1. Multiscale Analysis of Photon-Limited Astronomical Images Rebecca Willett

2. Photon-limited astronomical imaging NG2997Saturn

3. 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

4. Main question: how to best perform Poisson intensity estimation?

5. Test data Saturn Rosetta (Starck)

6. 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

7. Wavelet thresholding Sorted wavelet index Wavelet coefficient magnitude Wavelet coefficients of Saturn image Approximation using wavelet coeffs. > 0.3 Saturn image

8. Wavelet thresholding for denoising Sorted wavelet index Noise wavelet coefficient magnitude Wavelet coefficients of Noisy Saturn image Estimate using wavelet coeffs. > 0.3 Noisy Saturn image

9. 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 Avoid this difficulty by averaging over all different possible shifts; this can be done quickly with undecimated (redundant) wavelets

10. Wavelet thresholding results Haar wavelets

11. Variance stabilizing transforms Anscombe 1948

12. Anscombe transform results Haar wavelets

13. 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

14. Corrected Haar threshold results

15. Multiplicative Multiscale Innovation Models (aka Bayesian Multiscale Models) Timmermann & Nowak, 1999 Kolaczyk, 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 X 0,0,0 1,0,0 X 1,0,0 1,1,0 X 1,1,0 1,0,1 X 1,0,1 1,1,1 X 1,1,1

16. MMI-MAP estimation Basic idea: place Dirichlet prior distribution with parameters {  } on {  estimate {  by maximizing posterior distribution         0,0,0 X 0,0,0 1,0,0 X 1,0,0 1,1,0 X 1,1,0 1,0,1 X 1,0,1 1,1,1 X 1,1,1

17. MMI-MAP estimation results

18. 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         0,0,0 X 0,0,0 1,0,0 X 1,0,0 1,1,0 X 1,1,0 1,0,1 X 1,0,1 1,1,1 X 1,1,1

19. MMI - Complexity Regularization Kolaczyk & Nowak, 2004

20. MMI - Complexity Regularization pruning = aggregation = data fusion = robustness to noise

21. Complexity penalized estimator: set of all possible partitions Partitions selection |P| likelihood penalty (prior)

22. MMI-Complexity regularization results

23. 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!

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

25. Willett & Nowak, submitted to IEEE-Info.Th. ‘05 Platelet theory No other method can do significantly better asymptotically for this (smoother) class of images!

26. Platelet results

27. á trous wavelet transform Holschneider 1989 Starck 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

28. 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, 2 nd ed., unpublished) Compute variance stabilizing transform of each á trous coefficient Use level-dependent, wavelet-dependent, location-dependent thresholds (result on next slide)

29. á trous results

30. Truth Observations; 1.74 Corrected thresholds; 0.198Wavelets + Anscombe; 0.465Wavelet thresholding; 0.325 Platelets; 0.163MMI - Complexity Reg.; 0.173MMI - MAP; 0.245

31. Observations Wavelet thresholding MMI - MAPCorrected thresholdsWavelets + Anscombe A trousPlateletsMMI - Complexity Reg.

32. Observations Wavelet thresholding MMI - MAPCorrected thresholdsWavelets + Anscombe A trousPlateletsMMI - Complexity Reg.

33. MethodSpeedEffectiveness Wavelet thresholding FastPoor Wavelets + Anscombe FastPoor Corrected thresholds FastMedium MMI-MAPFastMedium MMI-EMC2MediumHigh; significance maps! MMI-Complexity regularization FastHigh PlateletsMedium-slowHigh A trousMediumHigh

34. Poisson inverse problems linear filter sumpenalized likelihood or MAP estimation m n P