Spectral Geometry Processing with Manifold Harmonics
Spectral Geometry Processing with Manifold Harmonics
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DOI：
10.1111/j.1467-8659.2008.01122.x
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发表时间：
2008-04
影响因子：
2.5
通讯作者：
B. Vallet;B. Lévy
中科院分区：
文献类型：
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作者：
B. Vallet;B. Lévy
我们提出了一种计算网格上傅里叶变换泛化的显式方法。众所周知，拉普拉斯贝尔特拉米算子（流形谐波）的特征函数定义了一个允许这种变换的函数基。然而，对于超过几千个顶点的网格来说，计算几个特征向量是遥不可及的，并且对于大型网格来说，存储这些特征向量是禁止的。为了克服这些限制，我们提出了一种逐带频谱计算算法和一种核外实现，可以为多达一百万个顶点的网格计算数千个特征向量。我们还提出了一种不需要存储特征向量的有限内存滤波算法。使用后一种算法，可以过滤特定的频段，而不需要计算整个频谱。最后，我们展示了我们的方法在交互式卷积几何滤波中的一些应用。这些技术成果是由基于离散外演算（DEC）的坚实而简单的理论框架所支持的。特别地，对算子的对称性和离散化问题进行了仔细的考虑。
We present an explicit method to compute a generalization of the Fourier Transform on a mesh. It is well known that the eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis allowing for such a transform. However, computing even just a few eigenvectors is out of reach for meshes with more than a few thousand vertices, and storing these eigenvectors is prohibitive for large meshes. To overcome these limitations, we propose a band‐by‐band spectrum computation algorithm and an out‐of‐core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. We also propose a limited‐memory filtering algorithm, that does not need to store the eigenvectors. Using this latter algorithm, specific frequency bands can be filtered, without needing to compute the entire spectrum. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering. These technical achievements are supported by a solid yet simple theoretic framework based on Discrete Exterior Calculus (DEC). In particular, the issues of symmetry and discretization of the operator are considered with great care.