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Multiscale Analysis for Intensity and Density Estimation

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. Poisson and Multinomial Processes. Why study Poisson Processes?. Astrophysics. Network analysis.

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Multiscale Analysis for Intensity and Density Estimation

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  1. 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

  2. Poisson and Multinomial Processes

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

  4. Multiresolution Analysis 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)

  5. 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?

  6. Computational Harmonic Analysis • Define Class of Functions to Model Signal • Piecewise Polynomials • Platelets • Derive basis or other representation • Threshold or prune small coefficients • Demonstrate near-optimality

  7. Approximating Besov Functions with Piecewise Polynomials

  8. Approximation with Platelets Consider approximating this image:

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

  10. Wedgelets Haar Wavelet Partition Original Image Wedgelet Partition

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

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

  13. Error Decay

  14. Platelet Approximation Theory Error decay rates: • Fourier: O(m-1/2) • Wavelets: O(m-1) • Wedgelets: O(m-1) • Platelets: O(m-min(a,b))

  15. A Piecewise Constant Tree

  16. A Piecewise Linear Tree

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

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

  19. Algorithm in Action

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

  21. Penalization

  22. Density Estimation - Blocks

  23. Density Estimation - Heavisine

  24. Density Estimation - Bumps

  25. Density Estimation Simulation

  26. Medical Imaging Results

  27. Inverse Problems Goal: estimate m from observations x ~ Poisson(Pm) EM algorithm (Nowak and Kolaczyk, ’00):

  28. Confocal Microscopy: An Inverse Problem

  29. Platelet Performance

  30. Confocal Microscopy: Real Data

  31. Hellinger Loss • Upper bound for affinity (like squared error) • Relates expected error to Lp approximation bounds

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

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

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

  35. 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

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

  37. Approximation Theory Results

  38. Why don’t we just find the MLE?

  39. MPLE Algorithm (1D)

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

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

  42. 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

  43. Astronomical Imaging

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