Multiscale PhotonLimited Solar Image Analysis

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Multiscale Photon-Limited Solar Image Analysis

• Rebecca Willett

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Photon-limited imaging of spatially and
temporally varying phenomena

• Complicated signal behavior
• (difficult to model a priori)
• Noisy and indirect measurements
• Limited system resources
• Image acquisition time
• Processing facilities

Priest et al, 1998
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Richardson-Lucy performance on photon-limited
image 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)
Solar data
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Wavelet thresholding
Solar image
Wavelet coefficients of solar image
Wavelet coefficient magnitude
Sorted wavelet index
Approximation using wavelet coeffs. gt 0.3
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Wavelet thresholding for denoising
Noisy solar image
Wavelet coefficients of noisy solar image
Noise wavelet coefficient magnitude
Sorted wavelet index
Estimate using wavelet coeffs. gt 0.3
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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

• Key idea
• Wavelet coefficients of noise will be small with
  very high probability.

If we had N obs. of 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)

• EMC2
• (Esch, Connors, Karovska, van Dyk 2004)
• hyperprior distribution on parameters ?
• use MCMC to draw samples from posterior
• Estimate posterior mean
• Estimate posterior variance

Timmermann Nowak, 1999Kolaczyk, 1999
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MMI-MAP estimation results
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Haar tree pruning estimation
Kolaczyk Nowak, 2004
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Haar tree pruning estimation
pruning aggregation data fusion robustness
to noise
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Partitions and estimators
Complexity penalized estimator
set of all possible partitions
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Haar tree pruning results
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Haar tree pruning 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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Implications for Image Acquisition Times
Willett Nowak, IEEE-TMI 03 Nowak, Mitra,
Willett, JSAC 04 Willett Nowak, submitted
IEEE-Info.Th. 05
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Platelet results
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á trous wavelet transform

• Compute á trous wavelet coefficients.
• Compute variance stabilizing transform of each á
  trous coefficient
• Use level-dependent, wavelet-dependent,
  location-dependent thresholds, set from
  coefficient histograms
• Compute inverse á trous transform

Starck Murtagh book, 2nd ed., unpublished
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Observations 1.63
Truth
Corrected thresholds 0.250
Wavelets Anscombe 0.441
Wavelet thresholding 0.395
Platelets 0.234
Haar tree pruning 0.236
MMI - MAP 0.331
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Wavelet thresholding
Observations
MMI - MAP
Corrected thresholds
Wavelets Anscombe
A trous
Platelets
Haar tree pruning
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Wavelet thresholding
Observations
MMI - MAP
Corrected thresholds
Wavelets Anscombe
A trous
Platelets
Haar tree pruning
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Photon-limited solar image reconstruction

• For very low photon counts, specialized
  reconstruction methods are essential
• Theory supports use of multiscale estimation
• Papers and software online at http//www.ee.duke.e
  du/willett/

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Translation invariance

• Approximate with Haar wavelets as on previous
  slide

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

Avoid this difficulty by averaging over all
different possible shiftsthis can be done
quickly with undecimated (redundant) wavelets
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