Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal

1. Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal 國立交通大學電子研究所 張瑞男 2008.12.11

2. Outline • Introduction of Wavelet Transform • Variance Stabilization Transform of a Filtered Poisson Process (VST) • Denoising by Multi-scale VST + Wavelets • Ridgelets & Curvelets • Conclusions

3. Introduction of Wavelet Transform(10/18) • Multiresolution Analysis • The spanned spaces are nested: • Wavelets span the differences between spaces wi. Wavelets and scaling functions should be orthogonal: simple calculation of coefficients.

4. Introduction of Wavelet Transform(11/18)

5. Introduction of Wavelet Transform(12/18) • Multiresolution Formulation. ( Scaling coefficients) ( Wavelet coefficients)

6. Introduction of Wavelet Transform(13/18) • Discrete Wavelet Transform (DWT) Calculation: • Using Multi-resolution Analysis:

7. Introduction of Wavelet Transform(14/18) • Basic idea of Fast Wavelet Transform (Mallat’s herringbone algorithm): • Pyramid algorithm provides an efficient calculation. • DWT (direct and inverse) can be thought of as a filtering process. • After filtering, half of the samples can be eliminated: subsample the signal by two. • Subsampling: Scale is doubled. • Filtering: Resolution is halved.

8. Introduction of Wavelet Transform(15/18) • A two-stage or two-scale FWT analysis bank and • its frequency splitting characteristics.

9. Introduction of Wavelet Transform(16/18) • Fast Wavelet Transform • Inverse Fast Wavelet Transform

10. Introduction of Wavelet Transform(17/18) A two-stage or two-scale FWT-1 synthesis bank.

11. From http://www.cerm.unifi.it/EUcourse2001/Gunther_lecturenotes.pdf, p.10 Introduction of Wavelet Transform(18/18) • Comparison of Transformations

12. VST of a Filtered Poisson Process(1/4) λ：intensity • Poisson process • Filtered Poisson process assume • Seek a transformation

13. VST of a Filtered Poisson Process(2/4) • Taylor expansion & approximation Solution

14. VST of a Filtered Poisson Process(3/4) • Square-root transformation • Asymptotic property • Simplified asymptotic analysis

15. VST of a Filtered Poisson Process(4/4) • Behavior of E[Z] and Var[Z]

16. Denoising by MS-VST + Wavelets(1/14) • Main steps • Transformation (UWT) • Detection by wavelet-domain hypothesis test • Iterative reconstruction (final estimation)

17. Denoising by MS-VST + Wavelets(2/14) • Undecimated wavelet transform (UWT)

18. Denoising by MS-VST + Wavelets(3/14) • MS-VST+Standard UWT

19. Denoising by MS-VST + Wavelets(4/14) • MS-VST+Standard UWT

20. Denoising by MS-VST + Wavelets(5/14) • Detection by wavelet-domain hypothesis test (hard threshold) p ：false positive rate (FPR) ：standard normal cdf

21. Denoising by MS-VST + Wavelets(6/14) • Iterative reconstruction (soft threshold) a constrained sparsity-promoting minimization problem

22. Denoising by MS-VST + Wavelets(7/14) • Iterative reconstruction hybrid steepest descent (HSD)

23. Denoising by MS-VST + Wavelets(8/14) significant coefficient original coefficient gradient component positive projection updated coefficient Iterative reconstruction hybrid steepest descent (HSD)

24. Denoising by MS-VST + Wavelets(9/14) • Algorithm of MS-VST + Standard UWT

25. Denoising by MS-VST + Wavelets(10/14) • Algorithm of MS-VST + Standard UWT

26. Denoising by MS-VST + Wavelets(11/14) • Applications and results Simulated Biological Image Restoration oringinal image observed photon-count image

27. Denoising by MS-VST + Wavelets(12/14) • Applications and results Simulated Biological Image Restoration denoised by Haar hypothesis tests MS-VST-denoised image

28. Denoising by MS-VST + Wavelets(13/14) • Applications and results Astronomical Image Restoration Galaxy image observed image

29. Denoising by MS-VST + Wavelets(14/14) • Applications and results Astronomical Image Restoration denoised by Haar hypothesis tests MS-VST-denoised image

30. Ridgelets & Curvelets (1/11) • Ridgelet Transform (Candes, 1998): • Ridgelet function: • The function is constant along lines. Transverse to these ridges, it is a wavelet.

31. Ridgelets & Curvelets (2/11) • The ridgelet coefficients of an object f are given by analysis of the Radon transform via:

32. Ridgelets & Curvelets (3/11) • Algorithm of MS-VST With Ridgelets

33. Ridgelets & Curvelets (4/11) • Results of MS-VST With Ridgelets Intensity Image Poisson Noise Image

34. Ridgelets & Curvelets (5/11) • Results of MS-VST With Ridgelets Intensity Image Poisson Noise Image

35. Ridgelets & Curvelets (6/11) • Results of MS-VST With Ridgelets denoised by MS-VST+UWT MS-VST + ridgelets

36. Ridgelets & Curvelets (7/11) Curvelets • Decomposition of the original image into subbands • Spatial partitioning of each subband • Appling the ridgelet transform

37. Ridgelets & Curvelets (8/11) • Algorithm of MS-VST With Curvelets

38. Ridgelets & Curvelets (9/11) • Algorithm of MS-VST With Curvelets

39. Ridgelets & Curvelets (10/11) • Results of MS-VST With Curvelets Natural Image Restoration Intensity Image Poisson Noise Image

40. Ridgelets & Curvelets (11/11) • Results of MS-VST With Curvelets Natural Image Restoration denoised by MS-VST+UWT MS-VST + curvelets

41. Conclusions • It is efficient and sensitive in detecting faint features at a very low-count rate. • We have the choice to integrate the VST with the multiscale transform we believe to be the most suitable for restoring a given kind of morphological feature (isotropic, line-like, curvilinear, etc). • The computation time is similar to that of a Gaussian denoising, which makes our denoising method capable of processing large data sets.

42. Reference • Bo Zhang, J. M. Fadili and J. L. Starck, "Wavelets, ridgelets, and curvelets for Poisson noise removal," IEEE Trans. Image Process., vol. 17, pp. 1093; 1093-1108; 1108, 07 2008. 2008. • R.C. Gonzalez and R.E. Woods, “Digital Image Processing 2nd Edition, Chapter 7”,Prentice Hall, 2002