2. Multiscale Analysis for Intensity and Density Estimation Rebecca Willett’s MS Defense Thanks to Rob Nowak, Mike Orchard, Don Johnson, and Rich Baraniuk Eric Kolaczyk and Tycho Hoogland

3. Poisson and Multinomial Processes

4. Why study Poisson Processes? Astrophysics Network analysis Medical Imaging

5. Examining data at different resolutions (e.g., seeing the forest, the trees, the leaves, or the dew) yields different information about the structure of the data. Multiresolution analysis is effective because it sees the forest (the overall structure of the data) without losing sight of the trees (data singularities) Multiresolution Analysis

6. Beyond Wavelets Multiresolution analysis is a powerful tool, but what about… Edges? Nongaussian noise? Inverse problems? Piecewise polynomial- and platelet- based methods address these issues. Non-Gaussian problems? Image Edges? Inverse problems?

7. Computational Harmonic Analysis I.Define Class of Functions to Model Signal A.Piecewise Polynomials B.Platelets II.Derive basis or other representation III.Threshold or prune small coefficients IV.Demonstrate near-optimality

8. Approximating Besov Functions with Piecewise Polynomials

9. Approximation with Platelets Consider approximating this image:

10. E.g. Haar analysis Terms = 2068, Params = 2068

11. Wedgelets Original Image Haar Wavelet Partition Wedgelet Partition

12. E.g. Haar analysis with wedgelets Terms = 1164, Params = 1164

13. E.g. Platelets Terms = 510, Params = 774

14. Error Decay

15. Platelet Approximation Theory Error decay rates: Fourier: O(m -1/2 ) Wavelets: O(m -1 ) Wedgelets: O(m -1 ) Platelets: O(m - min(,) )

17. A Piecewise Constant Tree

18. A Piecewise Linear Tree

19. Maximum Penalized Likelihood Estimation Goal: Maximize the penalized likelihood So the MPLE is

20. The Algorithm Data Const Estimate Wedge Estimate Platelet Estimate Wedged Platelet Estimate Inherit from finer scale

21. Algorithm in Action

22. Penalty Parameter Penalty parameter balances between fidelity to the data (likelihood) and complexity (penalty).  = 0  Estimate is MLE:  Estimate is a constant:

23. Penalization

24. Density Estimation - Blocks

25. Density Estimation - Heavisine

26. Density Estimation - Bumps

27. Density Estimation Simulation

28. Medical Imaging Results

29. Inverse Problems Goal: estimate  from observations x ~ Poisson(P) EM algorithm (Nowak and Kolaczyk, ’00):

30. Confocal Microscopy: An Inverse Problem

31. Platelet Performance

32. Confocal Microscopy: Real Data

33. Hellinger Loss Upper bound for affinity (like squared error) Relates expected error to L p approximation bounds

34. Bound on Hellinger Risk KL distance Approximation error Estimation error (follows from Li & Barron ’99)

35. Bounding the KL We can show: Recall approximation result: Choose optimal d

36. Near-optimal Risk Maximum risk within logarithmic factor of minimum risk Penalty structure effective:

37. Conclusions CHA with Piecewise Polynomials or Platelets Effectively describe Poisson or multinomial data Strong approximation capabilites Fast MPLE algorithms for estimation and reconstruction Near-optimal characteristics

38. Future Work Risk analysis for piecewise polynomials Platelet representations and approximation theory Shift-invariant methods Fast algorithms for wedgelets and platelets Risk Analysis for platelets Major Contributions

39. Approximation Theory Results

40. Why don’t we just find the MLE?

41. MPLE Algorithm (1D)

42. Multiscale Likelihood Factorization  Probabilistic analogue to orthonormal wavelet decomposition  Parameters   wavelet coefficients  Allow MPLE framework, where penalization based on complexity of underlying partition

43. Poisson Processes Goal: Estimate spatially varying function, (i,j), from observations of Poisson random variables x(i,j) with intensities (i,j) MLE of would simply equal x. We will use complexity regularization to yield smoother estimate.

44. Accurate Model Parsimonious Model Complexity Regularization Penalty for each constant region  results in fewer splits Bigger penalty for each polynomial or platelet region  more degrees of freedom, so more efficient to store constant if likely

45. Astronomical Imaging