Efficient processing of Minkowski functionals on a 3D binary image using binary decision diagrams

Abstract

The Morphological Image Analysis characterizes binary digitized 3D images in terms of shape (geometry) and connectivity (topology) by means of the Minkowski functionals known from integral geometry. In three dimensions, these functionals correspond to the enclosed volume, surface area, mean breadth and connectivity (Euler characteristic). To compute these functionals, it is necessary to count the number of open cubes, open faces, open edges and open vertices of the discretized object in the 3D image. In this paper we propose a new method to count the number of these geometric elements in a discretized binary image. We focus on the local configuration around a voxel and we report a fast algorithm for computing discrete Minkowski functionals with related topological conditions using binary decision diagrams. These diagrams could be applied to several binary image processing algorithms which evaluate a discrete function for small parts of this image. We also choose to create and implement a reduced and ordered triple-ADD adapted to our problem. We show that this algorithm is 17 times faster than the algorithm proposed recently in the literature by Michielsen. Moreover, large volumes of data, which become increasingly accessible and current,can be treated thanks to this algorithm.