Template Learning From Atomic Representations

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Template Learning from Atomic Representations
A Wavelet-based Approach to Pattern Analysis
Clay Scott and Rob Nowak
Electrical and Computer Engineering Rice
University www.dsp.rice.edu
Supported by ARO, DARPA, NSF, and ONR
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The Discrete Wavelet Transform

• prediction errors ? wavelet coefficients
• most wavelet coefficients are zero ? sparse
  representation

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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

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Pattern Recognition
Class 1
Class 2
Class 3
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Hierarchical Framework
Noisy observation of transformed pattern
Random transformation of pattern
Pattern template in spatial domain
Realization from wavelet-domain statistical model
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Wavelet-domain statistical model

• Sparsity ? can divide wavelet coefficients into
  significant and insignificant coefficients

• Model wavelet coefficients as independent
  Gaussian mixtures
• where

is significant

• Constraints

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Model Parameters

• Template parameters

where

• Finite set of pre-selected transformations
• model variability in location and orientation

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Pattern Synthesis
1. Generate a random template 2. Transform to
spatial domain 3. Apply random transformation
4. Add observation noise
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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)
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PMLE of ?, s, and ?

• PMLE ? maximize

• Complexity penalty function
• where is the number of
  significant coefficients

? Minimum description length (MDL) criterion

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

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TEMPLAR Template Learning from Atomic
Representations

• 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)
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Airplane Experiment
Picture of me gathering data
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Airplane Experiment

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

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Face Experiment
Training data for one subject, plus sequence of
template convergence
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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

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Classification
Given Templates for several patterns and an
unlabeled observation x
Classify

• Invariant to unknown transformations
• O(NT) complexity
• sparsity ? low-dimensional subspace classifier
• ? robust to background clutter

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Face Recognition
Results of Yale face test
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Image Registration
If I get results
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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