Maximum likelihood separation of spatially autocorrelated images

using a Markov model

Shahram Hosseini1, Rima Guidara1, Yannick Deville1

and Christian Jutten2

1. Laboratoire d’Astrophysique de Toulouse-Tarbes (LATT), Observatoire Midi-Pyrénées -Université Paul Sabatier Toulouse 3, 14 A. Edouard Belin, 31400

Toulouse, France. 2. Laboratoire des Images et des Signaux, UMR CNRS-INPG-UPS, 46 Avenue Félix

Viallet, 38031 Grenoble, France

MAXENT 2006, July 8-13, Paris-France 2

OUTLINE

Problem statement A maximum likelihood approach using Markov

model - Second-order Markov random field - Score function estimation - Gradient-based optimisation algorithm Experimental results Conlusion

MAXENT 2006, July 8-13, Paris-France 3

Problem statement

),( 21 nns ),(ˆ),( 2121 nnnn sy ),( 21 nnx 1ˆ ABA

Assumptions :

• Linear instantaneous mixture.

• K unknown independent source images, K observations, N=N1×N2 samples.

• Unknown mixing matrix A is invertible.

• Each source is spatially autocorrelated and can be modeled as a 2nd-order Markov random field.

Goal:

Compute B by a maximum likelihood (ML) approach.

Mixing matrix

Separating matrix

Problem statement A maximum likelihood approach using

Markov model Experimental results Conlusion

MAXENT 2006, July 8-13, Paris-France 4

Motivations Maximum likelihood approach: provides an asymptotically efficient

estimator (smallest error covariance matrix among unbiased estimators).

Modeling the source images by Markov random fields:

- Most of real images present a spatial autocorrelation within near pixels.

- Spatial autocorrelation can make the estimation of the model possible where the basic blind source separation methods cannot estimate it (if image sources are Gaussian but spatially autocorrelated).

- Markov random fields allow taking into account spatial autocorrelation without a priori assumption concerning the probability density of the sources.

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ML approach (1)

212111 1111 ,ΝΝ,...,x,ΝΝ,......,x,,...,x,xfC ΚΚx

K

iiisN NNssfC

i1

211

),(,...,1,1det

1

B

CMLB

B maxargˆ

Independence of sources

Problem statement A maximum likelihood approach using

Markov model Experimental results Conlusion

We denote the joint PDF of all the samples of all the components of the observation vector x.

Maximum likelihood estimate:

whereisf is the joint PDF of all the samples of

source si

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ML approach (2) Decomposition of the joint PDF of each source using Bayes rule Many possible sweeping trajectories that preserve continuity

and can exploit spatial autocorrelation within the image:

(1)

(2)

(3)

These different sweeping schemes being essentially equivalent, we chose the horizontal one.

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ML approach (3)

1,1,,1,,1,1,,,11,2

1,1,,1,1,11,1,2,13,11,12,11,1

,,...,1,1

21212

22

21

iiisiiis

iiisiiisiisis

iis

sNNsNNsfsNssf

sNsNsfsssfssfsf

NNssfF

ii

iiii

i

Bayes rule decomposition resulting from a horizontal sweeping:

To simplify F, sources are modeled by second-order Markov random fields Conditional PDF of a pixel given all remaining pixels of the image equals its conditional PDF given its 8 nearest neighbors.

),( 21 nns

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ML approach (4)

is the set of the predecessors of a pixel in the sense of the horizontal sweeping trajectory.

For a pixel not located on the boundary of the image, we obtain

211, ,\),(21

nlnknklksD inn Denote

21 ,nnD

1,1,,1,1,1,1,,)),(( 2121212121,21 21 nnsnnsnnsnnsnnsfDnnsf iiiiisnnis ii

If the image is quite large, pixels situated on the boundaries can be neglected. We can then write

1

1

2

22

1

22121212121 1,1,,1,1,1,1,,

N

n

N

niiiiis nnsnnsnnsnnsnnsfF

i

),( 21 nnsi

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ML approach (5) The initial joint PDF to be maximized:

1

1

2

22

1

2

.1

.N

n

N

nN NE

Taking the logarithm of C, dividing it by N and defining the spatial average operator

the log-likelihood can finally be written as

K

iiiiiisN nnsnnsnnsnnsnnsfEL

i1

21212121211 )1,1(),,1(),1,1(),1,(),((logdetlog B

212111 1111 ,ΝΝ,...,x,ΝΝ,......,x,,...,x,xfC ΚΚx

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ML approach (6)

),( 21 lnknsi

K

iiiiiisN

T nnsnnsnnsnnsnnsfEL

i1

21212121211 )1,1(),,1(),1,1(),1,(),(log

BB

B

Problem statement A maximum likelihood approach using

Markov model Experimental results Conlusion

Taking the derivative of L1with respect to the separating matrix B, we have

We define the conditional score function of the source si with respect to the pixel by:

),(

1,1,,1,1,1,1,,log,

21

212121212121

),(

lnkns

nnsnnsnnsnnsnnsfnnψ

i

iiiiislks

i

i

MAXENT 2006, July 8-13, Paris-France 11

Denoting:

- the column vector which has the conditional score fonctions of the K sources as components.

- the K-dimensional vector of observations.

we finally obtain

ML approach (7)

),( 21 nnx

)l,k(21

T21

)l,k(N

T1 ln,kn.n,nEL

xψBB s

),( 21),( nnlk

sΨ

)1,1(),0,1(),1,1(),1,0(),0,0( where

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Estimation of score functions

Conditional score fonctions must be estimated to solve our ML problem.

They may be estimated only via reconstructed sources yi(n1,n2). We used the method proposed in [D.-T.Pham, IEEE Trans. On Signal

Processing, Oct. 2004]

- A non-parametric kernel density estimator using third-order cardinal spline kernels.

- Estimation of joint entropies using a discrete Riemann sum.

Problem statement A maximum likelihood approach using

Markov model Experimental results Conlusion

No prior knowledge of the source distributions is needed.

Good estimation of the conditional score functions.

Very time consuming, especially for large-size images.

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An equivariant algorithm Initialize B=I.

Repeat until convergence : - Compute estimated sources y=Bx. Normalize to unit

power. - Estimate the conditional score functions - Compute the matrix

oldnew BGIB )(

),(2121

),( ,.,lk

TlkN lnknnnE yψG s

- Update B

Problem statement A maximum likelihood approach using

Markov model Experimental results Conlusion

MAXENT 2006, July 8-13, Paris-France 14

OUTLINE

Problem statement A maximum likelihood approach - Second-order Markov model - Score functions estimation - Gradient optimisation algorithm Experimental results Conclusion

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Experimental results Comparison with two classical methods: 1. SOBI algorithm - A second-order method - Joint diagonalisation of covariance matrices evaluated at

different lags. Exploits autocorrelation but ignores possible non-

Gaussianity2. Pham-Garat algorithm - A maximum likelihood approach - Sources are supposed i.i.d Exploits non-Gaussianity but ignores possible

autocorrelation.

Problem statement A maximum likelihood approach using

Markov model Experimental results Conlusion

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Artificial data (1) Generate two autocorrelated images :

1. Generate two independent white and uniformly distributed noise images

and .

2. Filter i.i.d noise images by 2 Infinite Impulse Response (IIR) filters

)1,1(3.0),1(5.0)1,1()1,(5.0),(),(

)1,1(3.0),1(5.0)1,1(4.0)1,(5.0),(),(

21221221222212212212

211211211211211211

nnsnnsnnsnnsnnenns

nnsnnsnnsnnsnnenns

ρ

)n,n(e 211 )n,n(e 212

Problem statement A maximum likelihood approach using

Markov model Experimental results Conlusion

In this case, generated images perfectly satisfy the working hypotheses : source images are stationary and second-order Markov random fields.

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Signal to Interference Ratio (SIR)

2

1i2

ii

2i

10 ])ys[(E

]s[Elog

2

1)dB(SIR

199.0

99.01A

Mixing matrix :

Problem statement A maximum likelihood approach using

Markov model Experimental results Conlusion

The mean of the SIR over 100 Monte Carlo simulations is computed and plotted as a function of the filter parameter ρ22 .

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Artificial data (2) Artificial data are generated by filtering i.i.d noise images by means of 2 Finite Impulse Response (FIR) Filters.

The source images are stationary but cannot be modeled by second-order Markov random fields.

The mean of the SIR over 100 Monte Carlo simulations is computed and plotted as a function of the selectivity of one of the filters.

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Astrophysical images (1)

13.0

3.011A

working hypotheses no longer true (non-stationary, non second-order Markov random field images)

199.0

99.012A

Weak mixture :

SIR=70 dB

Strong mixture:

Separation failed because of bad initial estimation of conditional score functions (estimated sources highly

different from actual sources).

Problem statement A maximum likelihood approach using

Markov model Experimental results Conlusion

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SIR Markov: 70 dB Pham-Garat: 13 dB SOBI: 36 dB

Solution : Initialize our method with a sub-optimal algorithm like SOBI to obtain a low mixture ratio.

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To conclude… A quasi-optimal maximum likelihood method taking into account

both non-Gaussianity and spatial autocorrelation is proposed.

Good performance on artificial and real data has been achieved.

Very time consuming : Solutions to reduce computational cost : - A parametric polynomial estimator of the conditional score

functions - A modified equivariant Newton optimization algorithm