We introduce a general extraction process for coherent surface and 3-D space curve extraction from volume data. Such volume input can be an inaccurate scalar or vecotr field, sampled densely or sparsely on a regular 3-D grid. It may be of poor resolution, and contain spurious noisy samples. for which various difficulties are posed to traditional iso-surface extraction techniques since an output is produced whenever an iso-value is satisfied. In this paper, we present a general-purpose methodology to extract surfaces or curves from a digital 3-D potenital vecotr field {(s,v-bar)}, in which each voexl holds a scalars designating strength, and a vector v-bar indicating direction. For scalar, sparse or low resolution data, we "tensorize" and "density" the volume by Tensor Voting to produce a dense vector field suitable as input to our algorithms, the Extremal Surface and Curve Algorithms. Both algorithms extract, with sub-voxel precison, coherent features representing local extrema in the given vector field. These coherent features are a hole-free triangulation mesh (in the surface case), and a set of connected. oriented, and non-intersecting polyline segments (in the curve case). We demonstrate the general usefulness of both extremal algorithms on a variety of real data by properly extracting their inherent extremal properties, such as (a) shock waves induced by abrupt velocity or direction changes in a flow field, (b) interacting vortex cores and vorticity lines in a velocity field, (c) crestlines and ridges implicit in a digital terrain map, and (d) grooves, anatomical lines and complex surfaces from noisy dental data.
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