Algorithm for Bicubic Nonuniform B-Spline Surface Interpolation

This Demonstration shows how to interpolate a set of matrix nets via a bicubic nonuniform B-spline surface and progressive-iterative approximation (PIA) technique. See Details for full implementation details.
  • Contributed by: Shutao Tang
  • (Northwestern Polytechnical University, Xi'an, China)


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We would like to use a bicubic nonuniform B-spline surface to interpolate a given set of matrix nets . We use the progressive-iterative approximation (PIA) algorithm rather than solving the control nets of a B-spline surface by a linear system. There are three main steps for the PIA algorithm.
1. Calculating the knot vectors and in two directions:
Let .
Then the knot vector is:
1.2 The knot vector :
In a similar way, the knot vector can be calculated.
2. The iterative process:
At the start of the iteration process, let
A bicubic nonuniform B-spline surface can be generated via the control nets by
Denote the first adjustment of the control net by
Then let
Again, a bicubic nonuniform B-spline surface can be defined by the control nets :
Generally, if the bicubic nonuniform B-spline surface was defined by iterations, denoting the adjustment of the control net as , then
In addition, let
So we could generate the bicubic nonuniform B-spline curve via the control nets .
Ultimately, the surface set can be generated, and H. Lin [1] has proved that this surface iteration format is convergent. Namely,
3. The error is given by
[1] 蔺宏伟, 王国瑾, 董辰世. 用迭代非均匀 B-spline 曲线(曲面)拟合给定点集[J]. 中国科学, 2003, 33(10), pp. 912–923.
H. Lin et al., "Use Iterative Non-Uniform B-Spline Curve (Surface) to Fitting Given Point Set [J]." China Science, 33(10), 2003 pp. 912–923 (in Chinese).
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