基于结构元的数学形态学在形态逻辑中的应用
Abstract Mathematical morphology based on structuring element: Application to morpho-logic
摘要：
在完全格理论的代数框架下，给出了数学形态学的一般定义。在这个框架中，处理确定性算子和增算子，膨胀（分别是侵蚀）是一个在上确界（分别是下确界）上分布的运算。从膨胀和侵蚀的这个简单定义，我们不能说太多关于它们的性质。然而，当它们形成一个adjunction，许多重要的性质可以推导出，如单调性，恒等式，和扩展性或反扩展性的组成，保持下确界和上确界等数学形态学已首先发展的设置集，然后扩展到其他代数结构，如图，超图或单纯复形。对于所有这些代数结构，腐蚀和膨胀通常基于结构元素。然后，目标是在给定对象上匹配这些结构元素，以扩张或侵蚀它们。基于结构元素定义腐蚀和膨胀的优点之一是这些操作是伴随的。基于这一观察，本文提出在范畴论的抽象层次上定义基于结构元素的腐蚀和膨胀。然后，我们定义的概念上的腐蚀和膨胀的定义形态范畴。然后，我们表明，topos，更准确地说，topos的预层是很好的候选人生成形态类别。然而，拓扑不允许考虑子结构之间包含的概念，而是由直到域同构的一元论定义。因此，我们定义了morpholizable范畴的概念，它允许生成morpho范畴，其中子结构沿着包含态射定义。这个框架的一个直接应用是将模态形态逻辑推广到简单集合以外的其他代数结构。
英文摘要：
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.
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