A Curvature Based Invariant for Object Recognition Derived from Covariance of Photometric Values

An invariant local descriptor of smooth object surfaces is Gaussian curvature. We present an object signature which is a condensed representation related to the object’s Gaussian curvature distribution over visible points. An invariant related to Gaussian curvature at an object point is developed based upon the covariance matrix of photometric values related to surface normals within a local neighborhood about the point. By employing three illumination conditions, two of which are completely unknown, we never need to explicitly know the surface normal at a point. The three-tuple of intensity values at a point is in one-to-one correspondence with the surface normal at that point. The determinant of the covariance matrix of these three-tuples in the local neighborhood of an object point is shown to be invariant with respect to rotation and translation. A way of combining these determinants over mutually illuminated object point regions to form a signature distribution is formulated that is rotation, translation, and, scale invariant. This signature is shown to be invariant over large ranges of poses of the same objects, while being significantly different between distinctly shaped objects. A new object recognition methodology is proposed by compiling signatures for only a few poses of a given object.