Event Location: NSH 1507
Bio: Matthew Trager is a second year PhD student at Inria/ENS Paris. In 2014 he received a master’s in applied mathematics (Degree in Mathematics, Machine Learning and Computer Vision “MVA”) from École Normale Supérieure de Cachan. Previously, he received a master’s in pure mathematics from Scuola Normale Superiore in Pisa. His research interests revolve around geometric and algebraic aspects of computer vision.

Abstract: Given n>=2 projective cameras, point correspondences in different images can be characterized using conditions that are “multilinear” in homogeneous image coordinates (e.g., epipolar and trifocal constraints). These constraints are at the core of any structure-from-motion (SfM) system, where they are mainly used in two tasks: selecting matching points in different pictures, and estimating the camera parameters from these correspondences. However, already for n=3 views, many fundamental questions, such as how many multilinear relations (and which ones) are necessary and/or sufficient to characterize correspondences, are by no means trivial. In order to investigate this type of problems we use some elementary tools from algebraic geometry. The (closure of the) joint image, which is formed by the n-tuples of matching points, is an algebraic variety and our goal reduces to studying its polynomial characterizations. With this framework, it is possible to explicitly characterize the spurious set of solutions arising from any incomplete set of conditions, and we can prove several new results. We clarify a few subtle aspects which may have led to confusion in the past (e.g., linear independence for trilinearities) and point out a rather surprising fact (classical point-point-point trilinearities do not yield a complete characterization of correspondences). Finally, our algebraic approach does not rely on analytic tools such as Grassman-Cayley algebras, tensor calculus, or Plucker coordinates, which have previously been used for studying multi-view constraints.