# Matrices dropping rank in codimension one and critical loci in computer vision

## TL;DR

This paper classifies matrices of size (n+1) x n that drop rank in codimension one, analyzing their role in critical loci for projective reconstruction in computer vision, and explores reconstruction instability experimentally.

## Contribution

It provides a complete classification of such matrices for n ≤ 3 and investigates the instability of reconstruction near non-linear critical loci components.

## Key findings

- Classified matrices of size (n+1) x n dropping rank in codimension one for n ≤ 3
- Analyzed the structure of critical loci in projective reconstruction
- Experimentally studied reconstruction instability near non-linear components

## Abstract

Critical loci for projective reconstruction from three views in four dimensional projective space are defined by an ideal generated by maximal minors of suitable $4 \times 3$ matrices, $N,$ of linear forms. Such loci are classified in this paper, in the case in which $N$ drops rank in codimension one, giving rise to reducible varieties. This leads to a complete classification of matrices of size $(n+1) \times n$ for $n \le 3,$ which drop rank in codimension one. Instability of reconstruction near non-linear components of critical loci is explored experimentally.