Morphology Analysis of 3D Scalar Fields Based on Morse Theory and Discrete Distortion

We investigate a morphological approach to the analysis and understanding of 3D scalar fields defined by volume data sets. We consider a discrete model of the 3D field obtained by discretizing its domain into a tetrahedral mesh. We use Morse theory as the basic mathematical tool which provides a segmentation of the graph of the scalar field based on relevant morphological features (such as critical points). Since the graph of a discrete 3D field is a tetrahedral hypersurface in 4D space, we measure the distortion of the transformation which maps the tetrahedral decomposition of the domain of the scalar field into the tetrahedral mesh representing its graph in R4, and we call it discrete distortion. We develop a segmentation algorithm to produce a Morse decompositions associated with the scalar field and its discrete distortion. We use a merging procedure to control the number of 3D regions in the segmentation output. Experimental results show the validity of our approach.