Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 023, 31 pages      arXiv:1303.3358      https://doi.org/10.3842/SIGMA.2013.023
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Object-Image Correspondence for Algebraic Curves under Projections

Joseph M. Burdis, Irina A. Kogan and Hoon Hong
North Carolina State University, USA

Received October 01, 2012, in final form March 01, 2013; Published online March 14, 2013; Proof of Theorem 4 corrected May 08, 2015; Equation (48) corrected March 10, 2019

Abstract
We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. To solve the latter problem we make an algebraic adaptation of signature construction that has been used to solve the equivalence problems for smooth curves. We introduce a notion of a classifying set of rational differential invariants and produce explicit formulas for such invariants for the actions of the projective and the affine groups on the plane.

Key words: central and parallel projections; finite and affine cameras; camera decomposition; curves; classifying differential invariants; projective and affine transformations; signatures; machine vision.

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