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Fast and stable evaluation of box-splines via the BB-form

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Abstract

To repeatedly evaluate linear combinations of box-splines in a fast and stable way, in particular along knot planes, the box-spline is converted to and tabulated as piecewise polynomial in BB-form (Bernstein–Bézier-form). We show that the BB-coefficients can be derived and stored as integers plus a rational scale factor and derive a hash table for efficiently accessing the polynomial pieces. This pre-processing, the resulting evaluation algorithm and use in a widely-used ray-tracing package are illustrated for splines based on two trivariate box-splines: the seven-directional box-spline on the Cartesian lattice and the six-directional box-spline on the face-centered cubic lattice.

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Authors and Affiliations

  1. Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, 32611-6120, USA

    Minho Kim & Jörg Peters

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  1. Minho Kim
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  2. Jörg Peters
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Correspondence to Minho Kim.

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Kim, M., Peters, J. Fast and stable evaluation of box-splines via the BB-form. Numer Algor 50, 381–399 (2009). https://doi.org/10.1007/s11075-008-9231-6

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  • DOI: https://doi.org/10.1007/s11075-008-9231-6

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