Birational geometry of critical loci in Algebraic Vision

TL;DR

This paper explores the algebraic structure of critical loci in the structure from motion problem, showing birational relations between two types of critical loci and introducing a unified locus to restore symmetry.

Contribution

It introduces a unified critical locus and proves birational relations between the two critical loci under mild smoothness conditions.

Findings

Two critical loci are birationally related under certain conditions.

A unified critical locus restores symmetry between the two critical loci.

The structure of critical loci changes with the number of views.

Abstract

In Algebraic Vision, the projective reconstruction of the position of each camera and scene point from the knowledge of many enough corresponding points in the views is called the structure from motion problem. It is known that the reconstruction is ambiguous if the scene points are contained in particular algebraic varieties, called critical loci. To be more precise, from the definition of criticality, for the same reconstruction problem, two critical loci arise in a natural way. In the present paper, we investigate the relations between these two critical loci, and we prove that, under some mild smoothness hypotheses, (some of) their irreducible components are birational. To this end, we introduce a unified critical locus that restores the symmetry between the two critical loci, and a natural commutative diagram relating the unified critical locus and the two single critical loci. For technical reasons, but of interest in its own, we also consider how a critical locus change when one increases the number of views.