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Approximating and intersecting surfaces from points

: Adamson, A.; Alexa, M.

Kobbelt, L. ; European Association for Computer Graphics -EUROGRAPHICS-:
Symposium on Geometry Processing 2003. Proceedings
New York: ACM, 2003
ISBN: 3-905673-06-1
Symposium on Geometry Processing (SGP) <1, 2003, Aachen>
Conference Paper
Fraunhofer IGD ()
ray tracing; object representation; curve representation; surface representation; solid representation

Point sets become an increasingly popular shape representation. Most shape processing and rendering tasks require the approximation of a continuous surface from the point data. We present a surface approximation that is motivated by an efficient iterative ray intersection computation. On each point on a ray, a local normal direction is estimated as the direction of smallest weighted co-variances of the points. The normal direction is used to build a local polynomial approximation to the surface, which is then intersected with the ray. The distance to the polynomials essentially defines a distance field, whose zero-set is computed by repeated ray intersection. Requiring the distance field to be smooth leads to an intuitive and natural sampling criterion, namely, that normals derived from the weighted co-variances are well defined in a tubular neighborhood of the surface. For certain, well-chosen weight functions we can show that well-sampled surfaces lead to smooth distance fields with non-zero gradients and, thus, the surface is a continuously differentiable manifold. We detail spatial data structures and efficient algorithms to compute ray-surface intersections for fast ray casting and ray tracing of the surface.