I. BLASQUEZ and J.-F. POIRAUDEAU

University of Limoges

LICN-IUT

Allée André Maurois

87 065 LIMOGES CEDEX

FRANCE

e-mail: isabelle.blasquez@unilim.fr, poiraudeau@unilim.fr |

**Keywords:**
Minkowski functionals, image processing, binary decision diagram,
mathematical morphology.

**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.