Anisotropic smoothing of posterior probabilities

Recently, we proposed an efficient image segmentation technique that anisotropically smoothes the homogeneous posterior probabilities before independent pixelwise MAP classification is carried out. In this paper, we develop the mathematical theory underlying the technique. We demonstrate that prior anisotropic smoothing of the posterior probabilities yields the MAP solution of a discrete MRF with a non-interacting, analog discontinuity field. In contrast, isotropic smoothing of the posterior probabilities is equivalent to computing the MAP solution of a single, discrete MRF using continuous relaxation labeling. Combining a discontinuity field with a discrete MRT is important as it allows the disabling of clique potentials across discontinuities. Furthermore, explicit representation of the discontinuity field suggests new algorithms that incorporate properties like hysteresis and non-maximal suppression.