| This dissertation extends and applies theoretical concepts of projective geometry to develop two new tools that have direct application to computer vision. These tools are: (i) the Omega invariant of a pair of conics, and (ii) a shadow model based on the theorem of Desargues. Invariance under projective transformations is a crucial property applied by vision researchers in order to identify objects from a single view. The Omega invariant appears to be tailor-made for the identification of a pair of conics in the plane. Omega reunites the algebra and the geometry, and it carries information about the relative positions of conics. We prove that Omega crosses zero exactly at positions of tangency. It has a wide variability range, which makes it suitable for fine recognition problems. We apply Omega to the practical problem of face recognition based on conics derived from key feature points on the image of a face. In the second part of this dissertation we introduce a novel invariant that serves as the bridge between 3D and 2D. It is derived from projective geometry reasoning over a shadow model based on the theorem of Desargues. We prove how relative positions of planes in space, like coplanarity, convexity or concavity transmit from a 3D scene to a single image that contains shadow information. This invariant works for parallel or non-parallel light rays, and it is independent of the position of the light source, point of view or shape of the objects. |
