Lines, Quadrics, and Cremona Transformations in Two-View Geometry

2023/08/05 by Erin Connelly, Connelly, Erin, Rekha R. Thomas +3
Engineering · Computer Science · #Advanced Numerical Analysis Techniques #Advanced Vision and Imaging #Digital Image Processing Techniques

paper · pdf · doi:10.48550/arxiv.2308.02757

Abstract

Given 7 ≤ k ≤ 9 points (x_i,y_i) ∈ ℙ² × ℙ², we characterize rank deficiency of the k × 9 matrix Z_k with rows x_i^\top ⊗ y_i^\top, in terms of the geometry of the point sets \x_i\ and \y_i\. This problem arises in the conditioning of certain well-known reconstruction algorithms in computer vision, but has surprising connections to classical algebraic geometry via the interplay of quadric surfaces, cubic curves and Cremona transformations. The characterization of rank deficiency of Z_k, when k ≤ 6, was completed in arXiv:2301.09826.

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