Extremal feature extraction from 3-D vector and noisy scalar fields

We are interested in feature extraction from volume data in terms of coherent surfaces and 3D space curves. The input can be an inaccurate scalar or vector field, sampled densely or sparsely on a regular 3D grid, in which poor resolution and the presence of spurious noisy samples make traditional iso-surface techniques inappropriate. In this paper, we present a general-purpose methodology to extract surfaces or curves from a digital 3D potential vector field {(s,v~)}, in which each voxel holds a scalar s designating the strength and a vector v~ indicating the direction. For scalar, sparse or low-resolution data, we "vectorize" and "densify" the volume by tensor voting to produce dense vector fields that are suitable as input to our algorithms, the extremal surface and curve algorithms. Both algorithms extract, with sub-voxel precision, 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) crest-lines and ridges implicit in a digital terrain map, and (d) grooves, anatomical lines and complex surfaces from noisy dental data.