Object-image correspondence for curves under finite and affine cameras

TL;DR

This paper introduces a new algorithmic approach using Cartan's moving frame method to determine if a planar curve is an image of a spatial curve under unknown projection parameters, improving computational efficiency.

Contribution

It develops a novel, efficient algorithm for the projection problem of curves using affine and projective equivalence, reducing parameter complexity.

Findings

Algorithm significantly reduces parameters to check for projection existence.

Method applies to both curves and finite point sets.

Provides a practical solution for object-image correspondence under unknown camera parameters.

Abstract

We provide criteria 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. These criteria reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. The latter problem can be addressed using Cartan's moving frame method. This leads to a novel algorithmic solution of the projection problem for curves. The computational advantage of the algorithms presented here, in comparison to algorithms based on a straightforward solution, lies in a significant reduction of a number of real parameters that has 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. The same 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.