The Elementary Equation of the Conjugate Transformation for the Hexagonal Grid

In this paper the conjugate transformation of the hexagonal grid is described and its elementary equation is defined. Two strategies are used to extend a matrix morphology into the conjugate transformation. First, the conjugate classification represents 128 structuring elements of the kernel form of the hexagonal grid to a tree of six levels. Each node of a given level is a class of structuring elements with a calculable index. Two conjugate nodes of the same level with the same index can be distinguished by two conjugate sets of 2 * n classes respectively. Second, by considering each element which has six neighbours as a state for any Boolean matrix of the hexagonal grid, it can be transformed into an index matrix relevant to a specific level of the classification. From the index matrix, two sets of Boolean matrices (feature matrices) can be constructed with the same number of classes on the level. Depending on simpler algebraic properties of feature matrices, dilation and erosion can be unified to one operation, reversion, in the elementary equation. The reversion has a self-duality property with a space of 22*n functions in which only a total of 2n+1 functions are dilation and erosion. In addition, several images generated by applying morphological operations using an implemented prototype of the conjugate transformation and their running complexities compared with a matrix morphology, are illustrated. Owing to the class representation, the new scheme has more than a 4–8 speed-up ratio for the general applications.