Maximum likelihood separation of spatially autocorrelated images using a Markov model Shahram Hosseini 1, Rima Guidara 1, Yannick Deville 1 and Christian Jutten 2 1. Laboratoire d’Astrophysique de Toulouse-Tarbes (LATT), Observatoire Midi-Pyrénées - Université Paul Sabatier Toulouse 3, 14 A. Edouard Belin, Toulouse, France. 2. Laboratoire des Images et des Signaux, UMR CNRS-INPG-UPS, 46 Avenue Félix Viallet, Grenoble, France

MAXENT 2006, July 8-13, Paris-France2 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-France3 Problem statement Assumptions : Linear instantaneous mixture. K unknown independent source images, K observations, N=N 1 ×N 2 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 matrixSeparating matrix Problem statement A maximum likelihood approach using Markov model Experimental results Conlusion

MAXENT 2006, July 8-13, Paris-France4 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.

MAXENT 2006, July 8-13, Paris-France5 ML approach (1)  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: whereis the joint PDF of all the samples of source s i

MAXENT 2006, July 8-13, Paris-France6 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.

MAXENT 2006, July 8-13, Paris-France7 ML approach (3)  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.

MAXENT 2006, July 8-13, Paris-France8 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 Denote If the image is quite large, pixels situated on the boundaries can be neglected. We can then write

MAXENT 2006, July 8-13, Paris-France9 ML approach (5) The initial joint PDF to be maximized: Taking the logarithm of C, dividing it by N and defining the spatial average operator the log-likelihood can finally be written as

MAXENT 2006, July 8-13, Paris-France10 ML approach (6) Problem statement A maximum likelihood approach using Markov model Experimental results Conlusion Taking the derivative of L 1 with respect to the separating matrix B, we have  We define the conditional score function of the source s i with respect to the pixel by:

MAXENT 2006, July 8-13, Paris-France11  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) where

MAXENT 2006, July 8-13, Paris-France12 Estimation of score functions Conditional score fonctions must be estimated to solve our ML problem. They may be estimated only via reconstructed sources y i (n 1,n 2 ). 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.

MAXENT 2006, July 8-13, Paris-France13 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 - Update B Problem statement A maximum likelihood approach using Markov model Experimental results Conlusion

MAXENT 2006, July 8-13, Paris-France14 OUTLINE Problem statement A maximum likelihood approach - Second-order Markov model - Score functions estimation - Gradient optimisation algorithm Experimental results Conclusion

MAXENT 2006, July 8-13, Paris-France15 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-Gaussianity 2. 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

MAXENT 2006, July 8-13, Paris-France16 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 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.

MAXENT 2006, July 8-13, Paris-France17 Signal to Interference Ratio (SIR) 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 ρ 22.

MAXENT 2006, July 8-13, Paris-France18 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.

MAXENT 2006, July 8-13, Paris-France19 Astrophysical images (1)  working hypotheses no longer true (non-stationary, non second-order Markov random field images)  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

MAXENT 2006, July 8-13, Paris-France20 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.

MAXENT 2006, July 8-13, Paris-France21 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