A Probabilistic Theory of Occupancy and Emptiness

This paper studies the inference of 3D shape from a set of n noisy photos. We derive a probabilistic framework to specify what one can infer about 3D shape for arbitrarily-shaped, Lambertian scenes and arbitrary viewpoint configurations. Based on formal definitions of visibility, occupancy, emptiness, and photo-consistency, the theoretical development yields a formulation of the Photo Hull Distribution, the tightest probabilistic bound on the scene’s true shape that can be inferred from the photos. We show how to (1) express this distribution in terms of image measurements, (2) represent it compactly by assigning an occupancy probability to each point in space, and (3) design a stochastic reconstruction algorithm that draws fair samples (i.e., 3D photo hulls) from it. We also present experimental results for complex 3D scenes.