MRF Parameter Estimation

A probabilistic distribution function has two essential elements: the form of the function and the parameters involved. For example, the joint distribution of an MRF is characterized by a Gibbs function with a set of clique potential parameters; and the noise by a zero-mean Gaussian distribution parameterized by a variance. A probability model is incomplete if not all the parameters involved are specified, even if the functional form of the distribution is known. While formulating the forms of objective functions such as the posterior distribution has long been a subject of research in image and vision analysis, estimating the parameters involved has a much shorter history. Generally, it is performed by optimizing a statistical criterion, for example, using existing techniques such as maximum likelihood, coding, pseudo-likelihood, expectation-maximization,or Bayesian estimation.

The problem of parameter estimation can have several levels of complexity. The simplest is to estimate the parameters, denoted by θ, of a single MRF, F, from the data d, which are due to a clean realization, f, of that MRF. Treatments are needed if the data are noisy. When the noise parameters are unknown, they have to be estimated, too, along with the MRF parameters. The complexity increases when the given data are due to a realization of more than one MRF (e.g., when multiple textures are present in the image data, unsegmented). Since the parameters of an MRF have to be estimated from the data, partitioning the data into distinct MRF’s becomes a part of the problem. The problem is even more complicated when the number of underlying MRF’s is unknown and has to be determined. Furthermore, the order of the neighborhood system and the largest size of the cliques for a Gibbs distribution can also be part of the parameters to be estimated.