Biointelligence Laboratory, Seoul National University Ch 6. Markov Random Fields 6.6 ~ 6.7 Adaptive Cooperative Systems, Martin Beckerman, 1997. Summarized by M.-O. Heo Biointelligence Laboratory, Seoul National University http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Contents 6.6 Simultaneous Autoregressive Models 6.6.1 Simultaneous Autoregressive Random Fields 6.6.2 Image Reconstruction 6.6.3 Fourier Computation and Block Circulant Matrices 6.7 The Method of Geman and Geman 6.7.1 Maximum A posteriori (MAP) Estimation 6.7.2 Clique Potentials 6.7.3 The Posterior Distribution 6.7.4 The Gibbs Sampler (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

Simultaneous Autoregressive Random Fields Simultaneous Autoregressive Models AR representations Finite support SAR Random Fields Defined with noise random variables Support Zero-mean independent random variable Real-valued coefficient over the lattice The set of acquaintances of (i,j) (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

Simultaneous Autoregressive Random Fields SAR random field is a MRF? Using joint probability distribution Find the following relation (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

Simultaneous Autoregressive Random Fields Toroidal lattice SAR field Defined by a pair of equations of the form For the interior region For the boundary (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Image Reconstruction Problem Formulation Notations D{f(x, y)} : point-spread function, linear, space invariant f(x, y) : ideal image g(x, y) : observed image n(x, y) : noise process (independent of the imaging process) Degredation process (or noise process) One dimensional process Two dimensional process (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Image reconstruction methods 1. Constrained least-squares estimation (LSE) Constrained least-squares estimation (LSE) Objective function and the corresponding solution Selecting an appropriate form of Q Subject to constraint Optimal estimate) Correlation matrix for the image f Correlation matrix for the noise n (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Image reconstruction methods 2. linear minimum mean square error (MMSE) estimation linear minimum mean square error (MMSE) estimation Define the error e Minimize the positive quantity Assuming that f transform into g under a linear operator L, Assumption of signal-independent noise Solution by this step) More assumptions Image f is stationary Rf is approximated by the block circulant covariance matrix Qf White noise process with common variance Solution Using SAR and GRF image models (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

Fourier Computation and Block Circulant Matrices Degredation with discrete Fourier transform Diagonal elements Tkk are the eigenvalues of the matrix R (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Using one dimensional convolution to the degredation model Discrete Fourier transform with W-1 The result The discrete convolution theorem Convolution of a pair of arrays in the spatial domain to element-by-element multiplication in the frequency domain, and vice versa. Two-dimensional image model (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

The method of Geman and Geman MAP estimation for image reconstruction 3 main points Use of Bayesian formalism to incorporate constraints Gibbs-Markov equivalence A second, dual lattice system containing discontinuity-preserving information (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

Maximum A Posteriori (MAP) Estimation Posterior distribution to maximize Degredation process Additive noise to be described by the multivariate gaussian Assumption for noise : uncorrelated and stationary with zero mean Likelihood Assumption: no blurring, point-spread function takes other effects (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Model the image as a MRF Prior distribution An Energy composed of a sum of local contributions Partition function Z The site potentials Strength parameter like the inverse of the temperature Clique potentials Cliques associated with the neighborhood system for the site (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Clique Potentials The general property of the lattice systems Piecewise smoothness Neighboring pixels tend to have similar grey values Ex) Four-level model for texture realization Four type of elements (a) (b) (c) (d) Encourages all elements within a given clique to have the same grey level. Controls the percentage of pixels of each region. And useful for textured image regions. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

The Posterior Distribution The energy If we use the quadratic difference potentials as clique potentials Gibbs distribution having the same potential as the posterior distribution Favors maximally smooth restorations, penalizing large pairwise contrasts in pixel values within a local neighborhood. Encourages restoration that are not too different from the data

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ The Gibbs Sampler Finding desired low-energy states using the SA Gibbs Sampler The sampling method on the Gibbs distribution In the exponential in the Posterior Gibbs distribution… The result of this sampler Controls the relative hardness of the constraints Annealing temperature (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ The Gibbs Sampler The general form (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/