Better Pasting Through Quasi-Interpolation

Abstract

Hierarchical surface pasting, developed by Barghiel, Bartels and Forsey, is an interactive surface modelling technique that is used to add local detail to a tensor product B-spline surface without increasing the overall complexity of the original surface. Surface pasting approximates displacement mapped surfaces by placing additional tensor product surfaces, called features, on the base surface. The features may be scaled, rotated, and translated arbitrarily over the base surface, and the composite surface can be evaluated using relatively little computational power. Because the feature only approximates a displacement map, a surface produced using standard surface pasting often has noticeable gaps at the edges of the pasted feature. The severity of the surface discontinuities may be made as small as desired via knot insertion, but this can result in an unacceptable degradation in the performance of the modelling software.

Quasi-interpolation is an approximation method that approximates curves with spline curves to a high degree of accuracy. The approximation is constructed by computing coefficients that are used to weight samplings of the curve to be approximated. The Lyche-Schumaker quasi-interpolant uses coefficients that are inexpensive to compute and samplings that are relatively expensive to compute. I propose an improved surface pasting technique that uses quasi-interpolation to set the feature's boundary control vertices. For surface pasting it is necessary to have quasi-interpolants that reproduce position and derivatives at the endpoints of the curve to be approximated. In addition, as the feature surface is moved, the curve samplings must be recomputed, but the coefficients remain fixed. Therefore, I have developed a variation of the quasi-interpolant whose coefficients are expensive to compute, but whose samplings are relatively inexpensive to compute.

The Paper

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Bibliography Entry

@mastersthesis{conrad99,
    author = "B. Conrad",
    title = "Better Pasting Through Quasi-Interpolation",
    year = "1999",
    school = "University of Waterloo",
    address = "Waterloo, Ontario, Canada N2L 3G1",
    note = "Available on WWW as
           mbox{ftp://cs-archive.uwaterloo.ca/cs-archive/CS-99-14/}"
}