We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions.

中文翻译：

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使用见证复合体重建

我们提出了一种新的重建算法，给定从对象 S 采样的输入点集，该算法构建一个单参数的复合物族，在不同尺度上近似于 S。在高层次上，我们的方法在精神上与 Chew 的表面网格划分算法非常相似，但有一个显着区别：受限的 Delaunay 三角剖分被证人复合体取代，这使得我们的算法适用于任何度量空间。为了证明它在曲线和曲面上的正确性，我们在 2d 和 3d 中强调了证人复合体和受限 Delaunay 三角剖分之间的关​​系。具体来说，我们证明在一些温和的采样假设下，两个复合体在 2d 中相等，在 3d 中密切相关。