Gray-scale Mathematical Morphology Research Articles

A mathematical morphology approach to Euclidean distance transformation

A distance transformation technique for a binary digital image using a gray-scale mathematical morphology approach is presented. Applying well-developed decomposition properties of mathematical morphology, one can significantly reduce the tremendous cost of global operations to that of small neighborhood operations suitable for parallel pipelined computers. First, the distance transformation using mathematical morphology is developed. Then several approximations of the Euclidean distance are discussed. The decomposition of the Euclidean distance structuring element is presented. The decomposition technique employs a set of 3 by 3 gray scale morphological erosions with suitable weighted structuring elements and combines the outputs using the minimum operator. Real-valued distance transformations are considered during the processes and the result is approximated to the closest integer in the final output image.

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

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.