Minkowski addition

Minkowski addition is a mathematical operation that involves taking the algebraic sum of two nonempty subsets, A and B, of a set Y. It is also known as the Hausdorff sum and involves adding each element of A to each element of B. This operation can also be extended to scalar multiplication, where tA represents the set obtained by multiplying each element of A by a scalar t.From: A unified scheme for scalarization in set optimization [2023]

Nonlinear Filters

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Published in Edward R. Dougherty, Digital Image Processing Methods, 2020

Jaakko Astola, Edward R. Dougherty

There are equivalent formulations of dilation that deserve mention. The historical Minkowski addition formulation isA⊕B=∪{A+b:b∈B} By commutativity, A⊕B=∪{B+a:a∈A} The formulation of Eq. (66) is often used for image-processing intuition: dilation by B is formed by translating B to all points of A and then forming the union. A third formulation of dilation involves translates of the reflected structuring element that “hit” (intersect) the input image: A⊕B={x:(−b+x)∩A≠∅} Our central interest is morphological filters. Generically, a binary-image filter is an operator V that takes as input a binary image and outputs a binary image. If A is the input image, Ψ(A) denotes the output. For instance, if Ψ is erosion by B, then Ψ (A) = A ⊖ B. The study of any class of filters is the study of the properties defining the class. We focus on four properties, two immediately and two subsequently.

Maximal pairs of convex sets and Zalgaller's minimal representation of dc-functions

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Published in Optimization, 2022

Jerzy Grzybowski, Robert Kolassa

First, in Section 2, we present properties of Minkowski addition and subtraction in the semigroup of closed convex sets sharing a fixed recession cone. We define a relation of equivalence between pairs of such sets and introduce the notion of the maximal pair in a quotient class. The notion of maximal pair is complementary to the notion of 0 – minimal pair of sets studied by Grzybowski and Przybycień and Urbański [19, 20].

Minkowski addition

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Nonlinear Filters

Maximal pairs of convex sets and Zalgaller's minimal representation of dc-functions