This paper proposes a Generalized Multilevel B-spline Approximation (GMBA) method, which addresses scattered data interpolation problems in image processing. Mathematically, the GMBA provides a better solution for the B-spline control lattice by superimposing identical level B-splines compared with traditional Multilevel B-spline Approximation (MBA). Specifically, the GMBA allows the spacing of next control lattice to be subdivided arbitrarily or remained unchanged, which is determined by a predefined spacing set or the current error level. These improvements bring higher approximation accuracy and more flexibility for algorithm design to avoid over-fitting. In this paper, basic GMBA algorithm and its refined algorithm are compiled for image processing. Finally, six relevant cases are involved to test the GMBA, including surface approximation, image enlargement, image completion, and Salt-and-Pepper (SAP) noise removal. The experimental results show that the GMBA has better performance than the MBA in surface approximation and image processing, performs comparatively fast with the best performance on more than half of the standard test images compared with traditional algorithms, and has partially better performance even than deep learning algorithms. The GMBA can effectively recover meaningful details in images contaminated with even extremely high SAP noise level (up to 99%).

中文翻译：

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图像处理中离散数据插值的广义多级B样条逼近


本文提出了一种广义多级B样条逼近（GMBA）方法，该方法解决了图像处理中的分散数据插值问题。从数学上讲，与传统的多级 B 样条逼近 (MBA) 相比，GMBA 通过叠加相同级别的 B 样条来为 B 样条控制格提供更好的解决方案。具体来说，GMBA允许下一个控制格的间距任意细分或保持不变，这是由预定义的间距集或当前误差水平确定的。这些改进为算法设计带来了更高的逼近精度和更大的灵活性，以避免过度拟合。本文针对图像处理编写了基本的GMBA算法及其细化算法。最后，涉及六个相关案例来测试GMBA，包括曲面近似、图像放大、图像补全和椒盐（SAP）噪声去除。实验结果表明，GMBA在曲面逼近和图像处理方面比MBA具有更好的性能，与传统算法相比，在超过一半的标准测试图像上表现较快，性能最好，并且部分性能甚至优于深度学习算法。 GMBA 可以有效地恢复受到 SAP 噪声水平极高（高达 99%）污染的图像中有意义的细节。