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

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 w i. 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)A two-stage or two-scale FWT analysis bank and (b)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 p.10http:// Introduction of Wavelet Transform(18/18) Comparison of Transformations

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

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 (1) Transformation (UWT) (2) Detection by wavelet-domain hypothesis test (3) 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)

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

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; ; 1108, R.C. Gonzalez and R.E. Woods, “Digital Image Processing 2nd Edition, Chapter 7”,Prentice Hall, 2002