Fuzzy Relational Mathematical Morphology: Erosion and Dilation

Application of Mathematical Morphology in Solving the Profile of Forming Grinding Wheel

“…Te maximum values (x p max ), (y p max ) and minimum values (x p min ), (y p min ) are extracted to construct a matrix. Te matrix element at the position (Z p gi , R p gi ) is "1" in equation (22). Te other element in the matrix is "0."…”

Aggregation of Operators of Fuzzy Relational Mathematical Morphology: Erosion and Dilation

“…Recall that an unary operator c : L → L is called negation if it is an order reversing involution, that is a ≤ b =⇒ b c ≤ a c and (a c ) c = a for all a, b ∈ L. A lattice L endowed with a negation that is the tuple (L, ≤, ∧, ∨, c ) is called a De Morgan algebra; respectively, the tuple (L, ≤, ∧, ∨, * , c ) will be referred to as a De Morgan quantale. In a De Morgan quantale a further binary operation, co-product ⊕, can be defined by setting a ⊕ b = (a c * b c ) c for all a, b ∈ L. Co-product is a commutative associative monotone operation and, in case (L, ≤, ∧, ∨, * ) is integral, 0 L acts as a zero, that is a ⊕ 0 L = a for every a ∈ L. Important properties of De Morgan quantales are collected in the next lemmas: [15] If operation * in a De Morgan quantale (L, ≤, ∧, ∨, * ) i is meet-distributive, then the corresponding operation ⊕ distributes over arbitrary joins:…”

“…However, in our opinion, this is not a natural interpretation, neither it is appropriate for different manipulations with fuzzy morphological operators, in particular in the process of aggregation. In order to overcome this problem, in [15] we proposed a structured version of fuzzy relational morphological operators. It assumes intermediate use of the product and co-product in the definition of erosion and dilation and this created inconvenience in the study and especially in the use of such operators.…”