Efficient detection of second-degree variations in 2D and 3D images

Estimation of local second-degree variation should be a natural first step in computerized image analysis, just as it seems to be in human vision. A p revailing obstacle is that the second derivatives entangle the three featur es, signal strength (i.e., magnitude or energy), orientation, and shape. To disentangle these features we propose a technique where the orientation of an arbitrary pattern f is identified with the rotation required to align t he pattern with its prototype p. This is more strictly formulated as solvin g the derotating equation. The set of all possible prototypes spans the sha pe space of second-degree variation. This space is one-dimensional for 2D i mages, two-dimensional for 3D images. The derotation decreases the original dimensionality of the response vector from 3 to 2 in the 2D-case and from 6 to 3 in the 3D case, in both cases leaving room only for magnitude and sh ape in the prototype. The solution to the derotation and a full understandi ng of the result requires (i) mapping the derivatives of the pattern f onto the orthonormal basis of spherical harmonics, and (ii) identifying the eig envalues of the Hessian with the derivatives of the prototype p. However, o nce the shape space is established, the possibilities of putting together i ndependent discriminators for magnitude, orientation, and shape are easy an d almost limitless. (C) 2001 Academic Press.