Abstract
The Hausdorff metric plays a fundamental role in the convergence of compact binary images, especially with regard to mathematical morphology. The present paper investigates an extension of the Hausdorff metric to bounded gray-scale signals possessing compact domains. The extension is made via truncated umbrae, which, under the conditions of the paper, can be employed in place of full umbrae in the representation of morphological operations. The salient point regarding truncated umbrae is that they are compact for bounded upper semicontinuous signals. Once the basic morphological and metric properties of truncated umbrae are developed, convergence and continuity relative to the gray-scale Hausdorff metric are investigated, especially as they impact mathematical morphology. Finally, there is extension of binary Hausdorff-metric sampling theory to signal Hausdorff-metric sampling.
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