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The distortion varieties of a given projective variety are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions. These are based on Chow polytopes and Gröbner bases. Multi-parameter distortions are studied using tropical geometry. The motivation for distortion varieties comes from multi-view geometry in computer vision. Our theory furnishes a new framework for formulating and solving minimal problems for camera models with image distortion.
KeywordsDistortion varieties Toric varieties Image distortion Minimal problems
Mathematics Subject Classification14Q15 13P15 68T45 14M25 13P25 14T05 68U10
This project started at the Algebraic Vision workshop (May 2016) at the American Institute of Mathematics (AIM) in San Jose. We are grateful to the organizers, Sameer Agarwal, Max Lieblich and Rekha Thomas, for bringing us together. Joe Kileel and Bernd Sturmfels were supported by the US National Science Foundation (DMS-1419018). Zuzana Kukelova was supported by the Czech Science Foundation (GACR P103/12/G084). Part of this study was carried out while she worked for Microsoft Research, Cambridge, UK. Tomas Pajdla was supported by H2020-ICT-731970 LADIO, the Robotics for Industry 4.0 Project CZ.02.1.01/0.0/0.0/15_003/0000470 and the European Regional Development Fund.
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