China
Africa
Explore chapters and articles related to this topic
Graph Morphology in Image Analysis
Published in Edward R. Dougherty, Mathematical Morphology in Image Processing, 2018
An operator is called a morphological filter if it is increasing and idempotent. Idempotence is an important property for an operator because it means that repeated application of such an operator has no further effect on the outcome. In a sense, one could argue that any operator (morphological or otherwise) designed to clean noise from an image has to be applied repeatedly until the result remains constant. In practice, such an iterative procedure results in idempotent operators; see Heijmans and Serra [12]. A formal theory for morphological filters has been designed by Matheron [16]. In this section we shall apply some of his results to the framework of gray-level graphs. Although we shall mainly be concerned with openings and closings and construction methods for such operators, we will consider alternating sequential filters at the end.
Logic Design Fundamentals
Published in Joseph Cavanagh, Computer Arithmetic and Verilog HDL Fundamentals, 2017
Idempotency relates to a nonzero mathematical quantity which, when applied to itself for a binary operation, remains unchanged. Thus, if x1 = 0, then x1 + x1 = 0 + 0 = 0 and if x1 = 1, then x1 + x1 = 1 + 1 = 1. Therefore, one of the elements is redundant and can be discarded. The dual is true for the operator •. The idempotent laws eliminate redundant variables in a Boolean expression and can be extended to any number of identical variables. This law is also referred to as the law of tautology, which precludes the needless repetition of the variable. For every x1 ∈ B,
An alternative construction of uninorms on bounded lattices
Published in International Journal of General Systems, 2023
In a general topology, by considering a nonempty set A and the set of all subsets of A, the closure operator (resp. interior operator) in is defined as an expansive, isotone and idempotent map (resp. a contractive, isotone and idempotent map ). Both of these operators can be used for constructing topologies in A in a general topology (Engelking 1989). More precisely, a one-to-one correspondence from the set of all topologies in A to the set of all closure (interior) operators in . That is, any topology in a nonempty set can induce the closure (interior) operator on its underlying powerset. It should be pointed out that closure and interior operators can be defined in a lattice of all subsets of a set A with the set union as the join and the set intersection as the meet. Hence, Everett (1944) extended the closure operator (resp. interior operator) in to a general lattice where the condition (resp. ) is omitted.
Analysis of P-time event graphs in (max,+) and (min,+) semirings
Published in International Journal of Systems Science, 2021
Pavel Špaček, Jan Komenda, Sébastien Lahaye
An idempotent semiring (also called dioid) is a set together with two operations (addition and multiplication), denoted, respectively, ⊕ and ⊗. with ⊕ forms an idempotent semigroup, i.e. ⊕ is commutative, associative, has a zero element ε ( for each ), and is idempotent: for each . Multiplication ⊗ is associative, has a unit element e, and distributes over ⊕. Moreover, ε is absorbing for ⊗, i.e. .
Formal analysis and control of timed automata with guards using (max, +) and (min, +) algebras
Published in International Journal of Systems Science, 2020
F. Ait Oumeziane, R. Kara, S. Amari
An idempotent semiring, also called Dioid, is a set D with two binary operations (addition and multiplication), denoted respectively by ⊕ and ⊗. The set D with ⊕ is an idempotent semigroup, i.e. ⊕ is commutative, associative, has a zero element ϵ (), and is idempotent: (). The operation ⊗ is associative, has a unit element e, and distributes over ⊕. Moreover, ϵ is absorbing for ⊗, i.e. .