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Non-parametric Bayesian techniques are considered for learning dictionaries for sparse image representations, with applications in denoising, inpainting and compressive sensing. The four main contributions of our paper are: The dictionary is learned using a beta process construction, and therefore the number of dictionary elements and their relative importance may be inferred non-parametrically. For the denoising and inpainting applications, we do not have to assume a priori knowledge of the noise variance or sparsity level. The spatial inter-relationships between different components in images are exploited by use of the Dirichlet process and a probit stick-breaking process. Using learned dictionaries, inferred off-line or in situ, the proposed approach yields CS performance that is markedly better than existing standard CS methods as applied to imagery. Non-Parametric Bayesian Dictionary Learning for Sparse Image Representations Mingyuan Zhou, Haojun Chen, John Paisley, Lu Ren, 1 Guillermo Sapiro and Lawrence Carin Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA 1 Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, USA {mz1, hc44, jwp4, lr, lcarin}@ee.duke.edu, {guille}@umn.edu Introduction Model and Inference Full likelihood of the Model Gibbs Sampling Inference Image denoising Image inpainting Applications Compressive sensing Corrupted image, 10.9475dB spectral band_100 Original image Restored image,25.8839dB Corrupted image, 10.9475dB spectral band_1 Original image Restored image,25.8839dB 80% Pixels Missing 50% Pixels Missing 480*321 RGB 80% Pixels Missing 150*150*210 Hyperspectral 95% Pixels Missing Image size: 480 by 320, 2400 8 by 8 patches, 153600 coefficients are estimated 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 x 10 4 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Number of Measurements Relative Reconstruction Error PSBP BP DP BP BP BCS Fast BCS Basis Pursuit LARS/Lasso OMP STOMP-CFAR 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 x 10 4 0 0.05 0.1 0.15 0.2 0.25 0.3 Number of Measurements Relative Reconstruction Error PSBP BP DP BP Online BP BP BCS Fast BCS Basis Pursuit LARS/Lasso OMP STOMP-CFAR Over-complete DCT Learned Dictionary 100 200 300 400 500 100 200 300 400 500 600 700 800 100 200 300 400 500 100 200 300 400 500 600 700 800 100 200 300 400 500 100 200 300 400 500 600 700 800 100 200 300 400 500 100 200 300 400 500 600 700 800 Spectral band 50 Spectral band 90 Original Restored 845*512*106 Hyperspectral, 98% pixels missing Original Scene Original Restored

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Page 1: Non-Parametric Bayesian Dictionary Learning for Sparse Image Representationspeople.ee.duke.edu/~mz1/Papers/MingyuanZhou_NIPS2009... · 2011-03-22 · dictionaries for sparse image

Non-parametric Bayesian techniques are considered for learningdictionaries for sparse image representations, with applications indenoising, inpainting and compressive sensing. The four maincontributions of our paper are:

The dictionary is learned using a beta process construction, andtherefore the number of dictionary elements and their relativeimportance may be inferred non-parametrically.For the denoising and inpainting applications, we do not have toassume a priori knowledge of the noise variance or sparsity level.The spatial inter-relationships between different components inimages are exploited by use of the Dirichlet process and a probitstick-breaking process.Using learned dictionaries, inferred off-line or in situ, theproposed approach yields CS performance that is markedly betterthan existing standard CS methods as applied to imagery.

Non-Parametric Bayesian Dictionary Learning for Sparse Image RepresentationsMingyuan Zhou, Haojun Chen, John Paisley, Lu Ren, 1Guillermo Sapiro and Lawrence Carin

Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA1Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, USA

{mz1, hc44, jwp4, lr, lcarin}@ee.duke.edu, {guille}@umn.edu

Introduction

Model and Inference

Full likelihood of the Model

Gibbs Sampling Inference

Image denoising

Image inpainting

Applications

Compressive sensing

Corrupted image, 10.9475dB

spectr

al band_100

Original imageRestored image,25.8839dB

Corrupted image, 10.9475dB

spectr

al band_1

Original imageRestored image,25.8839dB

80

%P

ixe

ls M

issi

ng

50

%P

ixe

ls M

issi

ng

480*321 RGB80% Pixels Missing

150*150*210 Hyperspectral

95% Pixels Missing

Image size: 480 by 320, 2400 8 by 8 patches, 153600 coefficients are estimated

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5

x 104

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Number of Measurements

Rela

tive R

econstr

uction E

rror

PSBP BP

DP BP

BP

BCS

Fast BCS

Basis Pursuit

LARS/Lasso

OMP

STOMP-CFAR

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5

x 104

0

0.05

0.1

0.15

0.2

0.25

0.3

Number of Measurements

Rela

tive R

econstr

uction E

rror

PSBP BP

DP BP

Online BP

BP

BCS

Fast BCS

Basis Pursuit

LARS/Lasso

OMP

STOMP-CFAR

Over-complete DCT Learned Dictionary

100 200 300 400 500

100

200

300

400

500

600

700

800

100 200 300 400 500

100

200

300

400

500

600

700

800

100 200 300 400 500

100

200

300

400

500

600

700

800

100 200 300 400 500

100

200

300

400

500

600

700

800

Spectral band 50 Spectral band 90

Original Restored

845*512*106 Hyperspectral, 98% pixels missing

Original Scene

Original Restored

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