CoarsetoFine Image Reconstruction

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Coarse-to-FineImage Reconstruction

• Rebecca Willett
• In collaboration with
• Robert Nowak and Rui Castro

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Poisson Data 14 photons/pixel MSE 0.0169
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Iterative reconstruction
(Willett Nowak, IEEE-TMI 03)
E-Step Compute conditional expectation of new
noisy image estimate given data and current
image estimate
Traditional Shepp-Vardi M-Step Maximum
Likelihood Estimation
Improved M-Step Complexity Regularized
Multiscale Poisson Denoising
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Wedgelet-based tomography
Shepp-Logan
MLE
Wedgelet-based reconstruction
Jeff Fesslers PWLS
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Tomography
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A simple image model
piecewise constant 2-d function with smooth
edges
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Measurement model
Access only to n noisy pixels
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Image space
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Kolmogorov metric entropy
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Dudley 74
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Minimax lower bound
approx. err
estimation err.
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Adaptively pruned partitions
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Tree pruning estimation
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Partitions and Estimators
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Complexity Regularization and the Bias-Variance
Trade-off
Complexity penalized estimator
set of all possible tree prunings
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The Li-Barron bound
Li Barron, 00 Nowak Kolaczyk, 01
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The Kraft inequality
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0
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0000
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Estimating smooth contours - Haar
Decorate each partition set with a constant
squared approximation error
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Approximating smooth contours - wedgelets
Donoho 99

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Approximating smoother contours
(Donoho 99)
Haar Wavelet Partition
Original Image
gt 850 terms
lt 370 terms
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Estimating smoother contours - wedgelets
Use wedges and decorate each partition set with a
constant
squared approximation error

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The problem with estimating smooth contours
Haar-based estimation
Wedgelet estimation
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Computational implications
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A solutionCoarse-to-fine model selection
two-step process involves search first over
coarse model space
from which one is selected
space of all signal models is very large
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Coarse-to-fine model selection
second step involves search over small subset of
models
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C2F wedgelets two-stage optimization
Start with a uniform partition
Stage 1 Adapt partition to the data
by pruning
Stage 2 Only apply wedges in the
small boxes that remain
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C2F wedgelets two-stage optimization
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Error analysis of two-stage approach
(Castro, Willett, Nowak, ICASSP 04)
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Controlling variance in the preview stage

• Start with a coarse partition in the first stage
• lowers the variance of the coarse resolution
  estimate
• with high probability, pruned coarse partition
  close to optimal coarse partition
• unpruned boxes at this stage indicate edges or
  boundaries

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Controlling bias in the preview stage

• Bias becomes large if a square containing a
  boundary fragment is pruned in the first stage
  (this may happen if a boundary is close to the
  side of the squares)
• Solution
• Compute TWO coarse partitions - one normal, and
  one shifted
• Refine any region unpruned in either or both
  shifts

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Computational implications
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Main result in action
noisy data MSE 0.0052
stage 1 result MSE 0.1214
stage 2 result O(n7/6), MSE 0.00046
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C2F limitations The ribbon
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C2F and other greedy methods
Matching pursuit
Boosting
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More general image models
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Platelet Approximation Theory
Twice continuously differentiable
Twice continuously differentiable

• m-term approximation error decay rate
• Fourier O(m-1/2)
• Wavelets O(m-1)
• Wedgelets O(m-1)
• Platelets O(m-2)
• Curvelets O(m-2)

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Confocal microscopy simulation
Noisy Image
Haar Estimate
Platelet Estimate
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C2F limitations complex images

• Images are edges many images consist almost
  entirely of edges
• C2F model still appropriate for many
  applications
• nuclear medicine
• feature classification
• temperature field estimation

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C2F in multiple dimensions
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Final remarks and ongoing work

• Careful greedy methods can perform as well as
  exhaustive searches, both in theory and practice
• Coarse-to-fine estimation dramatically reduces
  computational complexities
• Similar ideas can be used in other scenarios
• Reduce the amount of data required (e.g., active
  learning and adaptive sampling)
• Reduce number of bits required to encode model
  locations in compression schemes