Complete Endomorphisms in Computer Vision

TL;DR

This paper introduces a mathematical framework based on complete endomorphisms to model and analyze degenerate transformations in multiple view geometry, extending beyond traditional homographies and fundamental matrices.

Contribution

It develops a completion of bilinear maps via equivariant compactification, enabling robust handling of degenerate cases in view correspondence analysis.

Findings

Handles degenerate transformations in view geometry.

Extends the applicability of fundamental and essential matrices.

Provides a robust mathematical framework for multiple view analysis.

Abstract

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.