Neighborhood Decomposition of 3D Convex Structuring Elements for Morphological Operations

Abstract

Morphological operations with 3D images require a huge amount of computation. The decomposition of structuring elements used in the morphological operations such as dilation and erosion greatly reduces the amount of computation. This paper presents a new method for the decomposition of a 3D convex structuring element into a set of neighborhood structuring elements. A neighborhood structuring element is a convex structuring element consisting of a subset of a set consisting of the origin voxel and its 26 neighborhood voxels. First, we derive the set of decomposition conditions on the lengths of the original and the basis convex structuring elements, and then the decomposition problem is converted to linear integer optimization problem. The objective of the optimization is to minimize a cost function representing the optimal criterion for the implementation of morphological operations. Thus, our method can be used to obtain the different optimal decompositions minimizing the amount of computation in different cases.