Learning from Atomic Representations: A Wavelet-based Approach to Pattern Analysis

1. 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

2. The Discrete Wavelet Transform • prediction errors  wavelet coefficients • most wavelet coefficients are zero  sparse representation

3. 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

4. Pattern Recognition Class 1 Class 2 Class 3

5. Noisy observation of transformed pattern Random transformation of pattern Pattern template in spatial domain Realization from wavelet-domain statistical model Hierarchical Framework Noisy observation of transformed pattern Random transformation of pattern Pattern template in spatial domain Realization from wavelet-domain statistical model

6. 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:

7. Model Parameters • Template parameters: where • Finite set of pre-selected transformations • model variability in location and orientation

8. Pattern Synthesis 1. Generate a random template 2. Transform to spatial domain 3. Apply random transformation 4. Add observation noise

9. 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)

10. 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

11. 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)

12. Airplane Experiment Picture of me gathering data

13. Airplane Experiment • 853 significant coefficients out of 16,384 • 7 iterations

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

15. 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

16. 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

17. Face Recognition Results of Yale face test

18. Image Registration If I get results

19. 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