Birational Geometry of Critical Loci in Algebraic Vision

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 lie on 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 changes when one increases the number of views.