubly augmented Lagrangian (DAL)convex optimizations:�doubly�augmented�Lagrangian�(DAL)�method�[Rockafellar, 1976].
Given�the�problem:
The�DAL�method:
where�
The�inexact�DAL�method:
Hard�thresholding
Faster Algorithm: 0Faster Algorithm: 0 NormNormFaster�Algorithm:�0Faster�Algorithm:�0--NormNorm However,�the�inexact�DAL�method�does�not�seem�to�converge!!�
Nonetheless the sequence oscillates and is boundedNonetheless,�the�sequence�oscillates and�is�bounded. The�mean�doubly�augmented�Lagrangian�method�(MDAL)�
[Dong and Zhang, 2011]:
MDAL:
Comparisons: DeblurringComparisons: DeblurringComparisons:�DeblurringComparisons:�Deblurring Comparisons�of�best�PSNR�values�v.s.�various�noise�level
Comparisons: DeblurringComparisons: DeblurringComparisons:�DeblurringComparisons:�Deblurring Comparisons�of�computation�time v.s.�various�noise�level
Comparisons: DeblurringComparisons: DeblurringComparisons:�DeblurringComparisons:�Deblurring What�makes�“lena”�so�special?
1-norm 0-norm: PD 0-norm: MDAL
D f th it d f th Decay�of�the�magnitudes�of�the�wavelet�frame�coefficients�is�very�fast,�which�is�what�0-norm�prefers.�
Similar observation was made Similar�observation�was�made�earlier�by�[Wang and Yin, 2010].
With�the�Center�for�Advanced�Radiotherapy�and�Technology�(CART),�UCSD
APPLICATIONS IN CT APPLICATIONS IN CT RECONSTRUCTIONRECONSTRUCTIONRECONSTRUCTIONRECONSTRUCTION
3D Cone Beam CT3D Cone Beam CTCone Beam CT
3D�Cone�Beam�CT3D�Cone�Beam�CTCone�Beam�CTz v
g(u)
f(x)u*
xy0
n
x
un 0
xS
3D Cone Beam CT3D Cone Beam CTDiscrete
3D�Cone�Beam�CT3D�Cone�Beam�CTDiscrete
=
Animation�created�by�Dr.�Xun Jia
Cone�Beam�CT�Image�Cone�Beam�CT�Image�
Goal: solve
ReconstructionReconstruction Goal:�solve
Difficulties:�
Unknown�Image Projected�Image
• In�order�to�reduce�dose,�the�system�is�highly�underdetermined Hence the solution is not
Related work:
underdetermined.�Hence�the�solution�is�not�unique.•Projected�image�is�noisy.
Related�work: 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] 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�CT�Image�Reconstruction�with�Radon�Domain�Radon�Domain�InpaintingInpainting Idea: start with Idea:�start�with�
Instead�of�solving
We�find�both��������and���������such�that:
• is close to but with better qualityis�close�to�������but�with�better�quality
•
• Prior knowledge of them should be used
Benefits:
Prior�knowledge�of�them�should�be�used
S f l i i i d• Safely increase�imaging�dose• Utilizing�prior�knowledge�we�have�for�both�CT���
images�and�the�projected�images
CT�Image�Reconstruction�with�CT�Image�Reconstruction�with�Radon�Domain�Radon�Domain�InpaintingInpainting Model [Dong Li and Shen 2011] Model�[Dong, Li and Shen, 2011]
where
• p=1,�anisotropic• p=2,�isotropic
Algorithm:�alternative�optimization�&�split�BregmanBregman.
CT�Image�Reconstruction�with�CT�Image�Reconstruction�with�Radon�Domain�Radon�Domain�InpaintingInpainting Algorithm [Dong Li and Shen 2011]: block Algorithm�[Dong, Li and Shen, 2011]:�block�
coordinate�descend�method�[Tseng, 2001]Problem:ob e :
Algorithm:
Convergence�Analysis
Note: If each subproblem is solved exactly, then the convergence analysisNote: If�each�subproblem�is�solved�exactly,�then�the�convergence�analysis�was�given�by�[Tseng, 2001],�even�for�nonconvex�problems.
CT�Image�Reconstruction�with�CT�Image�Reconstruction�with�Radon�Domain�Radon�Domain�InpaintingInpainting Results: N denoting number of projections Results:�N�denoting�number�of�projections
N=15 N=20
CT�Image�Reconstruction�with�CT�Image�Reconstruction�with�Radon�Domain�Radon�Domain�InpaintingInpainting Results: N denoting number of projections Results:�N�denoting�number�of�projections
N=15
N=20
W/O�Inpainting With�Inpainting
Thank�YouThank�You
Collaborators:Mathematics�
Stanley�Osher,�UCLA Z i Sh NUS Zuowei Shen,�NUS Jia Li,�NUS Jianfeng Cai,�University�of�Iowa Yifei Lou UCLA/UCSD 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 Xun Jia,�Radiation�Oncology,�UCSD Aichi�Chien,�Radiology,�UCLA
