2020/02/20 by Javier Finat, Finat, Javier, Francisco Javier Delgado del Hoyo +2
Computer Science · Engineering · Mathematics · #Advanced Vision and Imaging #Algebra over a field #Bilinear interpolation #Compactification (mathematics) #Computer Vision and Pattern Recognition (cs.CV) #Degenerate energy levels #Endomorphism #Equivariant map #FOS: Computer and information sciences #Mathematics #Optical measurement and interference techniques #Pure mathematics #Robotics (cs.RO) #Robotics and Sensor-Based Localization #Vector space #cs.CV #cs.RO
paper · pdf · doi:10.48550/arxiv.2002.09003
22 pages, 2 figures
arxiv created 2020/02/20 · openalex publication_date 2020/02/20 · arxiv updated 2020/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Correspondences between k-tuples of points are key in multiple view geometry and motion analysis. Regular transformations are posed by homographies between two projective planes that serves as structural models for images. Such transformations can not include degenerate situations. Fundamental or essential matrices expand homographies with structural information by using degenerate bilinear maps. The projectivization of the endomorphisms of a three-dimensional vector space includes all of them. Hence, they are able to explain a wider range of eventually degenerate transformations between arbitrary pairs of views. To include these degenerate situations, this paper introduces a completion of bilinear maps between spaces given by an equivariant compactification of regular transformations. This completion is extensible to the varieties of fundamental and essential matrices, where most methods based on regular transformations fail. The construction of complete endomorphisms manages degenerate projection maps using a simultaneous action on source and target spaces. In such way, this mathematical construction provides a robust framework to relate corresponding views in multiple view geometry.