Changing Views on Curves and Surfaces
This work addresses a theoretical problem in computer vision for researchers in computational geometry and vision, but it appears incremental as it builds on existing algebraic geometry approaches.
The paper tackles the problem of understanding visual events in computer vision by analyzing how projected images of curves and surfaces change qualitatively when viewpoints cross visual event surfaces, deriving formulas for their degrees and showing how to compute exact representations using algebraic methods.
Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint. Qualitative changes in that curve occur when the viewpoint crosses the visual event surface. We examine the components of this ruled surface, and observe that these coincide with the iterated singular loci of the coisotropic hypersurfaces associated with the original curve or surface. We derive formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and show how to compute exact representations for all visual event surfaces using algebraic methods.