Image Deconvolution by Multiscale Methods Jean-Luc Starck Service d'Astrophysique, CEA-Saclay, France. [email protected] http://jstarck.free.fr

Collaborators: Fionn Murtagh, Eric Pantin, Mai Nguyen, Emmanuel Candes, David Donoho, Albert Bijaoui.

DECONVOLUTION METHODS IN ASTRONOMY Wiener Richardson Lucy method

Noise amplification

Maximum Entropy Method

Problem to restore stars

CLEAN Method

Problem to restore extended sources

SIGNAL PROCESSING DOMAIN

Markov Random Field

Multiscale Methods . The bi-orthogonal wavelet transfrom.

. The undecimated wavelet transfrom (à trous algorithm). . The isotropic wavelet transfrom (à trous algorithm). . The Multiscale Median Transform . New Constructions: - Ridgelet - Curvelet - Bandelet - Contourlet

The Othogononal Wavelet Transform (OWT) s l =

k

c J ,k Ë J , l k ƒ

J k

j=1

Í

j ,l

k wj ,k

Transformation H

2

C0 G c j ƒ1=

k

wjƒ 1=

k

2

h k

2l c j , k

g k

2l c j , k

C1 W1

Reconstruction: cj

,l = 2

k

– – c j ƒ 1, k h l ƒ 2k ƒ w j ƒ 1, k g l ƒ 2k

H

2

C2

G

2

W2

G H G

H

H

G

H

G

HH

GH

Smooth

Vertical

HG

GG

Horizontal

Diagonal

Undecimated Wavelet Transform

J

Isotropic Undecimated Wavelet Transform: I

x ,y =cJ x ,y ƒ

w j , x ,y j =1

J

Multiscale Median Transform:

I x ,y =cJ x ,y ƒ

w j , x ,y j =1

Multiresolution CLEAN (Wakker and Schwarz, 1998) Wavelet CLEAN (Starck and Bijaoui, 1994) Image

PSF WT

WT

CLEAN CLEAN CLEAN CLEAN CLEAN

Solution

Ideal PSF B

Reconstruction

WT

* * * * *

u ,v

FFT

º w c , j j j

B u ,v

- u ,v

2

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

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.

f=

i

·i Í

i

·i = f ,Í

i

Wavelet

Curvelet

Width = Length_