Characterizing three-dimensional open cell structures without segmentation

Foam cells, particle conglomerates, biological tissue slices and colloidal suspensions are just a few examples of collections that create an image with multiple touching or overlapping regions. The characterization of the open cell size of such a continuous structure is tedious and computationally intensive for large 3D data sets. Typically, it is accomplished by segmenting the cells with a watershed technique and aggregating the statistics of all regions found. This paper provides the mathematical foundation for a newly discovered relationship between the average pixel value of a Euclidean Distance Map (EDM) and the radius of a conic section. The implementation of this relationship allows for a computationally simple and accurate characterization of the aggregate diameter associated with these open cell structures without segmentation.