Hausdorff-metric continuity of projection-generated Fourier descriptors
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The present paper concerns Fourier descriptors resulting from waveforms generated by geometric projection: a pattern A is projected on a line of angle , and the pattern's waveform is given by Proj(A,), the length of the pattern's projection on the line. Rotating the line (varying ) generates the waveform and the pattern's descriptors are found by appropriately normalizing its DFT. Of interest is the behavior of projection- generated descriptors relative to the Hausdorff metric commonly employed in mathematical morphology, specifically, continuity of the descriptors relative to the Hausdorff metric. The fundamental proposition states that, as a mapping from the space of nonempty compact sets under the Hausdorff metric into the space of complex-valued sequences under the supremum norm, the projection-generated Fourier-descriptor transform is continuous. So long as we concern ourselves with nonempty compact sets, the basic morphological operations of erosion, dilation, opening, and closing are upper semicontinuous with respect to the Hausdorff metric; indeed, dilation is continuous. Hence, application of a morphological filter followed by computation of the projection-generated descriptors produces an upper semicontinuous operation (continuous in the case of dilation). Besides the general theory, the paper includes quantitative bounds on the descriptors for important morphological filters acting on noise images.
