A Probabilistic Theory of Occupancy and Emptiness

Abstract

Abstract. This paper studies the inference of 3D shape from a set of Ò 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.

Video-based, real-time multi-view stereo

“…Finally, a probabilistic approach to MVS has already been proposed in a number of papers [2,4,9,14,25]. However, while these methods model occlusion explicitly, our approach assumes probabilistic independence of the depth of different pixels and occlusion is implicitly modeled as another source of noise.…”

High Resolution Surface Reconstruction from Multi-view Aerial Imagery

“…These models do not make hard decisions about surface geometry and/or appearance; instead they explicitly represent uncertainties by assigning probabilities to multiple hypothesis within the volume. Early works along this line [4], [3] can be regarded as extensions of Space Carving [17], and more recently, algorithms based on generative models for the reverse image formation process have been introduced [20], [11]. Using Bayesian inference, these algorithms infer the maximum a-posteriori probabilities in the volume from the joint probability of all the images.…”

Multiperspective stereo matching and volumetric reconstruction

“…Another class of widely used 3D reconstruction algorithms are volumetric reconstruction [5,27,20,2,17]. Most existing algorithms in this category can be considered variations of the Space Carving framework by Kutulakos and Seitz [18].…”

“…For example, consider the projection of a poinṫ P (x, y, z) in two GLCs of identical types but translated by [−t x , −t y , 0]. The image ofṖ in the first GLC can be computed using Equation (2). To project P to the second GLC, we simply translateṖ by [t x , t y , 0].…”

“…The simplest approach to use the GLC decomposition for back-projecting a voxel is to go through every GLC in the tessellation and compute the image of the voxel using Equation (2). The search stops when the projected voxel lies inside the GLC triangle.…”