Wavelet Domain Image Separation Ali Mohammad-Djafari and Mahieddine ICHIR


Laboratoire des Signaux et Systèmes, Supélec, Plateau de Moulon, 91192 Gif-sur-Yvette, France Abstract. In this paper, we consider the problem of blind signal and image separation using a sparse representation of the images in the wavelet domain. We consider the problem in a Bayesian estimation framework using the fact that the distribution of the wavelet coefficients of real world images can naturally be modelled by an exponential power probability density function. The Bayesian approach which has been used with success in blind source separation gives also the possibility of including any prior information we may have on the mixing matrix elements as well as on the hyperparameters (parameters of the prior laws of the noise and the sources). In our knowledge, even the Bayesian approach has been used for blind source separation either in time and in Fourier domain, it has not yet been used in wavelet domain. We consider two cases: first the case where the wavelet coefficients are assumed to be i.i.d. and second the case where we model the correlation between the coefficients of two adjacent scales by a first order Markov chain. The estimation computations are done via a Monte Carlo Markov Chain (MCMC) procedure. Some simulations show the performances of the proposed method.

INTRODUCTION Blind source separation is an active area of research in signal and image processing. Different approaches have been proposed: Principal component analysis (PCA) [1], Independent factor analysis (IFA) [2, 3, 4], Independent component analysis (ICA) [5, 6, 7], Maximum likelihood estimation [8, 9, 10, 11, 12, 13, 14, 15] and Bayesian estimation [16, 17, 18, 19, 20, 20, 21, 22]. All these methods use in general independance, sparsity and diversity of the sources either in time or in Fourier domain. Wavelets have also been used in many fields of the signal and image processing. Donoho in [23], Wan and Nowak in [24] have used the multiscale representation of signals to solve the inverse problems. Antoniadis, Leporini and Pesquet in [25] have used wavelets for signal denoising. They established a close connection between the maximum a posteriori estimation approach and wavelet thresholding. Recently, some authors considered the diversity and sparsity (atomicity) of the wavelet domain coefficients of the sources for blind source separation [26]. In this paper, we transport the problem of image separation to the wavelet domain and propose to use the Bayesian estimation framework. This approach comes in a natural way by the fact that the distribution of the wavelet coefficients of a great class of real world images can naturally be modelled by an exponential power probabilty density function (pdf). Thus, independance, sparsity and diversity which are the main hypotheses of all the source separation technics are not required for the sources themselves, but rather for their wavelet coefficients. The Bayesian approach which has been used with success in blind source separation

gives also the possibility of including any prior information we may have on the mixing matrix elements as well as on the hyperparameters (parameters of the prior laws of the noise and the sources) of the problem. In our knowledge, even the Bayesian approach has been used either in time and Fourier domain, it has not yet been used in wavelet domain. In this work, we make use of the fast wavelet transform developed by Mallat [27] to have a non-redundant multiscale representation. This paper is organized as follows: In section 2, we first present the general source separation problem using a notations which can be used either in 1D, 2D or m-D case. Then, we write the same problem in wavelet domain and explicite our hypotheses about the prior distributions of the noise and wavelet coefficients. In section 3, we present the Bayesian approach and give the main expressions of the prior and posterior probability density functions. In section 4, first we give the basics of the MCMC algorithem and then apply it to our case. In section 5, we present a few simulation results to show the performances of the proposed method and give some comparison with other known and classical approaches. Finally, in section 6, we present our conclusions.

PROBLEM FORMULATION Blind image separation consists of estimating sources from a set of their linear mixtures.  

    which are instantaneous linear The observations consist of  images    mixtures unknown sources  , possibly corrupted by additive noise !" # $of     : % '&)(* (1) &,+.-0/12

where isthe  mixing  !matrix. To be able to consider 1D, 2D or even m-D signals, we assume that , and contain each 3 samples representing either 3 samples of pixels of an time series or 3   image or, more generally, 3 voxels of an m-D signal. Thus, is a 4576)38 matrix and and are 4  6)398 matrices. The blind source separation problem is to estimate both the mixing matrix : and the   sources from the data and some assumptions about noise distribution and some prior knowledge of sources distributions. Different approaches have been proposed: Principal component analysis (PCA) [28, 29] mainly assume the problem without noise and Gaussian distribution for sources, Independent component analysis (ICA) [28, 30] and Maximum likelihood estimation [29] assume again the problem without noise but different non Gaussian distributions for sources, Factor analysis (FA) methods take account of the noise, but assume Gaussian priors both for the noise and the sources. The Bayesian approach is a generalisation of the FA with the possibility of any non Gaussian priors for noise and sources as well as the possibilty of accounting any prior knowledge on the elements of the mixing matrix and the  hyperparameters of the problem. In addition, it allows us to jointly estimate the sources , the mixing matrix : and even the hyperparameters ; of the problem through the posterior: < 4 = :



;?>



8A@

< 4 >

= :



 ;B8 < 4 >C;A8 < 4":D>E;A8 < 4";A8

(2)



We have used this approach before with different priors < 4 >E;A8 such as Gaussian [31] and mixture of Gaussian [32, 33]. We also used this approach in multispectral image separation in astronomy for separating the cosmological microwave background (CMB) from other cosmological microwave activities [34, 35, 36, 37, 38, 39]. In this paper, we are going to use the same Bayesian approach, but doing the separation using the independance and diversity of the wavelet domain coefficients of the sources. Noting by the vector F the 3 samples of one of the sources, by G the discrete wavelet transform matrix, and by H the complete wavelet coefficients of the signal we have  F GIH (3) Now, using the fact discrete wavelet transform is a linear and unitary that the complete M GLGKJ 8 , the problem of source separation can be easily transoperator 4"GKJ5G ported to the wavelet domain and written as: NO '& NQP ( NSR 

(4)

The main advantage of using this last equation in place of the original source separation NTP problem is that we than for itself. For  can more easily assign simple prior laws for example, when contains discontinuity or nonstationary, still its wavelet coefficients distribution can be modeled by a simple generalized exponential (GE) probability density function (pdf) while it is harder to model appropriately signal samples distribution by a simple pdf. Indead, it has been reported by many authors that the distribution of the wavelet coefficients of real world images are well modeled by a GE pdf: < 4  >CU V 8 XW=Y 4U "V 8 Vh 

Z

V U\[]4 Vi

_^V 8

c^

exp `ab> Ud> egf

(5)

Z

Note that gives an exponential pdf and corresponds to a Gaussian pdf. We are going to use this prior probability law in our Bayesian estimation framework.

BAYESIAN FORMULATION In a first step, we assume that the sources and the noise wavelet coefficients are i.i.d. . Thus, to simplify the notations, we note, respectively, by jd4k8 , F#4"k8 and lm4"k8 the vectors containing the wavelet coefficients ( of the data, the sources and the noise for a given index k . Thus, we have jd4k8 :nFA4k8 lm4"ko8 . Hereafter, we omit the index k and note it only when needed. To proceed with the Bayesian approach, we have to assign the prior laws. In the following we assume: •

! W=Y " V The noise wavelet coefficients l are assumed independent and < 4 8 4U 8. Then -

< 45jp>C:

 F

 U

"V 8



q

V sr Z U=[]4 _^tV 8u exp va

-

w 

4x> y



 ^ a{z|:*F~} > U=8 e#

(6)



of the sources are also assumed independent and < 4€ 8

• The coefficients F W Y wavelet = "V

4U

8 . Then

1

•

E

… of the mixing matrix : The elements E E values † and variances ‡Bˆ :

Therfore, we may note by  ‘

< 4":D>E

where   and

œ †



Ž 8

Zt‰ 4

w

r Z U  []4 _^V 8 u exp v a 

< 4… E  8

•

1

V

q

< 4Fm>  U _"V  8





 ^  4‚>E€ > U 8 e„ƒ 

(7)

are assumed i.i.d. and Gaussian with mean

C ‡ ˆ 8ŠŒ‹ ˆ exp v a Z

 ‡ Žˆ 

45…

E

an†

E

8 ˆ 

(8)

ƒ

-“’ 1 q Zt‰ ‡ Ž ˆ  8 ŠŒ‹ ˆ  ’ 4 ƒ  ‘ ”x•—–™˜ a Z 4 Vect 4":Ia*š88 J Ž ŠŒ‹ 4 Vect 4":Ia*š88›

(9)

E  , Vect 4"š8 means a vector containing the elements of the matrix ‘ Ž

+-ž1/-ž12

All the hyperparameters 4 ¦¨‹ § Gamma prior distributions:





diag 4Ÿ‡ Žxˆ  ¡  ‡ Žxˆ  £¢



‡ Ž„ˆ ¤¥ 8

¦ ‹§ ƒ 8 are assumed independent and assigned standard ƒ

< 4Ÿy#8 'W 4 Z   8 y

”©•—–

4„aªy#8

The joint a posteriori law of the sources coefficients F , the mixing matrix : hyperparameters ; is then given by: < 4"F  :



;]>Cj]8«@

< 4j>EF  :

 

;B8 < 4"F«>E;B8 < 4":D>E;#8 < 4;B8 b­

where we noted all the hyperparameters ¬U e U e„ƒ

(10) and the (11)

by ; .

MCMC IMPLEMENTATION 



Once the expression of the joint a posteriori law < 4F : ;]>Cj]8 of all the unknowns has been derived, we can use it to infer them. However, in general, the computation of the normalization factor needs a huge dimensional integration. When the MAP estimation is chosen, this normalization factor is not needed, but it is formally needed for other estimation rule such as the posterior mean. The MCMC algorithms are then the basic tools to generate samples from the posterior law. The main idea is to generate successively

+c®2#¯

+.®‚2

< 4Fm>E:

+.®‚2 

+.®‚2g¯ 

< 4":D>EF

+.®‚2 

+c®‚2 

+.®‚2 ¯ 2 posterior +c®2 the samples from+.®‚the laws F ; j]8 , : ; j]8   < 4";]>CF : j]8 and then estimate their expected values by averaging these and ; samples. We use the Hasting-Metropolis algorithm combined to a Gibbs sampler to obtain an ergodic chain, and then approximate the ensemble expectation of any quantity ° by its ± empirical mean: 

4"°8m²

(µ

45³´ah3

+ 2  < 4 »#>  8 . where ° J  are samples VS from V0  Z Z

+ 2 4° J 8

w

8X¶ J¸·—¹dº

Noting that, when and , the posterior laws for the sources and for the elements of the mixing matrix are Gaussians, we can use these Gaussians as the trial (or instrumental) pdf. Thus, to simplify the presentation of the proposed algorithm, we give here the expressions of these Gaussian posterior laws: •



The trial posterior pdf of the sources is Gaussian ¼A4"Fm>C; j?8 F¾

and

‘ P ¾

where •

‘

Z Z : J:

U ˆ

+1/1¨2

¦¿

4"F¾

‘ P ¾ 8 with

‘ P ¾ : Jj

U ˆ 4

S½

(

(12)

Z ‘

¦ŠŒ¿ ‹ 8‚ŠŒ‹



diag 4U ˆ U ˆ ‹



ˆ

(13) 1 U ˆ 8 

(14)

The trial posterior pdf of the mixing matrix elements is Gaussian ¼A4 Vect 4:*8>E; j]8 ½ ‘ 4 Vect 4Áš À 8 ¾ Ž 8 with Vect 4  À 8



‘

and ‘

U ˆ

w

Vect 4 ®

Z

¾ Ž

Z

¾ ŽÃÂ

Â

diag Â

U ˆ

w ®

jd4k8F J 4"k88

F#4"k8F J 4k8 Ä

(S‘

(S‘

Ž ŠŒ‹ Vect 4"š8ÅÄ

Ž ŠŒ‹ Ä

(15)

ŠŒ‹

(16)

The proposed MCMC algorithm is then the following: 







F Æ :iÆ ; Æ and repeat the following steps until convergence Initialize F : ; to Á « Ç  • Sampling FA4k8 , for k :

•

ÈÊÉ

 Q½ ¼A4 È >E; j]8

4 F¾

‘ P ¾ 8

‘ P where F¾ and ¾ are given, respectively by eq. (12) and eq. (13) and +

2

F J¸Ë ‹ 4k8

˜

+ È2 F J 4k8



with probability Ì  with probability aiÌ

with Ì

min r





where < 4 È > jd4"k8 : • Sampling : : where  À

< 4 È C> jd4"k8  :  B ; 8 + 2   < 4"F k 8>Cjd4"ko8 : A ; 8ÎÍ J 4"



;B8 is given by eq.(7).  Q½ ¼A4 È >E; j]8

ÈÊÉ

4 Vect 4š À 8

¾ Ž 8

‘

with Ì

2

J¸Ë ‹ :



min r



˜



Mat+ 4 È 2 8 :

with probability Ì  with probability anÌ

J

< 4 È C> j  F  ;«8 + 2   < 4 Vect 4": ; 8 Í J 8_> j F B

where < 4 È > jd4"k8 : ;B8 is given by eq. (9). • Sampling ; + 2 ¦ ‹§ : W

; J¸Ë ‹ É

with …

Sampling ;

‘

and ¾ Ž are given, respectively by eq. (15) and eq. (16) and +

•

¼A+ 4 È2 8 ¼A4"F J 4 k 88 u

Ó



 Ç V (

Z

and

9 «

¦ ‹§ ƒ , for ƒ

ÏÐ

…



V

8 + 2 ¼ 4 Vect 4": J 8 8tu B

8

 : 2 ¸ J Ë ‹ É ;

Ç

 Ï

È

-]’ Ñ w   (µ ŠŒ‹   ’ ® > y 4"ko8ga{z|:*F#4"k8Ò} > e Ä



with

4…

¼A4

+

W

4…

 Ï 8

Ñ ( Z

and

ÏÐ

w ¬ ®

 (µ ­ ŠŒ‹ >E€ "4 k8> e„ƒ · ‹

SIMULATION RESULTS To illustrate the proposed method, we have mixed two images of ÔÎÕÖ6hÔÎÕ pixels (the pictures of lena and cameraman presented in figure (1)) with a mixing matrix: .ڗ ×Ø)Ù  Ù  Õ Ù  ÜÎ Z Ü Ù Ù Õ~Þ Ü

&'

.Û Û Û ÜÛ Ù .ÝÙ Ù  Ú Ü Ô*àß Ù Õ

R KÛ

and gaussian noise of zero mean and a variance ‡ ˆ a £ white  Z U Ô . Figure (2) shows the mixed images obtained. Ü Ü 10

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FIGURE 1. The original source images

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FIGURE 2. The mixed images with a white gaussian noise (áãâbä_åçæ~é è âbê )

First, we applied the proposed method directly on the mixed images where we assumed noises and original images to be i.i.d. and Gaussian. Figure estimatedb shows the separated images. Then, we accounted for the local correlation between neighboring pixels of the original images. Figure estimatedc shows the separated images. Then, we applied the method in the wavelet domain. What follows give more detailed description of the numerical experiments.

FIGURE 3. Estiamted source images obtained directly assuming i.i.d. Gaussian priors for noise and images

FIGURE 4. Estiamted source images obtained directly accounting for local spatial correlations

+ We 2 + apply 2 the MCMC algorithm described above to obtain ergodic chains of  and : J F J ; J . We used the following values for initialization: .Û cÛ + 2 + 2 + 2 + 2 š×Ø Ù .Û Ù cÛ  $ ð Z  ñ Æ Æ  ˆ C ë Ÿ ì í  ˆ E ë Ÿ ì í  ˆ E ë Ÿ ì í Æ : Æ ‡ ‡ ‡ # a ï î U U U Ù .Û Ù cÛ ˆ ˆ ‹ ‹ ßà Ù Ù

The final estimated values obtained by averaging the last 100 samples after 500 iterations are the following: 

š×Ø)Ù  Þ ÜÛ ÕÞ Þ Þ :¾ Ù Õ Î Z Õ Ù ÜÙ

cÛÝ Ô Ô Ù cݗ   Ù  Ù  ßà Ù Õ Ù Õ

U¾

Ü

¸ Ô

Z

 XÚ Á Û  Z  Õ Ù ÔÔ ÀU Ù ÀU ‹ ˆ

 Û ÕÎÕÕ Þ Ô Ù

We may also note  that cÝ the estimated  Ý values of U ‹ and U ˆ directly from the original Þ Ù Ü Ô Ù ,U Õ~Ô Ù Ù Ü Ô . images are U ‹ ‹ Figure (5) shows the estimated images. Figure (6) shows the convergence of the elements of the matrix : and figure (7) shows the convergence of the hyperparameters.

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FIGURE 5. Estiamted source images obtained in wavelet domain 1

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FIGURE 6. Convergence of the elements of ò 400

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FIGURE 7. Convergence of the hyperparameters ó : Left: ô ,

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Right: ôŒõ and ô

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è.

Figure (8) shows the histograms of the original, mixed and estimated images and Figure (9) shows the histograms of the wavelet coefficients of original, mixed and

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CONCLUSIONS In this contribution we proposed an approach to jointly estimate the mixing matrix and the original source images. We transported the problem to the wavelet domain using a Bayesian approach where the wavelet coefficients of real world images are naturally modeled by generalised exponential distributions. Independance of the wavelet coefficients of signals is more realistic than the independance of the signals themselves. In a first step, we assumed all the wavelet coefficients to be independent and identically distributed. However, inter-scale correlation can be modeled in the prior laws of the signals wavelet coefficients and the above algorithm can be extended and the equations rewritten for this case.

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