Geometric constraints and stereo disparity computation

Most stereo techniques compute disparity assuming that it varies slowly along surfaces. We quantify and justify this assumption, using weak assumptions about surface orientation distributions in the world to derive the density of disparity surface orientations. The small disparity change assumption is justified by the orientation density's heavy bias toward disparity surfaces that are nearly parallel to the image plane. In addition, the bias strengthens with smaller baselines, larger focal lengths, and as surfaces move farther from the cameras. To analyze current stereo techniques, we derive three densities from the first density, those of the disparity gradient magnitude, the directional derivative of disparity, and the difference in disparity between neighboring surface points. The latter may be used in Bayesian algorithms computing dense disparity fields. The directional derivative density and the disparity difference density both show that feature-based algorithms should strongly favor small disparity changes, contrary to several well-known algorithms. Finally, we use our original surface orientation density and the gradient magnitude density to derive two new ''surfaces-from-stereo'' techniques, techniques combining feature-based matching and surface reconstruction. The first uses the densities to severely restrict the search range for the optimum fit. The second incorporates the surface orientation density into the optimization criteria, producing a Bayesian formulation. Both algorithms are shown to be efficient and effective.