The Varieties of Bifocal Grassmann Tensors

TL;DR

This paper explores the algebraic and geometric properties of bifocal Grassmann tensors, which generalize fundamental matrices in multi-view geometry, analyzing their birational geometry and moduli spaces.

Contribution

It provides a comprehensive algebraic and geometric study of bifocal Grassmann tensors, including their birational classification and moduli space structure.

Findings

The variety of bifocal Grassmann tensors is birational to a homogeneous space.

There exists a dominant rational map from this variety to a Grassmannian.

The paper explicitly describes the duality via polarity in multi-view geometry.

Abstract

Grassmann tensors arise from classical problems of scene reconstruction in computer vision. In particular, bifocal Grassmann tensors, related to a pair of projections from a projective space onto view-spaces of varying dimensions, generalise the classical notion of fundamental matrices. In this paper we study in full generality the variety of bifocal Grassmann tensors focusing on its birational geometry. To carry out this analysis, every object of multi-view geometry is declined both from an algebraic and geometric point of view, e.g., the duality between the view spaces and the space of rays is explicitly described via polarity. Next, we deal with the moduli of bifocal Grassmann tensors, thus showing that this variety is both birational to a suitable homogeneous space and endowed with a dominant rational map to a Grassmannian.