When is one pinhole camera image equal to some other pinhole camera image?

TL;DR

This paper investigates when images from two different pinhole cameras are projectively equivalent, establishing a maximum of seven points for such equivalence and providing explicit geometric descriptions.

Contribution

It characterizes the conditions under which two different camera images are projectively equivalent, introducing the concept of centers-variety and using classical invariant theory.

Findings

Images of two point sets can be projectively equivalent only if each set has at most seven points.

Explicit descriptions of the centers-variety are provided for different point configurations.

The case of seven points involves a parametrization of the Goepel variety.

Abstract

Generically, one expects the images of two different point sets, in two different (projective) cameras, to be different. However, it can happen that the images are the same up to a projective transformation which is an instance of ill-posedness in computer vision. We prove that the images can become projectively equivalent only for point pairs with at most seven elements. In each case, we give explicit descriptions of the Zariski closure of the locus of camera centers which we call the centers-variety. To do this we use classical invariant theory and the geometry of moduli spaces of ordered points in the projective plane. The most involved case is that of seven points which uses a natural parametrization of the Goepel variety.