Multiresolution Analysis of 3D Images Based on Discrete Distortion

We consider a model of a 3D image obtained by discretizing it into a multiresolution tetrahedral mesh known as a hierarchy of diamonds. This model enables us to extract crack-free approximations of the 3D image at any uniform or variable resolution, thus reducing the size of the data set without reducing the accuracy. A 3D intensity image is a scalar field (the intensity field) defined at the vertices of a 3D regular grid and thus the graph of the image is a hypersurface in R⁴. We measure the discrete distortion, a generalization of the notion of curvature, of the transformation which maps the tetrahedralized 3D grid onto its graph in R⁴. We evaluate the use of a hierarchy of diamonds to analyze properties of a 3D image, such as its discrete distortion, directly on lower resolution approximations. Our results indicate that distortion-guided extractions focus the resolution of approximated images on the salient features of the intensity image.