Projections of Higher Dimensional Subspaces and Generalized Multiview Varieties

TL;DR

This paper generalizes multiview varieties to higher-dimensional subspaces, providing formulas for their dimensions, conditions for injectivity, calibration, and isomorphism, with applications in computer vision triangulation.

Contribution

It introduces a comprehensive framework for higher-dimensional multiview varieties, including dimension formulas and conditions for injectivity and calibration, advancing geometric understanding in computer vision.

Findings

Complete characterization of when the projection map is generically injective.

Formulas for the dimensions of higher-dimensional multiview varieties.

Conditions under which the multiview variety is isomorphic to its blowup.

Abstract

We present a generalization of multiview varieties as closures of images obtained by projecting subspaces of a given dimension onto several views, from the photographic and geometric points of view. Motivated by applications in Computer Vision for triangulation of world features, we investigate when the associated projection map is generically injective; an essential requirement for successful triangulation. We give a complete characterization of this property by determining two formulae for the dimensions of these varieties. Similarly, we describe for which center arrangements calibration of camera parameters is possible. We explore when the multiview variety is naturally isomorphic to its associated blowup. In the case of generic centers, we give a precise formula for when this occurs.