The connectivity of microstructures is directly related to the physical properties of materials. Currently, the Euler number is the most popular measure of connectivity. It is an elegant topological invariant, however, it does not provide information about cavities or the proximities and sizes of objects. In this 7, an alternative measure called contour tree connectivity (CTC) is developed and its applications for the analysis of microstructures are studied. CTC is derived from contour trees that are used in the first publication to represent complex binary images with simple graphs. By analyzing contour trees, CTC produces new connectivity information that is not provided by other approaches described in the literature. Contour tree representation of binary images and CTC can be computed for any dimensions of data and topology as explained in the second publication. Moreover, CTC is designed to be a scalar between 0 and 1, which makes it easy to use and understand. In this 7, the use of CTC for analyzing microstructures is presented in two studies. In the first study, the microstructure of trabecular bone is analyzed in relation to its mechanical strength. In the second study, the relationship between microstructures and the fluid flow within materials are examined. The results from these studies show that CTC contributes to the understanding of how the structural properties of materials are linked to their physical properties. To conclude, with its unique properties, CTC complements the structural information provided by currently used measures. This makes it an important image analysis tool for the study of the microstructures of materials such as soil, paper, filters and food products as well as biomaterials and biological tissues.
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