Invariants of objects and their images under surjective maps

TL;DR

This paper explores how differential invariants of objects relate to those of their images under surjective maps, especially in the context of reconstructing objects from multiple-view images in computer vision.

Contribution

It establishes a constructible isomorphism between invariants of images and gauge invariants of objects for projectable group actions, and describes residual effects for non-projectable cases.

Findings

Established isomorphism for projectable group actions.

Analyzed residual effects of non-projectable groups.

Applied results to multi-view image reconstruction.

Abstract

We examine the relationships between the differential invariants of objects and of their images under a surjective map. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the image, and the case when only a proper subgroup of the entire group acts projectably. In the former case, we establish a constructible isomorphism between the algebra of differential invariants of the images and the algebra of fiber-wise constant (gauge) differential invariants of the objects. In the latter case, we describe residual effects of the full transformation group on the image invariants. Our motivation comes from the problem of reconstruction of an object from multiple-view images, with central and parallel projections of curves from three-dimensional space to the two-dimensional plane serving as our main examples.