Stereo, Shading, and Surfaces: Curvature Constraints Couple Neural Computations

Vision problems are inherently ambiguous: Do abrupt brightness changes correspond to object boundaries? Are smooth intensity changes due to shading or material properties? For stereo: Which point in the left image corresponds to which point in the right one? What is the role of color in visual information processing? To answer these (seemingly different) questions we develop an analogy between the role of orientation in organizing visual cortex and tangents in differential geometry. Machine learning experiments suggest using geometry as a surrogate for high-order statistical interactions. The cortical columnar architecture becomes a bundle structure in geometry. Connection forms within these bundles suggest answers to the above questions, and curvatures emerge in key roles. More generally, our path through these questions suggests an overall strategy for solving the inverse problems of vision: decompose the global problems into networks of smaller ones and then seek constraints from these coupled problems to reduce ambiguity. Neural computations thus amount to satisfying constraints rather than seeking uniform approximations. Even when no global formulation exists one may be able to find localized structures on which ambiguity is minimal; these can then anchor an overall approximation.