Object-image correspondence for curves under projections

TL;DR

This paper introduces an efficient algorithm to determine if a planar curve is an image of a spatial curve under unknown projections, reducing parameters and using differential signatures for curve equivalence.

Contribution

The novel algorithm significantly reduces the parameters needed to verify curve projections and applies differential signature methods for affine and projective curve equivalence.

Findings

Reduces the number of parameters to check for projections.

Uses differential signatures based on Cartan's moving frame.

Applicable to point set correspondences in 3D and 2D.

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. 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 main 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. The latter problem is then solved by differential signature construction based on Cartan's moving frame method. A similar approach can be used to decide whether a given finite set of ordered points on a plane is an image of a given finite set of ordered points in R^3. 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.