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 Medical Imaging

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

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?

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

Approximating Besov Functions with Piecewise Polynomials

Approximation with Platelets Consider approximating this image:

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

Wedgelets Original Image Haar Wavelet Partition Wedgelet Partition

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

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

Error Decay

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

A Piecewise Constant Tree

A Piecewise Linear Tree

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

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

Algorithm in Action

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

Penalization

Density Estimation - Blocks

Density Estimation - Heavisine

Density Estimation - Bumps

Density Estimation Simulation

Medical Imaging Results

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

Confocal Microscopy: An Inverse Problem

Platelet Performance

Confocal Microscopy: Real Data

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

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

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

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

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

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

Approximation Theory Results

Why don’t we just find the MLE?

MPLE Algorithm (1D)

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

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.

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

Astronomical Imaging