It is essentially that from the observed data sets, we should be able to produce curves surfaces that have same characteristics as the original data sets. For instance, if the given data is positive, then the resulting curve or surface must be positive on entire given intervals i.e., everywhere. In this study, a new partial blended rational bi-quartic spline with C1 continuity is constructed through partially blended scheme. This rational spline is defined on four corners of the rectangular meshes. The sufficient condition for the positivity of rational bi-quartic spline is derived on four boundary curves network. There are eight (8) free parameters that can be used for shape modification. The first order partial derivatives are estimated by using numerical techniques. We also show that the proposed scheme is local quadratic reproducing such that it can exactly reproduce the quadratic surface. We test the proposed scheme to interpolate various types of positive surface data. Based on statistical indicators such as root mean square error (RMSE) and coefficient of determination (R2), we found that the proposed scheme is on par with some established schemes. In fact, it requires less CPU times (in seconds) to generate the interpolating surface on rectangular meshes. Furthermore, by combining the statistical indicators result and graphically visualizing of the test functions, the proposed method has the capability to reconstruct very comparable smoothing interpolating positive surfaces compared to some existing schemes. This finding is significant in producing a better interpolating surfaces for computer graphics applications since the proposed scheme has smaller error compared with existing schemes.

中文翻译：

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用保正性插值构造 C^1 有理双四次样条：数值结果与分析

从本质上讲，从观察到的数据集，我们应该能够生成与原始数据集具有相同特征的曲面。例如，如果给定的数据是正的，那么结果曲线或曲面在整个给定的区间，即任何地方都必须是正的。在这项研究中，通过部分混合方案构造了一种新的具有 C1 连续性的部分混合有理双四次样条。该有理样条定义在矩形网格的四个角上。在四边界曲线网络上推导出有理双四次样条为正的充分条件。有八 (8) 个自由参数可用于形状修改。使用数值技术估计一阶偏导数。我们还表明，所提出的方案是局部二次再现，因此它可以准确地再现二次曲面。我们测试了所提出的方案以插入各种类型的正表面数据。基于均方根误差（RMSE）和决定系数（R2）等统计指标，我们发现所提出的方案与一些既定方案相当。事实上，在矩形网格上生成插值表面需要更少的 CPU 时间（以秒为单位）。此外，通过结合统计指标结果和测试函数的图形可视化，与一些现有方案相比，所提出的方法能够重建非常相似的平滑插值正表面。