Intrinsically fuzzy approach to mathematical morphology
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Whereas gray-scale morphology has been formally interpreted in the context of fuzzy sets, heretofore there has not been developed a truly fuzzy mathematical morphology. Specifically, mathematical morphology is based on the notion of fitting, and rather than simply characterize standard morphological fitting in fuzzy terms, a true fuzzy morphology must characterize fuzzy fittings. Moreover, it should preserve the nuances of both mathematical morphology and fuzzy sets. In the present paper, we introduce a framework that satisfies these criteria. In contrast to the unusual binary or gray-scale morphology, herein erosion measures the degree to which one image is beneath (which is a subset type relation) another image, and it does so by employing an index for set inclusion. The result is a quite different `fitting' paradigm. Based on this new fitting approach, we define erosion, dilation, opening, and closing. The true fuzziness of the theory can be seen in a number of ways, one being that the dilation does not commute with union. (The commutativity lies at the heart of nonfuzzy lattice-based mathematical morphology.) However, we do arrive at a counterpart of Matheron's Representation Theorem for increasing translation-invariant mappings.
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Intelligent Robots and Computer Vision X: Algorithms and Techniques
