EDGE DETECTION AND GEOMETRIC METHODS IN COMPUTER VISION (DIFFERENTIAL TOPOLOGY, PERCEPTION, ARTIFICIAL INTELLIGENCE, LOW-LEVEL)

Basic problems of vision are studied from the viewpoint of modern differential topology and geometry; primarily: edge detection, stereo matching, picture representation at multiple scales, and motion. Some mathematical background is provided for the nonexpert.; A comprehensive review of edge detection is presented, including analyses from a mathematical perspective as well as evaluations of practical performance and promise.; Some new edge detection techniques are introduced, including a nonlinear operator based on a symmetry principle, a variational approach to global edge finding, and a least-squares localization method. A theorem is proved which shows that localizing edge position and orientation requires at least 2 orientation dependent families of convolution operators.; A unifying mathematical structure is presented for vision, notably stereo, motion stereo, optic flow (apparent flow of visual space under motion), and matching. The general matching problem is analyzed, and it is proved that generically, general matching is highly degenerate for monochrome pictures, but has a unique solution for 2 or more color dimensions. The result is extended to pictures with unknown bias and gain. Smale diagrams and level set topology are introduced as invariants useful for matching, reducing the problem to graph or tree matching. The level set topology tree is also proposed as a method of multi-scale description of the picture, and shown to be an invariant generalization of the "scale space" technique.; The motion problem is analyzed using Lie group methods, and a theorem is proved establishing that generically 6 simultaneous values of time derivative of the monochrome picture function are necessary and sufficient to uniquely specify the 3-dimensional rigid motion of a generic given object. For 2 or more color dimensions, this is reduced to values at 3 points in the picture.