Gaussian curvature using fundamental forms for binary voxel data

Local curvature characterizes every point of a surface and measures its deviation from a plane, locally. One application of local curvature measures within the field of image and geometry processing is object segmentation. Here, we present and evaluate a novel algorithm based on the fundamental forms to calculate the curvature on surfaces of objects discretized with respect to a regular three-dimensional grid. Thus, our new algorithm is applicable to voxel data, which are created e.g. from computed tomography (CT). Existing algorithms for binary data used the Gauss map, rather than fundamental forms. For the calculation of the fundamental forms, derivatives of a surface in tangent directions in every point of the surface have to be computed. Since the surfaces exist on grids with restricted resolution, these derivatives have to be discretized. In the presented method, this is realized by projecting the tangent plane onto the discrete object surface. The most important parameter of the proposed algorithm is the size of the chosen window for the calculation of the gradient. The size of this window has to be selected according to object size as well as with respect to distances between objects. In our experiments, an algorithm based on the Gauss map provided inconsistent values for simple test objects, whereas our method provides consistent values. We report quantitative results on various test geometries, compare our method to two algorithms working on gray value data and demonstrate the practical applicability of our novel algorithm to CT-reconstructions of Greenlandic firn.

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