On point configurations, Carlsson-Weinshall duality, and multi-view geometry

Matthew Trager (1) , Martial Hebert (2) , Jean Ponce (1)

1 WILLOW - Models of visual object recognition and scene understanding
2 CMU - Carnegie Mellon University [Pittsburgh]

This paper proposes projective point configurations as a natural setting for studying perspective projection in a geometric, coordinate-free manner. We show that classical results on the effect of permutations on point configurations give a purely synthetic formulation of the well known analytical Carlsson-Weinshall duality between camera pin-holes and scene points. We further show that the natural parameterizations of configurations in terms of subsets of their points provides a new and simple analytical formulation of Carlsson-Weinshall duality in any scene and image coordinate systems, not just in the reduced coordinate frames used traditionally. When working in such reduced coordinate systems, we give a new and complete characterization of multi-view geometry in terms of a reduced joint image and its dual. We also introduce a new parametrization of trinocular geometry in terms of reduced trilinearities, and show that, unlike trifocal tensors, these are not subject to any nonlinear internal constraints. This leads to purely linear primal and dual structure-from-motion algorithms , that we demonstrate with a preliminary implementation on real data.

Domaines: Vision par ordinateur et reconnaissance de formes [cs.CV]

Matthew Trager, Martial Hebert, Jean Ponce. On point configurations, Carlsson-Weinshall duality, and multi-view geometry. 2018. ⟨hal-01676732v1⟩