This paper offers computational theory and an algorithmic framework for perceptual organization of contours bounding opaque occluding surfaces of constant lightness. For any given visual scene, a sparse graph is constructed whose nodes are salient visual events such as contrast edges, and L-type and T-type junctions of contrast edges and whose arcs are coincidence and geometric configurational relations among node elements. An interpretation of the scene consists of choices among a small set of labels for graph elements reflecting physical events such as corners, visible surface occlusion, amodal continuation, and surface occlusion sans visible contrast edge (which perceptually give rise to illusory contours). Any given labeling induces an energy, or cost, associated with physical consistency and figural interpretation biases. Using the technique of deterministic annealing, optimization is performed such that local cues propagate smoothly to give rise to a global solution. We demonstrate that this approach leads to correct interpretations (in the sense of agreeing with human percepts) of popular simple "colorforms," figures known to induce illusory contours, as well as more difficult figures where interpretations acknowledging accidental alignment are preferred.