Multiscale Analysis for Intensity and Density Estimation

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Multiscale Analysis for Intensity and Density
Estimation

• Rebecca Willetts MS Defense
• Thanks to Rob Nowak, Mike Orchard,
• Don Johnson, and Rich Baraniuk
• Eric Kolaczyk and Tycho Hoogland

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Poisson and Multinomial Processes
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Why study Poisson Processes?

• Astrophysics

Network analysis
Medical Imaging
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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)
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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?
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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

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Approximating Besov Functions with Piecewise
Polynomials
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Approximation with Platelets
Consider approximating this image
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E.g. Haar analysis

• Terms 2068, Params 2068

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Wedgelets
Haar Wavelet Partition
Original Image
Wedgelet Partition
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E.g. Haar analysis with wedgelets
Terms 1164, Params 1164
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E.g. Platelets
Terms 510, Params 774
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Error Decay
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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))

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A Piecewise Constant Tree
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A Piecewise Linear Tree
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Maximum Penalized Likelihood Estimation
Goal Maximize the penalized likelihood
So the MPLE is
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The Algorithm

• Const Estimate

• Wedge Estimate

Data

• Platelet Estimate

• Wedged Platelet Estimate

• Inherit from finer scale

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Algorithm in Action
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Penalty Parameter
Penalty parameter balances between fidelity to
the data (likelihood) and complexity
(penalty). g 0 ? Estimate is MLE g
?? ? Estimate is a constant
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Penalization
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Density Estimation - Blocks
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Density Estimation - Heavisine
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Density Estimation - Bumps
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Density Estimation Simulation
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Medical Imaging Results
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Inverse Problems

• Goal estimate m from observations
• x Poisson(Pm)
• EM algorithm
• (Nowak and Kolaczyk, 00)

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Confocal Microscopy An Inverse Problem
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Platelet Performance
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Confocal Microscopy Real Data
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Hellinger Loss

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

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Bound on Hellinger Risk
(follows from Li Barron 99)
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Bounding the KL

• We can show
• Recall approximation result
• Choose optimal d

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Near-optimal Risk

• Maximum risk within logarithmic factor of minimum
  risk
• Penalty structure effective

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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

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Future Work
Major Contributions

• Risk analysis for piecewise polynomials
• Platelet representations and approximation theory

• Shift-invariant methods
• Fast algorithms for wedgelets and platelets
• Risk Analysis for platelets

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Approximation Theory Results
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Why dont we just find the MLE?
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MPLE Algorithm (1D)
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Multiscale Likelihood Factorization

• Probabilistic analogue to orthonormal wavelet
  decomposition
• Parameters ? ? wavelet coefficients
• Allow MPLE framework, where penalization based on
  complexity of underlying partition

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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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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
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Astronomical Imaging