Mathematical Morphology via Category Theory
This work provides a theoretical framework for image processing researchers, offering a novel categorical perspective that could unify and extend morphological methods, though it appears incremental as it builds on existing concepts without demonstrating practical applications or performance gains.
The paper tackles the problem of generalizing mathematical morphology operations like dilation and erosion by reformulating them using category theory, specifically through limit and co-limit preserving functors and tensor-hom adjunction, enabling new types of operations between images represented with different semirings beyond Boolean and (max,+) semirings.
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.