Template Learning from Atomic Representations:

Supported by ARO, DARPA, NSF, and ONR

Clay Scott and Rob Nowak

A Wavelet-based Approach to Pattern Analysis

Electrical and Computer EngineeringRice University

www.dsp.rice.edu

• prediction errors wavelet coefficients

• most wavelet coefficients are zero sparse representation

The Discrete Wavelet Transform

Wavelets as Atomic Representations

• Atomic representations: attempt to decompose images

into fundamental units or “atoms”

Examples: wavelets, curvelets, wedgelets, DCT

• Successes: denoising and compression

• Drawback: not transformation invariant

poor features for pattern recognition

Pattern Recognition

Class 1

Class 2

Class 3

Hierarchical Framework

Realization from wavelet-domain statistical model

Pattern template in spatial domain

Random transformation of pattern

Noisy observation of transformed pattern

Realization from wavelet-domain statistical model

Pattern template in spatial domain

Random transformation of pattern

Noisy observation of transformed pattern

Wavelet-domain statistical model

• Model wavelet coefficients as independent Gaussian mixtures

where

)()()( 200

211 1 i,i,ii,i,ii ,σμNs,σμNs~w

20

200 0 σσ,μ i,i,

• Constraints:

• Sparsity can divide wavelet coefficients into significant and insignificant coefficients

ii ws 1 is significant

•Template parameters:

TNss ),...,(

},,{

1

2011

s

Σμθ

where

),...,(diag

),...,(2

1,21,11

1,1,11

N

TN

Σ

μ

Model Parameters

• Finite set of pre-selected transformations

model variability in location and orientationL ,...,1

Pattern Synthesis

1. Generate a random template

2. Transform to spatial domain

3. Apply random transformation

4. Add observation noise

),(~ 2obsIyx N

}{DWT 1 wz

),|(~ sθww p

zy

Template Learning

Given: Independent observations of the same pattern

arising from the (unknown) transformations

Goal: Find , s, that “best describe” the observations

Approach: Penalized maximum likelihood estimation (PMLE)

),...,( 1 TxxΧ

),...,( 1 T

PMLE of , s, and

• Complexity penalty function

where is the number of significant

coefficients

Nkc log2)( s

isk

• PMLE maximize

)(),,|(log),,( ssθXsθ cpF

• Complexity regularization Find low-dimensional template that captures essential structure of pattern

Minimum description length (MDL) criterion

TEMPLAR: Template Learning from Atomic Representations

),,(maxarg

),,(maxarg

),,(maxarg

1

11

jjj

jjj

jjj

F

F

F

sθ

sθs

sθθ

s

θ

• Simultaneously maximizing F over , s, is intractable

• Maximize F with alternating-maximization algorithm

Non-decreasing sequence of penalized likelihood values

Each step is simple, with O(NLT) complexity

Converges to a fixed point (no cycling)

Airplane Experiment

Picture of me gathering data

Airplane Experiment

• 853 significant coefficients out of 16,384• 7 iterations

Face Experiment

Training data for one subject, plus sequence of template convergence

Why Does TEMPLAR Work?

• Wavelet-domain model for template is low-

dimensional (from MDL penalty and inherent

sparseness of wavelets)

• Low-dimensional template allows for improved

pattern matching by giving more weight to

distinguishing features

Classification

Given:

Templates for several patterns and an unlabeled observation x

Cccc 1},{ sθ

Classify:

)],,|(max[ maxarg* cc

cpc sθx

• Invariant to unknown transformations

• O(NT) complexity

• sparsity low-dimensional subspace classifier

robust to background clutter

Face Recognition

Results of Yale face test

Image Registration

If I get results

Conclusion

• Wavelet-based framework for representing pattern observations with unknown rotation and translation

• TEMPLAR: Linear-time algorithm for automatically learning low-dimensional templates based using MDL

• Low-dimensional subspace classifiers that are invariant to spatial transformations and background clutter