Binary image representation by contour trees

There is a growing need in medical image processing to analyze segmented objects. In this study we are interested in analyzing morphological properties of complex structures such as the trabecular bone. Although, there are various shape description approaches proposed in the literature, there is not an adequate method to represent foreground object(s) morphology with respect to the background. In this article, we propose a way of representing binary images of any dimensions using graphs that emphasize connectivity of level-sets to foreground and background. We start by calculating the euclidean distance transform (EDT) to create a scalar field. Then the contour tree of this scalar field is calculated using a modified version of the algorithm proposed by Carr[1]. Contour trees are mostly used to visualize high dimensional scalar fields as they can put on view the critical points, i.e: local min, max and saddle points; however, their use in representing complex shapes have not been studied. We demonstrate the use of our method on artificial 2D images having different topologies as well as 3D μ-CT images of two bone biopsies. We show that the application of contour trees to complex binary data particularly prove useful when interpreting pore-networks at micro-scale. Further work to quantify foreground and background interconnectivity using certain graph theoretical methods is still under research.