Spatial-scaling-compatible morphological granulometries on locally convex topological vector spaces

Eugene J. Kraus; Henk J. A. M. Heijmans; Edward R. Dougherty

doi:10.1117/12.60649

1 June 1992 Spatial-scaling-compatible morphological granulometries on locally convex topological vector spaces

Eugene J. Kraus, Henk J. A. M. Heijmans, Edward R. Dougherty

Proceedings Volume 1769, Image Algebra and Morphological Image Processing III; (1992) https://doi.org/10.1117/12.60649
Event: San Diego '92, 1992, San Diego, CA, United States

Abstract

Ganulometries are defined on function classes over topological vector spaces. The usual Euclidean property for scaling compatibility, set scaling in the binary case and graph scaling in the gray-scale case, is changed so that it is with respect to spatial (domain) scaling for function spaces. As in the binary case, scaling compatible granulometries possess representations as double suprema over scaled generating elements. Without further constraint on the generating elements, the double supremum involves, for each generating element, all scalings exceeding the parameter of the particular granulometric operator. The salient theorem of the present paper concerns necessary and sufficient conditions under which there is a reduction of the double-supremum representation to a single supremum over singularly scaled generating functions. Specifically, and in the context of locally convex topological vector spaces, there is a determination of when a domain-scaled function t*f is f-open for all t < 1 . Key roles are played by both topology and local convexity.

Citation Download Citation

Eugene J. Kraus, Henk J. A. M. Heijmans, and Edward R. Dougherty "Spatial-scaling-compatible morphological granulometries on locally convex topological vector spaces", Proc. SPIE 1769, Image Algebra and Morphological Image Processing III, (1 June 1992); https://doi.org/10.1117/12.60649

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KEYWORDS

Vector spaces

Platinum

Binary data

Image processing

Image filtering

Mathematical modeling

Probability theory

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