Edge Detection and Geometric Methods in Computer Vision

摘要

Basic problems of vision are studied form 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. 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 bias and gain. Small diagrams and level set topology are introduced as invariants usefull for matching, reducing the problem to graph or tree making. The level set topology tree is also proposed as a method of multi-scale description of the picture, and shown to be and invariant generalization of the 'scale space' technique.