The objective of this study is to present a shape-preserving non-linear subdivision scheme generalizing the exponential B-spline of degree 3, which is a piecewise exponential polynomial with the same support as the cubic B-spline. The subdivision of the exponential B-spline has a crucial limitation in that it can reproduce at most two exponential polynomials, yielding the approximation order two. Also, finding a best-fitting shape parameter in the exponential B-spline is a challenging and important problem. In this regard, we present a method for selecting an optimal shape parameter and then formulate it in the construction of new refinement rules. As a result, the new scheme provides an improved approximation order four while maintaining the same $C^2$ smoothness as the (exponential) B-spline of degree 3. Moreover, we show that the proposed method preserves geometrically important characteristics such as monotonicity and convexity, under some suitable conditions. Some numerical examples are provided to demonstrate the ability of the new subdivision scheme.

本研究的目的是提出一种形状保持的非线性细分方案，该方案推广了 3 次指数 B 样条，它是一个分段指数多项式，与三次 B 样条具有相同的支持。指数 B 样条的细分有一个关键限制，因为它最多可以再现两个指数多项式，产生近似阶数2。此外，在指数 B 样条中找到最佳拟合形状参数是一个具有挑战性且重要的问题。在这方面，我们提出了一种选择最佳形状参数的方法，然后在构建新的细化规则时对其进行公式化。因此，新方案提供了改进的近似四阶，同时保持相同$C^2$平滑度作为 3 次（指数）B 样条曲线。此外，我们表明，所提出的方法在某些合适的条件下保留了几何上重要的特征，例如单调性和凸性。提供了一些数值例子来证明新的细分方案的能力。