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Transactions on Machine Learning and Data Mining

(Print ISSN: 1865-6781) (ONLINE-ISSN: 2509-9337)

(ISBN: 978-3-942952-86-6)

## Volume 14 - Number 1 - July 2021 - Pages 3-30

### The Generalization of Mathematical Morphology to Non-numeric Sets

Xiaojin Ye^{1},

^{3}and Robert Haralick

^{3}

^{}

^{1}The Graduate Center, CUNY, New York,USA

^{2}Fordham University, Bronx, USA

^{3}The Graduate Center, CUNY, New York, USA

^{}

** Abstract**

It is well known that mathematical morphology plays an im-portant role in image analysis as it enables locating and detecting shapes as well as noise ﬁltering. This paper shows how many of the important properties in mathematical morphology hold in a much more general setting of symbolic or non-numeric sets. This includes the operations of dilation, erosion, opening and closing. For example, dilation of a union is the unions of dilations. Dilation is a union preserving operation. Erosion of an intersection is an intersection of erosions. Erosion is an intersec-tion preserving operation. If A is a subset of B, then the dilation of A is a subset of the dilation of B and the erosion of A is a subset of the erosion of B. There is a duality between dilation and erosion. Openings are formed by an erosion followed by dilation. Closings are formed by a dilation followed by erosion. Openings are idempotent: doing it more than once is the same as doing it once. Closings also are idempotent. And there are other properties of mathematical morphology that hold in the setting of arbitrary sets. Further that properties like idempotence of openings and closings happen in a setting of general sets whose elements are not numerical a nd where there are no numerical calculations and no orderings is surprising and unexpected.

** Keywords:**dilation, erosion, set operator, increasing operator, de-creasing operator, expansive operator, contractive operator, union pre-serving operator,
intersection preserving operator, set dilation operator, set erosion operator, dual operator, adjoint operator, opening operator, closing operator.

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