Boundary Detection by Constrained Optimization

We use a statistical framework for finding boundaries and for partitioning scenes into homogeneous regions. The model is a joint probability distribution for the a r ray of pixel gray levels and an a r ray of “labels.” In boundary finding, the labels a re binary, zero, or one, representing the absence o r presence of boundary elements. In partitioning, the label values a r e generic: two labels a re the same when the corresponding scene locations a r e considered to belong to the same region. The distribution incorporates a measure of disparity between certain spatial features of pairs of blocks of pixel gray levels, using the Kolmogorov-Smirnov nonparametric measure of difference between the distributions of these features. Large disparities encourage intervening boundaries and distinct partition labels. The number of model parameters is minimized by forbidding label configurations that a r e inconsistent with prior beliefs, such as those defining very small regions, o r redundant or blindly ending boundary placements. Forbidden configurations a re assigned probability zero. We examine the MAP (marimum a posterion’) estimator of boundary placements and partitionings. The forbidden states introduce constraints into the calculation of these configurations. Stochastic relaxation methods a re extended to accommodate constrained optimization, and experiments are performed on some texture collages and some natural scenes. i Zndex Terms-Annealing, Bayesian inference, boundary finding, constrained optimization, Gibbs distribution, MAP estimate, Markov random field, segmentation, stochastic relaxation, texture discrimina-