Multiscale Analysis of Photon-Limited Astronomical Images

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Multiscale Analysis of Photon-Limited
Astronomical Images

• Rebecca Willett

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Photon-limited astronomical imaging
NG2997
Saturn
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Richardson-Lucy performance on Saturn deblurring
Error performance of standard R-L algorithm
MSE of deconvolvedestimate
Iteration Number
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Main question how to best perform Poisson
intensity estimation?
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Test data
Rosetta (Starck)
Saturn
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Methods reviewed in this talk

• Wavelet thresholding
• Variance stabilizing transforms
• Corrected Haar wavelet thresholds
• Multiplicative Multiscale Innovation models
• MAP estimation
• EMC2 estimation
• Complexity Regularization
• Platelets
• á trous wavelet thresholding

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Wavelet thresholding
Saturn image
Wavelet coefficients of Saturn image
Wavelet coefficient magnitude
Sorted wavelet index
Approximation using wavelet coeffs. gt 0.3
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Wavelet thresholding for denoising
Noisy Saturn image
Wavelet coefficients of Noisy Saturn image
Noise wavelet coefficient magnitude
Sorted wavelet index
Estimate using wavelet coeffs. gt 0.3
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Translation invariance

1. Approximate with Haar wavelets as on previous
   slide

1. Shift image by 1/3 in each direction
2. approximate as before
3. shift back by 1/3

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Wavelet thresholding results
Haarwavelets
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Variance stabilizing transforms
Anscombe 1948
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Anscombe transform results
Haarwavelets
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Kolaczyks corrected Haar thresholds

• Basic idea
• Keep wavelet coeffs which correspond to
  signalThreshold wavelet coeffs which correspond
  to noise (or background)

If we had Gaussian noise (variance ?2) and no
signal
(j,k)th Gaussian wavelet coeff.
For Poisson noise, design similar bound for
background ?0 (noise)
(j,k)th Poisson wavelet coeff.
Threshold becomes
Background intensity level
Kolaczyk 1999
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Corrected Haar threshold results
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Multiplicative Multiscale Innovation Models (aka
Bayesian Multiscale Models)
Timmermann Nowak, 1999Kolaczyk, 1999
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MMI-MAP estimation
Basic idea place Dirichlet prior distribution
with parameters ? on ???estimate ???by
maximizing posterior distribution
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MMI-MAP estimation results
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MMI-EMC2

• Before (with MMI-MAP)
• place Dirichlet prior distribution with
  parameters ? on ???
• user sets parameters ?
• estimate ???by maximizing posterior distribution
• Now (with MMI-EMC2)
• place hyperprior distribution on parameters ?
• user only controls few hyperparameters
• prior information about intensity built into
  hyperprior
• use MCMC to draw samples from posterior
• Estimate posterior mean
• Estimate posterior variance

Esch, Connors, Karovska, van Dyk 2004
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MMI - Complexity Regularization
Kolaczyk Nowak, 2004
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MMI - Complexity Regularization
pruning aggregation data fusion robustness
to noise
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Partitions selection
Complexity penalized estimator
set of all possible partitions
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MMI-Complexity regularization results
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MMI-Complexity regularization theory
No other method can do significantly better
asymptotically for this class of images! This
theory also supports other Haar-wavelet based
methods!
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Platelet estimation
Donoho, Ann. Stat. 99 Willett Nowak, IEEE-TMI
03
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Platelet theory
No other method can do significantly better
asymptotically for this (smoother) class of
images!
Willett Nowak, submitted to IEEE-Info.Th. 05
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Platelet results
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á trous wavelet transform
1. Redefine wavelet as difference between scaling
functions at successive levels
2. Compute coeffs. at one level by filtering
coeffs at next finer scale
3. This means synthesis (getting image back from
wavelet coeffs.) is simple addition
Holschneider 1989Starck 2002
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Intensity estimation with á trous wavelets

• Method 1(Classical)
• Compute Anscombe transform of data
• Perform á trous wavelet thresholding as if iid
  Gaussian noise
• (same problems as other Anscombe-based approaches
  for very few photon counts)

• Method 2(Starck Murtagh, 2nd ed., unpublished)
• Compute variance stabilizing transform of each á
  trous coefficient
• Use level-dependent, wavelet-dependent,
  location-dependent thresholds
• (result on next slide)

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á trous results
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Observations 1.74
Truth
Corrected thresholds 0.198
Wavelets Anscombe 0.465
Wavelet thresholding 0.325
Platelets 0.163
MMI - Complexity Reg. 0.173
MMI - MAP 0.245
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Wavelet thresholding
Observations
MMI - MAP
Corrected thresholds
Wavelets Anscombe
A trous
Platelets
MMI - Complexity Reg.
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Wavelet thresholding
Observations
MMI - MAP
Corrected thresholds
Wavelets Anscombe
A trous
Platelets
MMI - Complexity Reg.
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Method Speed Effectiveness
Wavelet thresholding Fast Poor
Wavelets Anscombe Fast Poor
Corrected thresholds Fast Medium
MMI-MAP Fast Medium
MMI-EMC2 Medium High significance maps!
MMI-Complexity regularization Fast High
Platelets Medium-slow High
A trous Medium High
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Poisson inverse problems
P?
?
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