Image restoration, noise models, detection, deconvolution Outline : Image formation model Noise models Inverse problems in image processing Bayesian approaches Wiener filtering Maximum-entropy methode Shrinkage, Sparsity Applications os multiscale representations

NOISE MODELING For a positive coefficient: For a negative coefficient Given a threshold t, if P > t, the coefficient could be due to the noise. On the other habd, if P < t, the coefficient cannot be due to the noise, and a significant coefficient is detected.

NGC2997 MULTIRESOLUTION SUPPORT

DECONVOLUTION SIMULATION LUCY PIXON Wavelet

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.

Continuous Ridgelet Transform Ridgelet Transform (Candes, 1998): Ridgelet function: The function is constant along lines. Transverse to these ridges, it is a wavelet.

Digital Ridgelet Transform :

Example application of Ridgelets

SNR = 0.1

Undecimated Wavelet Filtering (3 sigma)

Ridgelet Filtering (5sigma)

Line detection by the ridgelet transform

NEWTON/XMM Image of the supernovae SN1604 Ridgelet Filtering

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. Algorithm :

The Curvelet Transform Wavelet Curvelet Width = Length^2

The Curvelet Transform J.L. Starck, E. Candès and D. Donoho, "Astronomical Image Representation by the Curvelet Transform, Astronomy and Astrophysics, 398, 785--800, 2003.

CURVELET FILTERING NOISE MODELING For a positive coefficient: For a negative coefficient Given a threshold t: if P > t, the coefficient could be due to the noise. if P < t, the coefficient cannot be due to the noise, and a significant coefficient is detected. Hard Thresholding:

Curvelet Undecimated WT Lena + Gaussian Noise

FILTERING

Algorithm :

a) Simulated image (gaussians+lines) b) Simulated image + noise c) A trous algorithm d) Curvelet transform e) coaddition c+d f) residual = e-b

a) A370 b) a trous c) Ridgelet + Curvelet Coaddition b+c

a) NGC2997 b) atrous c) Ridgelet d) Coaddition b+c

Galaxy SBS 0335-052 Ridgelet Curvelet A trous WT

= + = + Galaxy SBS 0335-052 10 micron GEMINI-OSCIR CTM Curvelet A trous WT Ridgelet + Residual CTM = Rid+Curvelet + Clean Data

Noise Standard Deviation PSNR Lena Curvelet Decimated wavelet Undecimated wavelet Noise Standard Deviation

Barbara Curvelet Decimated wavelet Undecimated wavelet

Curvelet Wavelet Curvelet

RESTORATION: HOW TO COMBINE SEVERAL MULTISCALE TRANSFORMS ? The problem we need to solve for image restoration is to make sure that our reconstruction will incorporate information judged as significant by any of our representations. Very High Quality Image Restoration, in Signal and Image Processing IX, San Diego, 1-4 August, 2001, Eds Laine, Andrew F.; Unser, Michael A.; Aldroubi, Akram, Vol. 4478, pp 9-19, 2001. Notations: Consider K linear transforms and the coefficients of x after applying : .

We propose solving the following optimization problem: min Complexity_penalty , subject to Where C is the set of vectors which obey the linear constraints: positivity constraint is significant The second constraint guarantees that the reconstruction will take into account any pattern which is detected by any of the K transforms.

DECONVOLUTION: We propose solving the following optimization problem: min Complexity_penalty , subject to Where C is the set of vectors which obey the linear constraints: positivity constraint is significant The second constraint guarantees that the reconstruction will take into account any pattern which is detected by any of the K transforms.

Multiscale Transforms Critical Sampling Redundant Transforms Pyramidal decomposition (Burt and Adelson) (bi-) Orthogonal WT Undecimated Wavelet Transform Lifting scheme construction Isotropic Undecimated Wavelet Transform Wavelet Packets Complex Wavelet Transform Mirror Basis Steerable Wavelet Transform Dyadic Wavelet Transform Nonlinear Pyramidal decomposition (Median) New Multiscale Construction Contourlet Ridgelet Bandelet Curvelet (Several implementations) Finite Ridgelet Transform Platelet (W-)Edgelet Adaptive Wavelet

The Curvelet Transform Wavelet Curvelet Width = Length^2

The Curvelet Transform Wavelet Curvelet Width = Length^2