Invariants of objects and their images under surjective maps
This work addresses a theoretical problem in differential geometry and computer vision for researchers, providing incremental advances in understanding invariants under projections.
The paper tackles the problem of relating differential invariants of objects to those of their images under surjective maps, establishing a constructible isomorphism for projectable transformation groups and describing residual effects for non-projectable subgroups, with applications in reconstructing objects from multiple-view images using projections from 3D to 2D.
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.