Maximum likelihood estimation of Gaussian Markov random field parameters

Addresses the problem of fitting Gaussian Markov random field (GMRF) models to natural textures through maximum likelihood (ML) estimation of the model parameters. In particular, the authors are interested in two implementation issues on ML estimation. First, the log-likelihood function to be maximized is not a concave function with respect to the parameters. Thus the conventional gradient maximization method is not computationally attractive. The second concern comes from the fact that the suppose-to-be covariance matrix expressed in terms of the estimated parameters must be non-negative definite. To resolve these difficulties one uses a version of the so-called multi-start algorithm, which is a variation of the deterministic relaxation algorithm. The algorithm aims at determining many of the local maxima of the likelihood function and determining a maximum among them. This algorithm always yields estimates that are in the allowable set of parameter values, yielding a non-negative definite covariance matrix. Some experimental results with natural textures show that often the best fit of natural images to GMRF models via ML estimation occurs at the boundary of the allowable parameter set