Mathematical Morphology via Category Theory

TL;DR

This paper introduces a category-theoretic framework for mathematical morphology, redefining dilation and erosion using functors, tensor products, and semirings, which broadens the scope of morphological operations in image processing.

Contribution

It develops a novel categorical approach to morphological operations, enabling new types of dilation and erosion with different semirings and linking morphology to linear logic.

Findings

Categorical formulation of dilation and erosion.

Extension of morphological operations to various semirings.

Insight into mathematical morphology as a model for linear logic.

Abstract

Mathematical morphology contributes many profitable tools to image processing area. Some of these methods considered to be basic but the most important fundamental of data processing in many various applications. In this paper, we modify the fundamental of morphological operations such as dilation and erosion making use of limit and co-limit preserving functors within (Category Theory). Adopting the well-known matrix representation of images, the category of matrix, called Mat, can be represented as an image. With enriching Mat over various semirings such as Boolean and (max,+) semirings, one can arrive at classical definition of binary and gray-scale images using the categorical tensor product in Mat. With dilation operation in hand, the erosion can be reached using the famous tensor-hom adjunction. This approach enables us to define new types of dilation and erosion between two images represented by matrices using different semirings other than Boolean and (max,+) semirings. The viewpoint of morphological operations from category theory can also shed light to the claimed concept that mathematical morphology is a model for linear logic.