Fast joint separation and segmentation of mixed images Hichem Snoussi and Ali Mohammad-Djafari
Laboratoire des Signaux et Systèmes (L2S), Supélec, Plateau de Moulon, 91192 Gif-sur-Yvette Cedex, France Abstract. We consider the problem of the blind separation of noisy instantaneously mixed images. The images are modelized by hidden Markov fields with unknown parameters. Given the observed images, we give a Bayesian formulation and we propose a fast version of the MCMC algorithm based on the Bartlett decomposition for the resulting data augmentation problem. We separate the unknown variables into two categories: . The parameters of interest which are the mixing matrix, the noise covariance and the parameters of the sources distributions. . The hidden variables which are the unobserved sources and the unobserved pixels segmentation labels. The proposed algorithm provides, in the stationary regime, samples drawn from the posterior distributions of all the variables involved in the problem leading to great flexibility in the cost function choice. Finally, we show the results for both synthetic and real data to illustrate the feasibility of the proposed solution.
I. INTRODUCTION AND MODEL ASSUMPTIONS
, each image The observations are images is defined on a finite set of ! sites , corresponding to the pixels of the image: . The observations are #%$&
$ ' defined on the same set noisy linear instantaneous mixture of " source images : ' (
)
$!,
$ .-
$ +* =
Ó è
C X8Q
(4)
the Gamma
ÿ
))
;
C
leading to the following -priors on è
8 /
CBT
? ¤
The expressions of the averaged divergences between the ) ) â ê
â
is the marginal probability of the label Õ and 9
9 ñ
8
Ó ¿ §
-
è S ;
/ ;
Tr
B
;
))
KHB
Ñ K
B
parameters are: è Ò
KLB
E
:
=GF
Q
; K
B
¿ J/
è K B
=
q $ ' K 2
q ñ
is the wishart distribution of an "ÅÈ[" matrix:
H /
?³q K
JI Ì' & NË M LK q
PO ?
+
(
Tr
Ñ K
H
RQ Ò
Therefore, the -prior is a normal inverse Wishart prior (conjugate prior). The mixing matrix and the noise covariance are not a priori independent. In fact, the covariance
è
matrix of ; is the noise to signal ratio K KÃB . We note a multiplicative term which is a power of the determinant of the a priori expectation of the source covariance ñ /
£
K ! . This term can be injected in the prior and thus the ; ® K B parameters and 2
Ø
the parameters are a priori independent. è è ; K K B ) for We note that the precision matrix for the mixing matrix ( is ² Û the product of the confidence term å Ü in the reference parameters and the signal to è å noise ratio. Therefore, the resulting precision of the reference matrix ; is not only our a priori coefficient æç but the product of this coefficient and the signal to noise ratio.
IV. MCMC IMPLEMENTATION We divide the vector of unknown variables into two sub-vectors: The hidden variables WL/~#
and the parameter ° and we consider a Gibbs sampler: repeat until convergence, 1.
#S xUT z /V# S Ux T z DW IWÃ/~#]q£l/ S Ux T draw z ° S xXT W q£l/YW S xXT z /D# S xUT z
W
2. draw °
z
°
S Xx T
z
z This Bayesian sampling [39] produces a Markov chain ° , ergodic with stationxXT â\[ T q£
S
è ary distribution ° . After Z iterations (warming up), the samples ° can q£
be considered to be drawn approximately from their a posteriori distribution ° . Then, by the ergodic theorem, we can approximate a posteriori expectations by empirical expectations: z M
WL/~#
Sampling obtained by, 1. draw
S
W
N] O
°
_^ U
1q
7
)
T
`
] S Ux T â [ T °
: The sampling of the hidden fields
from
Wîq£l/ °
?³
´q5WÃ/ °
WÃ/X#
(5) from
WL/~#]q£l/ °
is
knmLIWR
!a
W
In this expression, we have two kinds of dependencies: are independent IWR b ' Wø$VW knmWY$
IWY$&
$@ across components, but each discrete image has !aca5
W a Markovian structure. , the fields are independent through the set Given q /
µqWÃ/ 1b @ & D , ° É ° but dependent through the components because S q / ( D +¿ /
D  Á of the mixing operation is É ° ;²À Áá ;²K ; KÃB where É ¾ 0 =¦ / 8_8:8:/5¦ _Í pÊ~Ë '!Ê Ì the vector label on the site , ÀÁ  and K Áá the diagonal matrix »s4 / 8_8:8_/ A
pC Ê Ë A
'c C ÊÌ . *9Î
2. draw
S
#]q W
S
from I#]q£
/XWÃ/ °
?¢¡ @ &.¾
edg f ßih Í /j f ßih Í
+¿ E
where the a posteriori mean and covariance are easily computed [40],
j f ßih Í
d f ßih Í
O
S ;
K
j f ßih Í
B
-
Ñ
; S ;
U Áá K
K
-
D B
Á Â À K
Áá Ò
WÃ/X#
Sampling ° : Given the observations and the samples , the sampling of the parameter ° becomes an easy task (this represents theq£ principal reason of introducing /XWÃ/~#T
the hidden sources). The conditional distribution ° is factorized into two conditional distributions,
q£l/~WÃ/X# ?³ °
/ ;
/
q
KLB
/X#
lk
k /
À
q5#º/~WR
/
|
leading to a separate sampling of ; K®B and À . Choosing the -priors developed in the previous section, the a posteriori distributions are: Inverse Wishart for the noise covariance and Inverse Gamma for sources variances. • Normal for the mixing matrix and for the sources means.
•
The expressions of these distributions are developed in L theFappendix A. We give below /
| the expressions for ; KÃB in the particular case when (Jeffreys prior):
W
e fg
K
¢ B ;
q
KÃB
4
DW
?
Hnm? / / ß
¾
;
ß
lp /
ß
?
/
oH
' /
ß
;
ß
C
K
ß
qp
/
. ~K !
C
ß
.>. K
. K K .S
K
K
(6)
! ]KLB
S
S
D D ,K . D E and where we define the empirical statistics K .>. S # E E (the sources are generated in the first step of the Gibbs sampling). K We note that the covariance matrix of ; is proportional to the noise to signal ratio. This explains the fact noted in [41] concerning the slow convergence of the EM algorithm.
FAST MCMC IMPLEMENTATION A critical aspect of the above implementation is the computational cost of the sampling steps. Indeed, the convergence of the MCMC sampling may require a great number of iterations to ensure the convergence. Therefore, we need fast steps in the proposed algorithm to obtain a great number of iterations with a reasonable computational cost.# We investigated this direction by avoiding the sources sampling. In fact, the sources are sampled in the MCMC algorithm but only the statistics K . and K ! are used in the / generation of the parameters ; K²B (see equation (6)). Therefore we avoid the sampling # of the sources and we sample directly the statistic matrices K . and K ! . We show in the following how these simulations are easily performed in our problem# formulation. W , the multidimensional source images are classified After the drawing of the labels ` F` 8_8:8 `
' Ê Ê © defined by: È È into regions Ê
\j02
qpÆdp0
Fob
In each region Ê , the sources are Gaussians with mean and covariance:
j
Ê
d
j
Ê
S ;
xÊ
.
xÊ
.>. K
;
K
D B
z
Ê
K
xÊ
. @ &er
and K
er
er
@ &
er
@ &
z
z
S ;
xÊ
K
B
z
xÊ
Ê
We then define the statistic matrices K K
K
Ê K
À
Ê
(7)
z Ê
on the region D E
r
S E
er
E
D
as:
S
(8)
D S
z the expressions (7) and (8) and some algebraic manipulations, the statistics K From xÊ and K can be decomposed as follows: z
xÊ
K
. xÊ
K
j
z
!
ts®'S ¯7u -
K
tj Ê 7s 'V¯ -
K
u Ê
C
Ê S
;
K B
xÊ
.
z
S
s ts
ts
- ®S
®S 'V¯ 'V¯ B CÊ 'V¯ C
K ;
S
us v Ê S u u Ê
S
u Ê u Ê . The matricess K and where Ê s '¯ K C ares not v random matrices and are updated V ' ¯ at each iteration. The matrices , and are random matrices and have the C following distributions: s s
'¯
W
'¯
W
4
5r
|/
5r
¾
C
s vxW F F
|/
¾ 4
q '
7/ H[
? H
Ê
z
xÊ
qQ/
K
.>. .
K
è -
K
;
RÙ 4 4 ;
-
è
4 ;
. 9K K .
&`
f RÙ ; K
K
è
è K
!
;
è
#
;
4
×
Q
S
The statistics K . and K are computed from the sampled sources or directly sampled according è to their a posteriori distributions in the fast version of the MCMC implementation. K is the a priori expectation of the matrix K : è
K
!
ñ
2 â K
3
- a posteriori
The same computations as in the previous section lead to a Normal Inverse Gamma for the means and variances of the univariate Gaussians: q£l/~#Ï/XWR ? p¦ § /5Ó.§
p¦ ¾
§
¿ ¦ ß
/Ó ß
9
e ß !/ B
pÓ. ¿ §
ß
The parameters of these distributions are updated according to the following equations:
e
f
g
where
§
[ eev v 67â â 6 ß w#6 ¡ 6â Ó 6 4 ev â ß w6 ¡ 6 ß w#6 ¡ ev 6â ev â B ß C6 4 â ' Xz ¢, ¤£ Â\¥C ¦ 6 x C w 6 p0
@ ! , C 6 , C ¦
w
C
6 eevv 6ââ , ¢ 6¡ 6
w w
²¦ è
C
¢
§ , C
is the region of the image Ô such that the sampled label is equal to Õ :
§
\j0Ç2
§
§
q
§
q
qpÆ
£09
Õ
b
APPENDIX B: BARTLETT DECOMPOSITION
F
7H
/
Let ¨ an ¸È matrix distributed from a Wishart distribution Q ? . A direct simulation from this distribution consists in sampling -variate normal vectors © §Gª ? |+/
w and then computing: ¾
¨ Hî««
8
¬«
© §© ? §
×
« ×
×
where . This method involves ? simulations from univariate Normal distribution leading to a high computational cost when ? increases. An alternative is to use the Bartlett decomposition which can be summarized in the following theorem: F «¯j°j × « × /H
H« × j and . Put ¨ , where Ó is Theorem Let ¨ be Q ? 8 =$
