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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

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

why study poisson processes
Why study Poisson Processes?

Astrophysics

Network analysis

Medical Imaging

multiresolution analysis
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
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
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
approximation with platelets
Approximation with Platelets

Consider approximating this image:

e g haar analysis
E.g. Haar analysis

Terms = 2068, Params = 2068

wedgelets
Wedgelets

Haar Wavelet Partition

Original Image

Wedgelet Partition

e g haar analysis with wedgelets
E.g. Haar analysis with wedgelets

Terms = 1164, Params = 1164

e g platelets
E.g. Platelets

Terms = 510, Params = 774

platelet approximation theory
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))
maximum penalized likelihood estimation
Maximum Penalized Likelihood Estimation

Goal: Maximize the penalized likelihood

So the MPLE is

the algorithm
The Algorithm
  • Const Estimate
  • Wedge Estimate

Data

  • Platelet Estimate
  • Wedged Platelet Estimate
  • Inherit from finer scale
penalty parameter
Penalty Parameter

Penalty parameter balances between

fidelity to the data (likelihood) and complexity (penalty).

g = 0 Estimate is MLE:

g   Estimate is a constant:

inverse problems
Inverse Problems

Goal: estimate m from observations

x ~ Poisson(Pm)

EM algorithm

(Nowak and Kolaczyk, ’00):

hellinger loss
Hellinger Loss
  • Upper bound for affinity

(like squared error)

  • Relates expected error to Lp approximation bounds
bound on hellinger risk

KL distance

Approximation error

Estimation

error

Bound on Hellinger Risk

(follows from Li & Barron ’99)

bounding the kl
Bounding the KL
  • We can show:
  • Recall approximation result:
  • Choose optimal d
near optimal risk
Near-optimal Risk
  • Maximum risk within logarithmic factor of minimum risk
  • Penalty structure effective:
conclusions
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

Future Work

Major Contributions

multiscale likelihood factorization
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
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.
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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