New Multiscale Methods for 2D and 3D Astronomical Data Set

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1 New Multiscale Methods for 2D and 3D Astronomical Data Set Jean Luc Starck Dapnia/SEDI SAP, CEA Saclay, France.

2 Collaborators: D.L. Donoho, O. Levi, Department of Statistics, Stanford E. Candès, department of Applied Mathematics, California Institute of Technology P. Querre, R. Teyssier, CEA, V. Martinez, Valencia Observatori The Curvelet Transform for Image Denoising, IEEE Transaction on Image Processing, 11, 6, Astronomical Image Representation by the Curvelet Transform, Astronomy and Astrophysics, in press, Experiments: stat.stanford.edu/~jstarck

3 New Transforms 2D Ridgelet Transform 2D Curvelet Transform 3D Ridgelet Transform 3D Beamlet Transform

4 The wavelet transform is used in many astronomical domains: galaxies counting: large scale structures analysis, void detection,... Gamma: Gamma ray burst detection X ray images (ROSAT, XMM, CHANDRA): extended sources and filaments detection Optics: asteroid and planetary ring detection, deconvolution Infrared (ISO project): calibration, source detection, deconvolution,... Cosmic Microwave Background (Planck): fluctuation analysis Radio: aperture synthesis image deconvolution and for all kind of astrophysics, from solar to CMB! ADS Abstract Service keyword in title: wavelet multiscale multiresolution more than 500 papers!

5 Problems related to the WT 1) Edges representation: if the WT performs better than the FFT to represent edges in an image, it is still not optimal. 2) There is only a fixed number of directional elements independent of scales. 3) Limitation of existing scale concepts: there is no highly anisotropic elements.

6 SNR = 0.1

8 Undecimated Wavelet Filtering (3 sigma)

9 Ridgelet Filtering (5sigma)

10 Non Linear Approximation Wavelet Curvelet

11 Rate of Approximation Suppose we have a function f which has a discontinuity across a curve, and which is otherwise smooth, and consider approximating f from the best m terms in the Fourier expansion. The squarred error of such an m term expansion obeys: f f m F In a wavelet expansion, we have 2 m 1 2, m f f m W 2 m 1, m In a curvelet expansion (Donoho and Candes, 2000), we have f f m C 2 log m 3 m 2, m

12 Ridgelet Transform Ridgelet Transform (Candes, 1998): R f a,b,ϑ = ψ a,b, ϑ x f x dx 1 Ridgelet function: ψ a,b, ϑ x =a 2 ψ x cos ϑ x sin ϑ b 1 2 a The function is constant along lines. Transverse to these ridges, it is a wavelet.

13 The ridgelet coefficients of an object f are given by analysis of the Radon transform via: R f a,b,ϑ = R f ϑ,t ψ t b a dt FFT IMAGE FFT2D FFF1D 1 WT1D Radon Transform Ridgelet Transform Angle Frequency

14 Local Ridgelet Transform The ridgelet transform is optimal to find only lines of the size of the image. To detect line segments, a partitioning must be introduced. The image is decomposed into blocks, and the ridgelet transform is applied on each block. Partitioning Image Ridgelet transform

15 In practice, we use overlap to avoid blocking artifacts. Smooth partitioning The partitioning introduces a redundancy, as a pixel belongs to 4 neighboring blocks.

16 Line detection by the ridgelet transform

17 NEWTON/XMM Image of the supernovae SN1604 Ridgelet Filtering

18 The Curvelet Transform The curvelet transform opens us the possibility to analyse an image with different block sizes, but with a single transform. The idea is to first decompose the image into a set of wavelet bands, and to analyze each band by a ridgelet transform. The block size can be changed at each scale level. à trous wavelet transform partitionning ridgelet transform. Radon Transform. 1D Wavelet transform

19 IMAGE WT2D FFT FFT2D FFF1D 1 WT1D Radon Transform Ridgelet Transform Angle Frequency

20 CONTRAST ENHANCEMENT Modified curvelet coefficient Curvelet coefficient

21 Contrast Enhancement

22 F

23 FILTERING

24 3D MULTISCALE TRANSFORMS 1) 3D WAVELET TRANSFORM: Isotropic Structures 2) 3D RIDGELET TRANSFORM: Sheet like Structures 3) 3D BEAMLET TRANSFORM: Filaments

33 Wavelet Beamlet Ridgelet Beamlet Ridgelet Wavelet Ridgelet Beamlet Wavelet

37 Z=5 Z=3 Z=1 Z=2 Z=0.5 Z=0

Image Processing by the Curvelet Transform

Image Processing by the Curvelet Transform Jean Luc Starck Dapnia/SEDI SAP, CEA Saclay, France. jstarck@cea.fr http://jstarck.free.fr Collaborators: D.L. Donoho, Department of Statistics, Stanford E. Candès,