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 prevailing obstacle is that the second derivatives entangle the three features, 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 the pattern with its prototype p. This is more strictly formulated as solving the derotating equation. The set of all possible prototypes spans the shape space of second-degree variation. This space is one-dimensional for 2D images, 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 shape in the prototype. The solution to the derotation and a full understanding of the result requires (i) mapping the derivatives of the pattern f onto the orthonormal basis of spherical harmonics, and (ii) identifying the eigenvalues of the Hessian with the derivatives of the prototype p. However, once the shape space is established, the possibilities of putting together independent discriminators for magnitude, orientation, and shape are easy and almost limitless. (C) 2001 Academic Press.