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, StanfordE. Candès, department of Applied Mathematics, California Institute of Technology

Experiments: http://jstarck.free.fr http://www−stat.stanford.edu/~jstarck

The Astronomical Image Representation by the Curvelet Transform, Astronomy and Astrophysics, in press.

The Curvelet Transform for Image Denoising, IEEE Transaction on Image Processing, 11, 6, 2002.

Gray and Color Image Contrast Enhancement by the Curvelet Transform,IEEE Transaction on Image Processing, in press.

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.

Non Linear Approximation

Wavelet Curvelet

Width = Lengh²

Rate of Approximation

Suppose we have a function f which has a discontinuity across a curve, andwhich is otherwise smooth, and consider approximating f from the best m−terms in the Fourier expansion. The squarred error of such an m−termexpansion obeys:

fB fm

F 2∝mB1

2 , m→+∞

fB ˜fm

W 2∝mB1 , m→+∞

In a wavelet expansion, we have

fB fm

C 2∝ log m 3 mB2 , m→+∞

In a curvelet expansion (Donoho and Candes, 2000), we have

The Curvelet Transform

The curvelet transform can be seen as a combination of reversible transformations:

− à trous 2D isotropic wavelet transform− partitionning− ridgelet transform . Radon Transform . 1D Wavelet transform

NGC2997

I x,y =cJ

x,y +∑j=1

J

w j,x,yA trous algorithm:

PARTITIONING

The ridgelet coefficients of an object f are given by analysis of the Radon transform via: R

fa,b,ϑ =∫R f ϑ ,t ψ

tBb

adt

FFT2D

FFF1D−1

WT1D

Frequency

Ang

le

Radon Transform Ridgelet Transform

FFT

IMAGE

CONTRAST ENHANCEMENT

Curvelet coefficient

Modifiedcurvelet coefficient

yc

x,σ =1 if x<cσ

yc

x,σ =xBcσ

cσm

cσ

p

+2cσBx

cσif x<2cσ

yc

x,σ = m

x

p

if 2cσ≤x<m

yc

x,σ = m

x

s

if x>m

I=C R yc

C T I

Contrast Enhancement

F

Color Images

"R,G,B ===> L, U, V"We apply the curvelet transform to the three components"At each scale and at each position, we calculate:

" Coefficients correction

"Inverse curvelet transform"L,UV to RGB

e= cL

2+cu

2+cv

2

cL,

cu,

cv= y

ce c

L,y

ce c

u,y

ce c

v

NOISE MODELING

For a positive coefficient: P=Prob W >w

For a negative coefficient P=Prob W <w

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.

FILTERING

y=C Rδ C T yHard Thresholding:

Soft Thresholding:

δ c =c if c ≥ t

=0 if c< t

δ c =sgn c c B t +

FILTERING

Lenna

PSNR

Noise Standard Deviation

Curvelet

Decimated wavelet

Undecimated wavelet

Curvelet

Undecimated wavelet

Decimated wavelet

Barbara

Wavelet

Curvelet

Curvelet

The problem we need to solve for image restoration is to makesure that our reconstruction will incorporate information judged as significant by any of our representations.

Notations:

T1,

...,TK

Consider K linear transforms and the coefficients of x after applying : .T

K

αK

αk=T

ks, s=T

k

B1αk

RESTORATION: HOW TO COMBINE SEVERAL MULTISCALE TRANSFORMS ?

We propose solving the following optimization problem:

min Complexity_penalty , subject to s∈Cs

Where C is the set of vectors which obey the linear constraints:

s>0, positivity constraintT

ksBT

ks

l≤e, if T

ks

lis significant

The second constraint guarantees that the reconstruction will take into account any pattern which is detected by any of the K transforms.

We propose solving the following optimization problem:

min Complexity_penalty , subject to s∈Cs

Where C is the set of vectors which obey the linear constraints:

s>0, positivity constraintT

ksBT

kP∗s

l≤e, if T

ks

lis significant

The second constraint guarantees that the reconstruction will take into account any pattern which is detected by any of the K transforms.

DECONVOLUTION: s=P∗s+N

Morphological Component Analysis

Given a signal s, we assume that it is the result of a sparse linear combination of atoms from a known dictionary D.

s=∑γαγφγ

Or an approximate decomposition:

s=∑γαγφγ+R

φγ γ∈Γ

A dictionary D is defined as a collection of waveforms , and the goal isto obtain a representation of a signal s with a linear combination of a small number of basis such that:

38

Example � Composed Signal

−0.1

−0.05

0

0.05

0.1

−0.1

−0.05

0

0.05

0.1

φ1

φ2

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

+

×1.0

×0.3

×0.5

×0.05

0

0.5

1

0

0.5

1

φ3

φ4

0 64 128

0 20 40 60 80 100 12010−10

10−8

10−6

10−4

10−2

100

|T{φ1+0.3φ2}|

DCT Coefficients

|T{φ1+0.3φ2+0.5φ3+0.05φ4}|

39

Example � Desired Decomposition

0 40 80 120 160 200 24010

−10

10−8

10−6

10−4

10−2

100

DCT Coefficients Spike (Identity) Coefficients

Formally, the sparsest coefficients are obtained by solving the optimizationproblem:

(P0) Minimize subject to α 0 s=φα

It has been proposed to replace the l° norm by the l¹ norm (Chen, 1995):

(P1) Minimize subject to α 1 s=φα

It can be seen as a kind of convexification of (P0).

It has been shown (Donoho and Huo, 1999) that for certain dictionary, it thereexists a highly sparse solution to (P0), then it is identical to the solution of (P1).

We consider now that the dictionary is built of a set of L dictionaries relatedto multiscale transforms, such wavelets, ridgelet, or curvelets.

a solution α is obtained by minimizing a functional of the form:

J α = sB∑ k =1

Lφ

kα

k 2

2

+λ α p

s=φα=∑ k =1

Lφ

kα

k

Considering L transforms, and the coefficents relative to the kth transform :αk

φ= φ1,

...,φL

, α= α1,

...,αL

,

An efficient algorithm is the Block−Coordinate Relaxation Algorithm(Sardy, Bruce and Tseng, 1998):

"Initialize all to zero

"Iterate j=1,...,M

− Iterate k=1,..,L

− Update the kth part of the current solution by fixing all other parts and minimizing:

J αk= sB∑ i=1, i≠ k

Lφ

iα

iBφ

kα

k 2

2

+λ αk 1

Which is obtained by a simple soft thresholding of :

sr= sB∑ i=1, i≠ k

Lφ

iα

i

αk

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

Galaxy SBS 0335−052

Curvelet A trous WT

Ridgelet

=Curvelet A trous WTRidgelet

+ ++

Residual

CTM

=Rid+Curvelet

+Clean Data

Galaxy SBS 0335−05210 micronGEMINI−OSCIR

+

Galaxy SBS 0335−05220 micronGEMINI−OSCIR

Clean Data Residual

Rid+Curvelet A trous WT