The eigenfunctions of the Laplace-Beltrami operator have widespread applications in a number of disciplines of engineering, computer vision/graphics, machine learning, etc. These eigenfunctions or manifold harmonics, provide the means to smoothly interpolate data on a manifold. They are highly effective, specifically as it relates to geometry representation and editing; manifold harmonics form a natural basis for multi-resolution representation (and editing) of complex surfaces and functioned defined therein. In this paper, we seek to develop the framework to exploit the benefits of manifold harmonics for shape reconstruction. To this end, we develop a highly compressible, multi-resolution shape reconstruction scheme using manifold harmonics. The method relies on subdivision basis sets to construct both boundary element isogeometric methods for analysis and surface finite elements to construct manifold harmonics. We pair this technique with the volumetric source reconstruction method to determine an initial starting point. Examples presented highlight efficacy of the approach in the presence of noisy data, including significant reduction in the number of degrees of freedom for complex objects, the accuracy of reconstruction, and multi-resolution capabilities.

中文翻译：

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细分曲面上基于Laplace-Beltrami的多分辨率形状重构

Laplace-Beltrami运算符的本征函数在工程，计算机视觉/图形，机器学习等许多学科中都有广泛的应用。这些本征函数或流形谐波提供了在流形上平滑地插值数据的方法。它们非常有效，特别是在几何图形表示和编辑方面。歧管谐波为复杂表面的多分辨率表示（和编辑）奠定了自然基础，并在其中定义了功能。在本文中，我们寻求开发一种框架，以利用流形谐波在形状重构中的优势。为此，我们开发了使用歧管谐波的高度可压缩，多分辨率形状重构方案。该方法依靠细分基集来构造分析的边界元等几何方法和构造歧管谐波的曲面有限元。我们将此技术与体积源重构方法配对以确定初始起点。所提供的示例突出显示了在存在嘈杂数据的情况下该方法的有效性，包括显着减少了复杂对象的自由度数量，重建的准确性以及多分辨率功能。