1

Dependent Hierarchical Beta Process

for Image Interpolation and Denoising

1Mingyuan Zhou, 2Hongxia Yang, 3Guillermo Sapiro,2David Dunson and 1Lawrence Carin

1Department of Electrical & Computer Engineering, Duke University2Department of Statistical Science, Duke University

3Department of Electrical & Computer Engineering, University of Minnesota

AI & Statistics, Ft. Lauderdale, April 12 2011

Zhou 2011

2

Outline

• Introduction

• Dependent Hierarchical Beta Process (dHBP)

• Dictionary Learning with dHBP

• Image Interpolation and Denoising

• Conclusions

3

Introduction: Background

• Dictionary learning and sparse coding

• Sparse factor analysis model

(Factor/feature/dish/dictionary atom)

• Indian Buffet process and beta process

11

minN

iFi

λ=

− + ∑D,W

X DW w

4

Introduction: Background

• Beta process and Bernoulli process (Thibaux &

Jordan AISTATS2007)

• Indian Buffet process (Griffiths & Ghahramani 2005)

5

Introduction: Motivation

• Exchangeability assumption is not true

Image interpolation with BPFA (Zhou et al. 2009)

80% pixels are missing at random

6

Introduction: Covariate Dependence

• Dependent Dirichlet process

MacEachern (1999), Duan et al. (2007), Griffin & Steel

(2006)

• Non-exchangeable IBP

Phylogenetic IBP (Miller, Griffiths & Jordan 2008)

Dependent IBP (Williamson, Orbanz & Ghahramani 2010)

• Bayesian density regression (Dunson & Pillai 2007)

7

Review of Beta Process (Thibaus & Jordan 2007)

• A beta process is a positive Levy process, whose Levy

measure lives on and can be expressed as

• If the base measure is continuous, then

is drawn from a degenerate beta

distribution parameterized by

• If , then

[ ]0,1Ω ⊗

0B

1

, and kk w k

k

B p pδ∞

=

=∑

01

kk wk

B q δ∞

=

=∑

1

, and ~ Beta( , (1 ))kk w k k k

k

B p p cq c qδ∞

=

= −∑

c

8

Dependent Hierarchical Beta Process

• Random walk matrix

• dHBP

• Covariate-dependent correlations

9

Dictionary Learning with dHBP

BP

dHBP

10

• Missing data

Full data:

Observed: , Missing:

• Sparse spiky noise

• Recoverd data:

Missing Data and Outliers

11

• Independence chain Metropolis-Hastings

• Slice sampling

• Gibbs sampling

MCMC Inference

12

Experiments

• Image interpolation

Missing pixels

Locations of missing pixels are known

• Image denoising

WGN + sparse spiky noise

Amplitudes unknown

Locations of spiky noise are unknown

• Covariates: patch spatial locations

13

Image Interpolation: BP vs. dHBP

BP: 26.9 dB dHBP: 29.92 dB

80% pixels missing at random

14

BP atom usage dHBP atom usage

BP dictionary dHBP dictionary Dictionary atom activation probability map

dHBP recoveryOriginaldHBP recovery

BP recoveryObserved

Image Interpolation: BP vs. dHBP

15

Observed (20%) BP recovery

dHBP recovery Original

dHBP recovery

Image Interpolation: BP vs. dHBP

16

Image Interpolation: BP vs. dHBP

Observed (20%) BP recovery

dHBP recovery Original

dHBP recovery

17

Spiky Noise Removal: BP vs. dHBP

BP denoised imageNoisy image (WGN + Sparse

Spiky noise)BP dictionary

dHBP denoised imageOriginal image dHBP dictionary

18

Spiky Noise Removal: BP vs. dHBP

BP denoised imageNoisy image (WGN + Sparse

Spiky noise)

dHBP denoised imageOriginal image dHBP dictionary

BP dictionary

19

Future Work

• Landmark-dHBP

J landmarks

J << N

• Locality constraint for manifold learning

Covariates: cosine distance between samples

Dictionary atoms look like the data

• Variational inference, online learning

• Other applications

Super-resolution

Deblurring

Video background & foreground modeling

20

• A dependent hierarchical beta process is

proposed

• Efficient hybrid MCMC inference is presented

• Encouraging performance is demonstrated on

image-processing applications

Conclusions