Multiscale Analysis of Photon-Limited Astronomical Images

Rebecca Willett

Photon-limited astronomical imaging

NG2997 Saturn

Richardson-Lucy performance on Saturn deblurring

Iteration Number

MSE of

deconvolved

estimate

Error performance of standard R-L

algorithm

Error performance of R-L algorithm with regularization

Main question: how to best perform Poisson intensity estimation?

Test data

Saturn Rosetta (Starck)

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

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Wavelet thresholding

Sorted wavelet indexWavelet coefficient magnitude

Wavelet coefficients of Saturn

image

Approximation using wavelet coeffs. >

0.3

Saturn image

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Wavelet thresholding for denoising

Sorted wavelet indexNoise wavelet coefficient magnitude

Wavelet coefficients of Noisy

Saturn image

Estimate using wavelet coeffs. >

0.3

Noisy Saturn image

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 1/3Avoid this difficulty by averaging over all different possible shifts;

this can be done quickly with undecimated (redundant) wavelets

Wavelet thresholding resultsHaar

wavelets

Variance stabilizing transforms

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Anscombe 1948

Anscombe transform resultsHaar

wavelets

Kolaczyk’s corrected Haar thresholds

Kolaczyk 1999

Basic idea:Keep wavelet coeffs which correspond to signal;Threshold 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

Corrected Haar threshold results

Multiplicative Multiscale Innovation Models (aka Bayesian Multiscale Models)

Timmermann & Nowak, 1999Kolaczyk, 1999

• Recursively subdivide image into squares• Let { denote the ratio between child and parent intensities

• Knowing {Knowing {• Estimate {} from empirical estimates based on counts in each partition square

0,0,0 X0,0,0

1,0,0 X1,0,01,1,0 X1,1,01,0,1 X1,0,11,1,1 X1,1,1

MMI-MAP estimation

Basic idea: place Dirichlet prior distribution with parameters {} on {estimate {by maximizing posterior distribution

0,0,0 X0,0,0

1,0,0 X1,0,01,1,0 X1,1,01,0,1 X1,0,11,1,1 X1,1,1

MMI-MAP estimation results

MMI-EMC2Before (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

0,0,0 X0,0,0

1,0,0 X1,0,01,1,0 X1,1,01,0,1 X1,0,11,1,1 X1,1,1

MMI - Complexity Regularization

Kolaczyk & Nowak, 2004

MMI - Complexity Regularization

pruning = aggregation = data fusion = robustness to noise

Complexity penalized estimator:

set of all possible partitions

Partitions selection

|P|

likelihood

penalty(prior)

MMI-Complexity regularization results

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!

Platelet estimation

Donoho, Ann. Stat. ‘99Willett & Nowak, IEEE-TMI ‘03

Willett & Nowak, submitted to IEEE-Info.Th. ‘05

Platelet theory

No other method can do significantly better asymptotically for this (smoother) class of images!

Platelet results

á trous wavelet transform

Holschneider 1989Starck 2002

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

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)

á trous results

Truth Observations; 1.74

Corrected thresholds; 0.198Wavelets + Anscombe; 0.465Wavelet thresholding; 0.325

Platelets; 0.163MMI - Complexity Reg.; 0.173MMI - MAP; 0.245

Observations Wavelet thresholding

MMI - MAPCorrected thresholdsWavelets + Anscombe

A trousPlateletsMMI - Complexity Reg.

Observations Wavelet thresholding

MMI - MAPCorrected thresholdsWavelets + Anscombe

A trousPlateletsMMI - Complexity Reg.

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