In [46], Haralick proposes the use of zero crossings of the second directional derivative of the image brightness function. This is theoretically the same as using maxima in the first directional derivatives, and in one dimension is the same as the LoG filter. The zero crossings are found by fitting two dimensional functions to the image brightness surface and then analyzing the resulting fit (see the following section). The functions used are ``discrete orthogonal polynomials of up to degree three''. There is a problem with ``phantom edges'' created by the second directional derivative at ``staircase structures''. Connectivity at junctions is poor.
In [39] Fleck describes the use of the second directional derivative for edge finding, with various extensions to the basic use of zero crossings. The problem of phantom edges is reduced with a test using the first and third derivatives. Image noise is reduced using topological sums; smoothing is achieved at potential edge points by measuring the local support they have for their ``edge point type'' classification. The overall algorithm (and in particular the noise reduction part) is computationally expensive.