Efficient Detection of Second-Degree Variations in 2D and 3D Images Contents

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 2Dimages, two-dimensional for 3D-images. The derotation decreases the original dimensionality of the response vector from three to two in the 2D-case and from six to three 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 . But once the shape-space is established the possibilities to put together independent discriminators for magnitude, orientation, and shape are easy and almost limitless.