Bayesian segmentation via asymptotic partition functions

Asymptotic approximations to the partition function of Gaussian random fiel ds are derived. Textures are characterized via Gaussian random fields induc ed by stochastic difference equations determined by finitely supported, sta tionary, linear difference operators, adjusted to be nonstationary at the b oundaries. It is shown that as the scale of the underlying shape increases, the log-normalizer converges to the integral of the log-spectrum of the op erator inducing the random field. Fitting the covariance of the fields amou nts to fitting the parameters of the spectrum of the differential operator- induced random field model. Matrix analysis techniques are proposed for han dling textures with variable orientation. Examples of texture parameters es timated from training data via asymptotic maximum-likelihood are shown. Iso tropic models involving powers of the Laplacian and directional models invo lving partial derivative mixtures are explored. Parameters are estimated fo r mitochondria and actin-myocin complexes in electron micrographs and clutt er in forward-looking infrared images. Deformable template models are used to infer the shape of mitochondria in electron micrographs, with the asympt otic approximation allowing easy recomputation of the partition function as inference proceeds.