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

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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