Multiscale Analysis for Intensity and Density Estimation Rebecca
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
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)
Beyond Wavelets Multiresolution analysis is a powerful tool, but what about… Non-Gaussian problems? Edges? Image Edges? Nongaussian noise? Inverse problems? Piecewise polynomial- and plateletbased methods address these issues.
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, b))
A Piecewise Constant Tree
A Piecewise Linear Tree
Maximum Penalized Likelihood Estimation Goal: Maximize the penalized likelihood So the MPLE is
The Algorithm Data n Const Estimate n Wedge Estimate n Platelet Estimate n n Wedged Platelet Estimate Inherit from finer scale
Algorithm in Action
Penalty Parameter Penalty parameter balances between fidelity to the data (likelihood) and complexity (penalty). g = 0 Estimate is MLE: g Estimate is a constant:
Penalization
Density Estimation - Blocks
Density Estimation - Heavisine
Density Estimation - Bumps
Density Estimation Simulation
Medical Imaging Results
Inverse Problems Goal: estimate m from observations x ~ Poisson(Pm) 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 Lp approximation bounds
Bound on Hellinger Risk (follows from Li & Barron ’ 99) KL distance Approximation error Estimation error
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
Major Contributions • Risk analysis for piecewise polynomials • Platelet representations and approximation theory Future Work • Shift-invariant methods • Fast algorithms for wedgelets and platelets • Risk Analysis for platelets
Approximation Theory Results
Why don’t we just find the MLE?
MPLE Algorithm (1 D)
Multiscale Likelihood Factorization Ø Ø Ø 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, 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.
Accurate Model Complexity Regularization Parsimonious Model 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
- Slides: 44