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D論発表会スライド 2014/06/12

- 信号処理・画像処理における凸最適化6ヶ月前 by Shunsuke Ono
- MIRU2016 チュートリアル3ヶ月前 by Shunsuke Ono

- A Study of Priors and Algorithms for Signal Recovery

by Convex Optimization Techniques

Shunsuke Ono

Yamada Lab.

Dept. Communications and Integrated Systems

Tokyo Institute of Technology

2014/06/12 - General Introduction

2 - Signal Recovery Problem

Signal recovery is a fundamental problem in signal processing

• signal reconstruction

• image restoration

• compressed sensing

• tensor completion

...

signal recovery problems = inverse problems of the form:

observation

unknown signal

Goal: estimate from and

noise contamination

linear degradation

How to resolve this ill-posed/ill-conditioned problem?

3 - Prior and Convex Optimization

Some a priori information on signal of interest, e.g.,

• sparsity

• l1-norm [Donoho+ ‘03; Candes+ ‘06]

• smoothness

• total variation (TV) [Rudin+ ‘92; Chambolle ‘04]

• low-rankness

• nuclear norm [Fazel ‘02; Recht et al. ‘11]

should be taken into consideration.

A powerful approach: convex optimization [see, e.g., 1-3]

convex function

desired signal

convex set

advantage: 1. local optimal = global optimal

2. flexible framework

a-priori information convex function =: prior

1. D. P. Palomar and Y. C. Eldar, Eds., Convex Optimization in Signal Processing and Communications, Cambridge University Press, 2009.

2. J.-L. Starck et al., Sparse Image and Signal Processing: Wavelets, Curvelets,Morphological Diversity. Cambridge University Press, 2010.

3. H. H. Bauschke et al., Eds., Fixed-Point Algorithm for Inverse Problems in Science and Engineering, Springer-Verlag, 2011.

4 - Optimization Algorithms for Signal Recovery

Optimization algorithms for signal recovery must deal with

• useful priors = nonsmooth convex function

• problem scale = often more than 10^4

proximal splitting methods [e.g., Gabay+ ‘76; Lions+ ‘79; Combettes+ ‘05; Condat ‘13]

• first-order (without Hessian)

• nonsmooth functions

• multiple constraints

Efficient

Useful priors

algorithms

Why do we need more?

5 - Motivation & Goal

# prior: signal-specific properties are NOT fully exploited.

=> undesired results, e.g., texture degradation, color artifact, …

# algorithm: CANNOT deal with sophisticated constraints.

=> only the intersection of projectable convex sets.

Goal: design priors and algorithms to resolve them.

6 - Structure of The Dissertation

Chap. 1 General Introduction

Chap. 2 Preliminaries

priors

Chap. 3 Image restoration with component-wise

use of priors

Chap. 4 Blockwise low-rank prior for cartoon-texture

image decomposition and restoration

Main

Chap. 5 Priors for color artifact reduction

chapters

in image restoration

Chap. 6 A hierarchical convex optimization algorithm

with primal-dual splitting

Chap. 7 An efficient algorithm for signal recovery with

sophisticated data-fidelity constraints

Chap. 8 General conclusion

algorithms

7 - Structure of The Dissertation

Chap. 1 General Introduction

Chap. 2 Preliminaries

priors

Chap. 3 Image restoration with component-wise

use of priors

Chap. 4 Blockwise low-rank prior for cartoon-texture

image decomposition and restoration

Main

Chap. 5 Priors for color artifact reduction

chapters

in image restoration

Chap. 6 A hierarchical convex optimization algorithm

with primal-dual splitting

Chap. 7 An efficient algorithm for signal recovery with

sophisticated data-fidelity constraints

Chap. 8 General conclusion

algorithms

8 - Chap. 4 Blockwise low-rank prior for cartoon-texture

image decomposition and restoration

9 - Background

10 - Cartoon-Texture Decomposition Model

Assumption: image = sum of two components

image

cartoon

texture

optimization problem [Meyer ‘01; Vese+ ‘03; Aujol+ ’05; ~ Schaeffer+ ‘13 ]

priors for each component data-fidelity to

advantage: 1. prior suitable to each component

2. extraction of texture

11 - Cartoon-Texture Decomposition Model

image

cartoon

texture

optimization problem [Meyer ‘01; Vese+ ‘03; Aujol+ ‘05; ~ Schaeffer+ ‘13 ]

total variation (TV) [Rudin 92]: suitable for cartoon

12 - Cartoon-Texture Decomposition Model

image

cartoon

texture

optimization problem [Meyer ‘01; Vese+ ‘03; Aujol+ ‘05; ~ Schaeffer+ ‘13 ]

Texture is rather difficult to model…

13 - Existing Texture Priors

2001 ~

2005 ~

2013

G norm

frame

prior

[Meyer ‘01]

[Daubechies+ ‘05] [Schaeffer+ ‘13]

capability

[Aujol+ ‘05] [Elad+ ’05]

[Ng+ ‘13]

[Fadili+ ‘10]

noise

->

fine pattern

local adaptivity

14 - Contribution

2001 ~

2005 ~

2013

G norm

frame

Proposed

[Meyer ‘01]

[Daubechies+ ‘05] [Schaeffer+ ‘13]

prior

[Aujol+ ‘05] [Elad+ ’05]

[Ng+ ‘13]

[Fadili+ ‘10]

Noise

Fine pattern

Local adaptivity

Propose a prior for a better interpretation of texture.

15 - SVD, Rank and Nuclear Norm

• singular value decomposition (SVD)

nonzero singular values

•

number of nonzero singular values

• nuclear norm

rank

nuclear norm

tightest convex relaxation of rank [Fazel 02]

* applications to robust PCA (sparse + low-rank)

[Gandy & Yamada ‘10; Candes+ ‘11; Gandy, Recht, Yamada ‘11]

16 - Proposed Method

17 - How To Model Texture?

globally dissimilar but locally well-patterned

Any block is approximately low-rank after suitable shear.

18 - Proposed Prior: Block Nuclear Norm (1/2)

Any block is approximately low-rank after suitable shear.

Definition: pre-Block-nuclear-norm (pre-BNN)

positive weight nuclear norm

Important property of pre-BNN

Pre-BNN is tightest convex relaxation of

weighted blockwise rank

* Generalization of [Fazel ‘02]

19 - Proposed Prior: Block Nuclear Norm (2/2)

Any block is approximately low-rank after suitable shear.

BNN becomes small,

i.e., good texture prior.

Definition: Block Nuclear Norm (BNN)

periodic expansion operator (overlap) shear operator

20 - Cartoon-Texture Decomposition Using BNN

proposed cartoon-texture decomposition model

various shear angles

texture (K=3)

sub-texture 1

sub-texture 2

sub-texture 3

• Patterns running in different directions are separately extracted.

• Proximal splitting methods can solve the problem after reformulation.

21

image

cartoon

texture - Experimental Results

CASE 1: pure decomposition

compared with a state-of-the-art decomposition [Schaeffer & Osher, 2013]

cartoon

cartoon

image

texture

texture

[Schaeffer & Osher 2013] “A low patch-

rank interpretation of texture,” SIAM J.

Imag. Sci.

[Schaeffer & Osher 2013]

proposed

22 - Experimental Results

CASE 2: blur+20%missing pixels

compared also with [Schaeffer & Osher 2013]

observation

[Schaeffer & Osher 2013]

proposed

PSNR: 23.20

PSNR: 23.75

SSIM: 0.6613

SSIM: 0.6978

23 - Experimental Results

CASE 2: blur+20%missing pixels

Compared with a state-of-the-art decomposition [Schaeffer & Osher, 2013]

observation

[Schaeffer & Osher 2013]

proposed

PSNR: 23.20

PSNR: 23.75

SSIM: 0.6613

SSIM: 0.6978

24 - Chap. 5 Priors for color artifact reduction

in image restoration

25 - Background

26 - Color Artifact in Image Restoration

observation

restored by an existing prior

color artifact

original

27 - Color-Line Property

# color-line: RGB entries are linearly distributed in local regions.

[Omer & Werman ‘04]

restored by

original

an existing prior

color-line

corrupted

28 - Contribution

Propose a prior for promoting color-line property.

restored by

existing +

original

an existing prior

proposed prior

color-line

corrupted

reconstructed

29 - Proposed Method

30 - Mathematical Modeling of Color-Line

Define matrices for every local region of a color image.

B G R

-th local region

(e.g., block)

Vectorize

color image

matrix for

-th local region

31 - Proposed Prior: Local Color Nuclear Norm

Local Color Nuclear Norm (LCNN)

number of local regions

key principle

color-line property low-rankness of

exact cases

rank( ) = 1

32 - Proposed Prior: Local Color Nuclear Norm

Local Color Nuclear Norm (LCNN)

key principle

color-line property small singular values of

practical cases

rank(

) ≠ 1

but is small

Suppressing LCNN promotes the color-line property.

33 - Application to Denoising

: color image contaminated by impulsive noise

optimization problem

smoothness [VTV]

color-line

dynamic range

data-fidelity robust to

Impulsive noise

Proximal splitting methods are applicable after reformulation.

[VTV] Bresson et al. “Fast dual minimization of the vectorial total variation norm and

34

applications to color image processing”, Inverse Probl. Img., 2008. - Experimental Results

22.95, 7.59

24.30, 5.80

25.12, 3.07

27.22, 2.58

observation

VTV

VTV+LCNN

original

(PSNR, D2000)

35

(PSNR, D2000) - Chap. 6 A hierarchical convex optimization algorithm

with primal-dual splitting

36 - Background

37 - Solutions of Convex Optimization Problems

Solution set of a convex optimization problem

contains infinitely many solutions non-strict convexity of .

Solutions could be considerably different in another criterion.

Strictly convex

NOT strictly convex

.

Unique

NOT unique

38 - Hierarchical Convex Optimization

ideal strategy: hierarchical convex optimization:

selector: smooth convex function

highly involved (≠the intersection of projectable convex sets)

proximal splitting methods cannot solve the problem.

via fixed point set characterization

[e.g., Yamada ‘01; Ogura & Yamada‘03; Yamada, Yukawa, Yamagishi ‘11]

computable nonexpansive mapping on a certain Hilbert space

Definition: nonexpansive mapping

39 - Hierarchical Convex Optimization

fixed point set characterized problem

Hybrid Steepest Descent Method (HSDM) [e.g., Yamada ‘01; Ogura & Yamada ‘03]

nonexpansive mapping

gradient of selector

Q. What kinds of are available?

40 - Nonexpansive Mappings for

Two characterizations underlying proximal splitting methods

are given in [Yamada, Yukawa, Yamagishi ‘11].

• Forward-Backward Splitting (FBS) method [Passty ’79; Combettes+ ‘05]

• Douglas-Rachford Splitting (DRS) method [Lions+ ‘79; Combettes+ ‘07]

Q. Can we deal with a more flexible formulation?

Definition: proximity operator [Moreau ‘62]

41 - Nonexpansive Mappings for

Two characterizations underlying proximal splitting methods

are given in [Yamada, Yukawa, Yamagishi ‘11].

• Forward-Backward Splitting (FBS) method [Passty ’79; Combettes+ ‘05]

• Douglas-Rachford Splitting (DRS) method [Lions+ ’79; Combettes+ ‘07]

• Primal-Dual Splitting (PDS) method [Condat ‘13; Vu ‘13]

42 - Contribution

hierarchical convex optimization by HSDM

• reveal convergence properties

• modify gradient computation

incorporate • extract operator-theoretic idea from [Condat 13]

• reformulate in a certain product space

• Primal-Dual Splitting (PDS) method [Condat ‘13; Vu ‘13]

43 - Proposed Method

44 - Outline

Reformulate in the canonical product space with dual problem

Extract & incorporate fixed point set characterization from [Condat ‘13]

Install another inner product for nonexpansivity of by [Condat ‘13]

Apply HSDM with modified gradient computation w.r.t.

45 - Reformulation in The Canonical Product Space

solution set of the first stage problem (=primal problem)

solution set of the dual problem of the first stage problem

By letting

Note:

46 - Incorporation of PDS Characterization

Extract the PDS fixed point characterization from [Condat ‘13]

47 - Activation of Nonexpansivity

is nonexpansive NOT on the canonical product space

BUT on the following space with another inner product [Condat ‘13]

where

: strongly positive bounded linear operator

Definition: canonical inner product of

48 - Solver via HSDM

We can apply HSDM [e.g., Yamada ‘01; Ogura & Yamada ‘03]

NOTE:

49 - Convergence of HSDM with PDS

Assumptions:

Convergence 1:

Convergence 2:

Recall

Definition: distance function

50 - Application to Signal Recovery

observation model: unknown signal

degradation

Gaussian noise

first stage problem: non-strictly convex

data-fidelity

numerical range

prior

hierarchical convex optimization problem:

another prior

to specify

a better solution

Definition: indicator function

51 - Application to Signal Recovery

observation model: unknown signal

degradation

Gaussian noise

first stage problem: non-strictly convex

hierarchical convex optimization problem:

another prior

to specify

a better solution

Definition: indicator function

52 - Experimental Results

original

observed

no-

hierarchical

proposed

53 - General Conclusion

We have developed novel priors and algorithms for signal recovery.

priors: to model signal-specific properties

Chap. 3 Image restoration with component-wise

use of priors

Chap. 4 Blockwise low-rank prior for cartoon-texture

image decomposition and restoration

Chap. 5 Priors for color artifact reduction

in image restoration

Chap. 6 A hierarchical convex optimization algorithm

with primal-dual splitting

Chap. 7 An efficient algorithm for signal recovery with

sophisticated data-fidelity constraints

algorithms: to deal with involved constraints

54 - Related Publications

# Journal Papers

[J1] S. Ono, T. Miyata, I. Yamada, and K. Yamaoka, "Image Recovery by

Decomposition with Component-Wise Regularization,"

IEICE Trans. Fundamentals, vol. E95-A, no. 12, pp. 2470-2478, 2012.

(Best Paper Award from IEICE)

[J2] S. Ono, T. Miyata, and I. Yamada, "Cartoon-Texture Image Decomposition

Using Blockwise Low-Rank Texture Characterization,"

IEEE Trans. Image Process., vol. 23, no. 3, pp. 1028-1042, 2014.

[J3] S. Ono and I. Yamada, "Hierarchical Convex Optimization with Primal-Dual

Splitting,“ submitted to IEEE Trans. Signal Process (accepted conditionally

in May. 2014).

[J4] S. Ono and I. Yamada, "Signal Recovery Using Complicated Data-Fidelity

Constraints,“ in preparation.

55 - Related Publications

# Articles in Proceedings of International Conferences (reviewed)

[C1] S. Ono, T. Miyata, and K. Yamaoka, "Total Variation-Wavelet-Curvelet

Regularized Optimization for Image Restoration," IEEE ICIP 2011.

[C2] S. Ono, T. Miyata, I. Yamada, and K. Yamaoka, "Missing Region Recovery by

Promoting Blockwise Low-Rankness," IEEE ICASSP 2012.

[C3] S. Ono and I. Yamada, "A Hierarchical Convex Optimization Approach for High

Fidelity Solution Selection in Image Recovery,' APSIPA ASC 2012, (Invited).

[C4] S. Ono and I. Yamada, "Poisson Image Restoration with Likelihood Constraint

via Hybrid Steepest Descent Method," IEEE ICASSP 2013.

[C5] S. Ono, M. Yamagishi, and I. Yamada, "A Sparse System Identification by Using

Adaptively-Weighted Total Variation via A Primal-Dual Splitting Approach,"

IEEE ICASSP 2013.

[C6] S. Ono and I. Yamada, "A Convex Regularizer for Reducing Color Artifact in

Color Image Recovery,“ IEEE Conf. CVPR 2013.

[C7] I. Yamada and S. Ono, "Signal Recovery by Minimizing The Moreau Envelope

over The Fixed Point Set of Nonexpansive Mappings," EUSIPCO 2013, (invited).

[C8] S. Ono and I. Yamada, “Second-Order Total Generalized Variation Constraint,”

IEEE ICASSP 2014.

[C9] S. Ono and I. Yamada, “Decorrelated Vectorial Total Variation,” IEEE Conf. CVPR

2014 (to appear).

56 - Other Publications

# Journal Papers

[J5] S. Ono, T. Miyata, and Y. Sakai, "Improvement of Colorization Based Coding by

Using Redundancy of The Color Assignment Information and Correct Color

Component," IEICE Trans. Information and Systems, vol. J93-D, no. 9, pp.

1638-1641, 2010 (in Japanese).

[J6] H. Kuroda, S. Ono, M. Yamagishi, and I. Yamada, "Exploiting Group Sparsity in

Nonlinear Acoustic Echo Cancellation by Adaptive Proximal Forward-Backward

Splitting," IEICE Trans. Fundamentals, vol.E96-A, no.10, pp.1918-1927, 2013.

[J7] T. Baba, R. Matsuoka, S. Ono, K. Shirai, and M. Okuda, "Image Composition Using

A Pair of Flash/No-Flash Images by Convex Optimization,“ IEICE Transactions on

Information and System, 2014 (in Japanese, to appear)

57 - Other Publications

# Articles in Proceedings of International Conference (reviewed)

[C10] S. Ono, T. Miyata, and Y. Sakai, "Colorization-Based Coding by Focusing on

Characteristics of Colorization Bases," PCS 2010.

[C11] M. Yamagishi, S. Ono, and I. Yamada, "Two Variants of Alternating Direction

Method of Multipliers without Inner Iterations and Their Application to Image

Super-Resolution,' IEEE ICASSP 2012.

[C12] S. Ono and I. Yamada, "Optimized JPEG Image Decompression with Super-

Resolution Interpolation Using Multi-Order Total Variation," IEEE ICIP 2013

(top 10% of all accepted papers).

[C13] K. Toyokawa, S. Ono, M. Yamagishi, and I. Yamada, "Detecting Edges of

Reflections from a Single Image via Convex Optimization,“ IEEE ICASSP 2014.

[C14] T. Baba, R. Matsuoka, S. Ono, K. Shirai, and M. Okuda, "Flash/No-flash Image

Integration Using Convex Optimization,“ IEEE ICASSP 2014.

58

* Many other articles in proceedings of domestic conferences