Entropy-constrained overcomplete -based coding of natural images

Entropy-constrained overcomplete-based coding of natural images

André F. de Araujo, Maryam Daneshi, Ryan Peng

Stanford University

2EE398A Project – Winter 2010/2011 Mar. 10, 2011

Outline

Motivation Overcomplete-based coding: overview Entropy-constrained overcomplete-based coding Experimental results Conclusion Future work


Motivation (1)

Study of new (and unusual) schemes for image compression

Recently, new methods have been developed using the overcomplete approach

Restricted scenarios for compression Did not fully exploit this approach’s characteristics for compression


Motivation (2)

Why? Sparsity on coefficients better overall RD


Overcomplete coding: overview (1)

K > N implies: Bases are not linearly independent Example:

8x8 blocks: N = 64 basis functions are needed to span the space of all possible signals

Overcomplete basis could have K = 128

Two main tasks:1. Sparse coding2. Dictionary learning



1. Sparse coding (“atom decomposition”)

Compute the representation coefficients x based on the signal y (given) and dictionary D (given)

overcomplete D Infinite solutions approxim.

Commonly used algorithms: Matching Pursuits (MP), Orthogonal Matching Pursuits (OMP)



Sparse coding (OMP)

Input: Dictionary , signal , number of non-zero coefficients (NNZ) (or error target ε)Output: Coefficient vector x

1. Set r = (r: residual)

2. Project r on every basis of

3. Select from with maximum projection

4.

5.

6. Stop if (or ||r||2 < ε). Otherwise, go to 2



2. Dictionary learning Two basic stages (analogy with K-means)i. Sparse coding stage: use a pursuit algorithm to compute x (OMP is

usually employed)ii. Dictionary update stage: adopt a particular strategy for updating the

dictionary

Convergence issues: as first stage does not guarantee best match, cost can increase and convergence cannot be assured



2. Dictionary learning Most relevant algorithms in the literature: K-SVD and MOD Sparse coding stage is done in the same way Codebook update stage is different:

MOD Update entire dictionary using optimal adjustment for a given

coefficients matrix K-SVD

Update each basis one at a time using SVD formulation Introduces change in dictionary and coefficients


Entropy-const. OC-based coding (1)

We introduce a compression scheme which employs entropy-constrained stages

RD-OMP Introduced by Gharavi-Alkhansar (ICIP 1998), uses the

Lagrangian cost with variable NNZ coefficients to select basis vectors

EC Dictionary Learning Introduced in this work, uses a framework inspired in EC

VQ to select basis vectors



RD-OMP – key ideas Introduction of Lagrangian cost

Estimation of rate cost: ( is fixed)

Stopping criterion/variable NNZ coefficients Once no more improvement is reached on the

Lagrangian cost, algorithm stops



Input: Dictionary , Input signal Output: coefficient vector

1. For every basis k (from 1 to K)

1. calculate 2. Pick coefficient with smallest 3. 4. Stop if , otherwise go to 1.

RD-OMP



EC Dictionary Learning – key ideas Dictionary update strategy

K-SVD modifies dictionary and coefficients - reduction in Lagrangian cost is not assured.

We use MOD, which provides the optimal adjustment assuming fixed coefficients

Introduction of “Rate cost update” stage Analogous to ECVQ algorithm for training data Two pmfs must be updated: indexes and coefficients



EC-Dictionary Learning

Input: input signal yOutput: Dictionary

1. Initialize from 2. Sparse coding stage:

RD-OMP find coefficient 3. Rate cost update stage:

1. pmfs update (indexes and coefficients)2. Codeword length update:

4. Dictionary update stage: MOD dictionary update

5. Stop when , Otherwise go to 2


Experiments (Setup)

Rate calculation: optimal codebook (entropy) for each subband

Test images: Lena, Boats, Harbour, Peppers Training dictionary experiments

Training data: 18 Kodak downsampled (to 128x128) images (does not include images being coded)

Use of downsampled images to 128x128, due to very high computational complexity (for other experiments, higher resolutions were employed: 512x512, 256x256)


Experiments (Sparse Coding)

Comparison of Sparse coding methods


Experiments (Dict. learning)

Comparison of dictionary learning methods


Experiments (Compression schemes) (1)

1: Training and coding for the same image (dictionary is sent)

2: Training with a set of natural images and applying to other images






Conclusion

Improvement of sparse coding: RD-OMP

Improvement of dictionary learning Entropy-constrained overcomplete dictionary learning

Better overall performance compared to standard techniques


Future work

Extension of implementation to higher resolution images

Further investigation of trade-off between K and N

Evaluation against directional transforms

Low complexity implementation of the algorithms

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Entropy-constrained overcomplete -based coding of natural images