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A general definition of mathematical morphology has been defined within thealgebraic framework of complete lattice theory. In this framework, dealing withdeterministic and increasing operators, a dilation (respectively an erosion) isan operation which is distributive over supremum (respectively infimum). Fromthis simple definition of dilation and erosion, we cannot say much about theproperties of them. However, when they form an adjunction, many importantproperties can be derived such as monotonicity, idempotence, and extensivity oranti-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 orsimplicial complexes. For all these algebraic structures, erosion and dilationare usually based on structuring elements. The goal is then to match thesestructuring elements on given objects either to dilate or erode them. One ofthe advantages of defining erosion and dilation based on structuring elementsis that these operations are adjoint. Based on this observation, this paperproposes to define, at the abstract level of category theory, erosion anddilation based on structuring elements. We then define the notion ofmorpho-category on which erosion and dilation are defined. We then show thattopos and more precisely topos of presheaves are good candidates to generatemorpho-categories. However, topos do not allow taking into account the notionof inclusion between substructures but rather are defined by monics up todomain isomorphism. Therefore we define the notion of morpholizable categorywhich allows generating morpho-categories where substructures are defined alonginclusion morphisms. {A direct application of this framework is to generalizemodal morpho-logic to other algebraic structures than simple sets.

Abstract Mathematical morphology based on structuring element: Application to morpho-logic