Characterizing ∇2G filtered images by their zero crossings

This paper considers the characterization of\nabla^{2}Gfiltered images by their zero crossings. It has been suggested that\nabla^{2}Gfiltered images might be characterized by their zero crossings [1]. It is shown here that\nabla^{2}Gfiltered images, filtered in 1-D or 2-D are not, in general, uniquely given within a scalar by their zero crossing locations. Two theorems in support of such a suggestion are considered. We consider the differences between the requirements of Logan's theorem and\nabla^{2}Gfiltering, and show that the zero crossings which result from these two situations differ significantly in number and location. Logan's theorem is therefore not applicable to\nabla^{2}Gfiltered images. A recent theorem by Curtis [8] on the adequacy of zero crossings of 2-D functions is also considered. It is shown that the requirements of Curtis' theorem are not satisfied by all\nabla^{2}Gfiltered images. An example of two different\nabla^{2}Gfiltered images with the same zero crossings is presented.