Image Invariants for Smooth Reflective Surfaces

Image invariants are those properties of the images of an object that remain unchanged with changes in camera parameters, illumination etc. In this paper, we derive an image invariant for smooth surfaces with mirror-like reflectance. Since, such surfaces do not have an appearance of their own but rather distort the appearance of the surrounding environment, the applicability of geometric invariants is limited. We show that for such smooth mirror-like surfaces, the image gradients exhibit degeneracy at the surface points that are parabolic. We leverage this result in order to derive a photometric invariant that is associated with parabolic curvature points. Further, we show that these invariant curves can be effectively extracted from just a few images of the object in uncontrolled, uncalibrated environments without the need for any apriori information about the surface shape. Since these parabolic curves are a geometric property of the surface, they can then be used as features for a variety of machine vision tasks. This is especially powerful, since there are very few vision algorithms that can handle such mirror-like surfaces. We show the potential of the proposed invariant using experiments on two related applications object recognition and pose estimation for smooth mirror surfaces.