Geometry Seminar

Date: September 22, 2023

Time: 4:00PM - 5:00PM

Location: BLOC 302

Speaker: Timothy Duff, U. Washington Seattle

Title: Geometry of two, three, or four cameras

Abstract: I will introduce a line of work that aims to characterize the set of all valid algebraic constraints that relate any number of perspective cameras, 3D points, and their 2D projections. More formally, this framework involves the study of certain multigraded vanishing ideals. This leads to several new results, as well as new proofs of old results about the well-studied multiview ideal. For example, a "folklore theorem" from geometric computer vision states: "all algebraic constraints on the 2D projections of 3D points can be obtained from those involving 2, 3, or 4 cameras." (joint w/ S. Agarwal, E. Connelly, J. Loucks-Tavitas, R. Thomas) I will also discuss a complementary line of work focused on practical estimation methods. Incremental 3D reconstruction systems usually focus on estimating the relative orientation of two cameras. This in turn requires solving systems of algebraic equations with (very) special structure. I will describe recent progress extending the domain of such solvers to problems involving three or four cameras, including non-perspective cameras with lens distortion. The key players are numerical homotopy continuation methods, and the Galois/monodromy groups that capture their inherent complexity. (joint w/ P. Hruby, K. Kohn, V. Korotynskiy, V. Larsson, A. Leykin, L. Oeding, T. Pajdla, M. Pollefeys, M. Regan)