Application of the Hausdorff metric in gray-scale mathematical morphology via truncated umbrae

doi:10.1016/1047-3203(91)90007-3

Journal of Visual Communication and Image Representation

Volume 2, Issue 2, June 1991, Pages 177-187

https://doi.org/10.1016/1047-3203(91)90007-3 Get rights and content

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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Regular articleApplication of the Hausdorff metric in gray-scale mathematical morphology via truncated umbrae

Abstract

Comput. Vision Graphics Image Process

Comput. Vision Graphics Image Process

Random Sets and Integral Geometry

Image Analysis and Mathematical Morphology

The digital morphological sampling theorem

Fractal modeling of real world images

Robust Morphologically Continuous Fourier Descriptors Generated by Geometric Projections

Hausdorff-Metric Interpretation of Convergence in the Matheron Topology for Binary Mathematical Morphology

Regular article
Application of the Hausdorff metric in gray-scale mathematical morphology via truncated umbrae