When is it possible to identify 3D objects from single images using class constraints?

One approach to recognizing objects seen from arbitrary viewpoint is by extracting invariant properties of the objects from single images. Such properties are found in images of 3D objects only when the objects are constrained to belong to certain classes (e.g., bilaterally symmetric objects). Existing studies that follow this approach propose how to compute invariant representations for a handful of classes of objects. A fundamental question regarding the invariance approach is whether it can be applied to a wide range of classes. To answer this question it is essential to study the set of classes for which invariance exists. This paper introduces a new method for determining the existence of invariant functions for classes of objects together with the set of images from which these invariants can be computed. We develop algebraic tests that determine whether the objects in a given class can be identified from single images. These tests apply to classes of objects undergoing affine projection. In addition, these tests allow us to determine the set of views of the objects which are degenerate. We apply these tests to several classes of objects and determine which of them is identifiable and which of their views are degenerate.