# The Rank of Trifocal Grassmann Tensors

## TL;DR

This paper investigates the rank of trifocal Grassmann tensors in computer vision, providing a canonical form and explicit formulas under general conditions, and explores tensor rank behavior in degenerate configurations.

## Contribution

It introduces a canonical form for projection matrices and derives a closed-form formula for the tensor rank under general position assumptions.

## Key findings

- Derived a canonical form for combined projection matrices.
- Provided a closed formula for the rank of trifocal Grassmann tensors.
- Analyzed tensor rank behavior in degenerate projection configurations.

## Abstract

Grassmann tensors arise from classical problems of scene reconstruction in computer vision. Trifocal Grassmann tensors, related to three projections from a projective space of dimension k onto view-spaces of varying dimensions are studied in this work. A canonical form for the combined projection matrices is obtained. When the centers of projections satisfy a natural generality assumption, such canonical form gives a closed formula for the rank of the trifocal Grassmann tensors. The same approach is also applied to the case of two projections, confirming a previous result obtained with different methods in [6]. The rank of sequences of tensors converging to tensors associated with degenerate configurations of projection centers is also considered, giving concrete examples of a wide spectrum of phenomena that can happen.