Characterizing ∇2G filtered images by their zero crossings

This paper considers the characterization of filtered images by their zero crossings. It has been suggested that filtered images might be characterized by their zero crossings [1]. It is shown here that filtered 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 filtering, 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 filtered 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 filtered images. An example of two different filtered images with the same zero crossings is presented.