Algorithms for generating B-nets and graphically displaying spline surfaces on three- and four-directional meshes

doi:10.1016/0167-8396(91)90032-7

Computer Aided Geometric Design

Volume 8, Issue 6, December 1991, Pages 479-493

https://doi.org/10.1016/0167-8396(91)90032-7 Get rights and content

Abstract

This paper is concerned with the description of an algorithm for generating the B-nets of box splines and an algorithm based on subdivision of these B-nets for graphically displaying spline surfaces on a three- or four-directional mesh. The combination of these two algorithms is shown to compare favorably over the other existing algorithms for this purpose.

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Box splines provide smooth spline spaces as shifts of a single generating function on a lattice and so generalize tensor-product splines. Their elegant theory is laid out in classical papers and a summarizing book. This compendium adds a succinct but exhaustive survey of the important sub-space of symmetric low-degree box splines on symmetric lattices with special focus on two and three variables. Tables contrast the complexity in terms of support size and polynomial degree, analytic and reconstruction properties, and list available implementations and code.
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Other families of evaluation techniques are based on Bernstein–Bézier (BB) forms to which each polynomial piece is converted. Chui and Lai [14] proposed a technique to generate BB-coefficients for a specific box-spline family. Kim and Peters [15] proposed an analytic evaluation technique for integer direction box-splines based on rational BB-coefficients.
We propose a new volume reconstruction technique based on the six-direction cubic box-spline $M_{6}$ . $M_{6}$ is $C^{1}$ continuous and possesses an approximation order of three, the same as that of the tri-quadratic B-spline but with much lower degree. In fact, $M_{6}$ has the lowest degree among the symmetric box-splines on $Z^{3}$ with at least $C^{1}$ continuity. We analyze the polynomial structure induced by the shifts of $M_{6}$ and propose an efficient analytic evaluation algorithm for splines and their derivatives (gradient and Hessian) based on the high symmetry of $M_{6}$ . To verify the evaluation algorithm, we implement a real-time GPU (graphics processing unit) isosurface raycaster which exhibits interactive performance (54.5 fps (frames per second) with $24 1^{3}$ dataset on $51 2^{2}$ framebuffer) on a modern graphics hardware. Moreover, we analyze $M_{6}$ as a reconstruction filter and state that it is comparable to the tri-cubic B-spline, which possesses a higher approximation order.
Multilevel refinable triangular PSP-splines (Tri-PSPS)
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A multi-level spline technique known as partial shape preserving splines (PSPS) (Li and Tian, 2011) has recently been developed for the design of piecewise polynomial freeform geometric surfaces, where the basis functions of the PSPS can be directly built from an arbitrary set of polygons that partitions a giving parametric domain. This paper addresses a special type of PSPS, the triangular PSPS (Tri-PSPS), where all spline basis functions are constructed from a set of triangles. Compared with other triangular spline techniques, Tri-PSPS have several distinctive features. Firstly, for each given triangle, the corresponding spline basis function for any required degree of smoothness can be expressed in closed-form and directly written out in full explicitly as piecewise bivariate polynomials. Secondly, Tri-PSPS are an additive triangular spline technique, where the spline function built from a given triangle can be replaced with a set of refined spline functions built on a set of smaller triangles that partition the initial given triangle. In addition, Tri-PSPS are a multilevel spline technique, Tri-PSPS surfaces can be designed to have a continuously varying levels of detail, achieved simply by specifying a proper value for the smoothing parameter introduced in the spline functions. In terms of practical implementation, Tri-PSPS are a parallel computing friendly spline scheme, which can be easily implemented on modern programmable GPUs or on high performance computer clusters, since each of the basis functions of Tri-PSPS can be directly computed independent of each other in parallel.
Symmetric box-splines on root lattices
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Root lattices are efficient sampling lattices for reconstructing isotropic signals in arbitrary dimensions, due to their highly symmetric structure. One root lattice, the Cartesian grid, is almost exclusively used since it matches the coordinate grid; but it is less efficient than other root lattices. Box-splines, on the other hand, generalize tensor-product $B$ -splines by allowing non-Cartesian directions. They provide, in any number of dimensions, higher-order reconstructions of fields, often of higher efficiency than tensored $B$ -splines. But on non-Cartesian lattices, such as the BCC (Body-Centered Cubic) or the FCC (Face-Centered Cubic) lattice, only some box-splines and then only up to dimension three have been investigated.
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Sampling and reconstruction of generic multivariate functions is more efficient on non-Cartesian root lattices, such as the BCC (Body-Centered Cubic) lattice, than on the Cartesian lattice. We introduce a new $n \times n$ generator matrix $A^{*}$ that enables, in $n$ variables, efficient reconstruction on the non-Cartesian root lattice $A_{n}^{*}$ by a symmetric box-spline family $M_{r}^{*}$ . $A_{2}^{*}$ is the hexagonal lattice and $A_{3}^{*}$ is the BCC lattice. We point out the similarities and differences of $M_{r}^{*}$ with respect to the popular Cartesian-shifted box-spline family $M_{r}$ , document the main properties of $M_{r}^{*}$ and the partition induced by its knot planes and construct, in $n$ variables, the optimal quasi-interpolant of $M_{2}^{*}$ .
Quasi-interpolation projectors for box splines
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^*: Research supported by NSF under Grant # DMS-89-0-01345 and SDIO/IST managed by the U.S. Army Research Office under Contract No. DAAL03-90-G-0091.

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Algorithms for generating B-nets and graphically displaying spline surfaces on three- and four-directional meshes

Abstract

J. Comput. Appl. Math.

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer-Aided Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Subdividing multivariate splines

Computer-Aided Design

A survey of curve and surface methods in CAGD

Computer Aided Geometric Design

B-splines from parallelepipeds

J. Analyse Math.

Bivariate cardinal interpolation by splines on a three-direction mesh

Illinois J. Math.

Convergence of bivariate cardinal interpolation

Constr. Approx.