论文标题
基于结构元素的抽象数学形态：应用于形态逻辑
Abstract Mathematical morphology based on structuring element: Application to morpho-logic
论文作者
论文摘要
在完整晶格理论的代数框架内定义了数学形态的一般定义。在此框架中，处理确定性和增加的操作员，扩张（分别是侵蚀）是一种分布超过超级（分别为immum）的操作。从扩张和侵蚀的简单定义中，我们不能对它们的特性说太多。但是，当它们形成邻接时，可以得出许多重要的特性，例如单调性，势力，扩展性或反延伸性的组成，保留原子和至上等等。数学形态在集合的环境中首先开发，然后在集合的环境中扩展，然后扩展到其他代数结构，例如其他构图，例如图形，超级或简单的复杂物。对于所有这些代数结构，侵蚀和扩张通常基于结构元素。然后，目标是匹配给定对象上的这些结构元素以扩张或侵蚀它们。基于结构元素定义侵蚀和扩张的优点之一是这些操作是伴随的。基于这一观察，本文提议在类别理论，基于结构元素的侵蚀和扩张的抽象层面上定义。然后，我们定义了定义侵蚀和扩张的形态类别的概念。然后，我们证明Topos和更精确的预示例是生成形态类别的好候选者。但是，TOPOS不允许考虑子结构之间包含的概念，而是由Monics定义为域等构象。因此，我们定义了可将形态化类别的概念，该类别允许在沿纳入形态定义的子结构中生成形态类别。 {此框架的直接应用是将模态形态逻辑概括到其他代数结构中而不是简单集合。
A general definition of mathematical morphology has been defined within the algebraic framework of complete lattice theory. In this framework, dealing with deterministic and increasing operators, a dilation (respectively an erosion) is an operation which is distributive over supremum (respectively infimum). From this simple definition of dilation and erosion, we cannot say much about the properties of them. However, when they form an adjunction, many important properties can be derived such as monotonicity, idempotence, and extensivity or anti-extensivity of their composition, preservation of infimum and supremum, etc. Mathematical morphology has been first developed in the setting of sets, and then extended to other algebraic structures such as graphs, hypergraphs or simplicial complexes. For all these algebraic structures, erosion and dilation are usually based on structuring elements. The goal is then to match these structuring elements on given objects either to dilate or erode them. One of the advantages of defining erosion and dilation based on structuring elements is that these operations are adjoint. Based on this observation, this paper proposes to define, at the abstract level of category theory, erosion and dilation based on structuring elements. We then define the notion of morpho-category on which erosion and dilation are defined. We then show that topos and more precisely topos of presheaves are good candidates to generate morpho-categories. However, topos do not allow taking into account the notion of inclusion between substructures but rather are defined by monics up to domain isomorphism. Therefore we define the notion of morpholizable category which allows generating morpho-categories where substructures are defined along inclusion morphisms. {A direct application of this framework is to generalize modal morpho-logic to other algebraic structures than simple sets.
