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

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.