Chung Moo K, Seo Seongho, Adluru Nagesh, Vorperian Houri K
Department of Biostatistics and Medical Informatics, University of Wisconsin, Madison, WI 53706, USA; Waisman Laboratory for Brain Imaging and Behavior, University of Wisconsin, Madison, WI 53706, USA; Vocal Tract Development Laboratory, Waisman Center, University of Wisconsin, Madison, WI 53706, USA; Department of Brain and Cognitive Sciences, Seoul National University, Korea.
Department of Brain and Cognitive Sciences, Seoul National University, Korea.
Mach Learn Med Imaging. 2011 Sep;7009:225-232. doi: 10.1007/978-3-642-24319-6_28.
The second eigenfunction of the Laplace-Beltrami operator follows the pattern of the overall shape of an object. This geometric property is well known and used for various applications including mesh processing, feature extraction, manifold learning, data embedding and the minimum linear arrangement problem. Surprisingly, this geometric property has not been mathematically formulated yet. This problem is directly related to the somewhat obscure in differential geometry. The aim of the paper is to raise the awareness of this nontrivial issue and formulate the problem more concretely. As an application, we show how the second eigenfunction alone can be used for complex shape modeling of tubular structures such as the human mandible.
拉普拉斯 - 贝尔特拉米算子的第二个特征函数遵循物体整体形状的模式。这种几何特性是众所周知的，并用于各种应用，包括网格处理、特征提取、流形学习、数据嵌入和最小线性排列问题。令人惊讶的是，这种几何特性尚未在数学上得到公式化表述。这个问题与微分几何中有些晦涩的内容直接相关。本文的目的是提高对这个重要问题的认识，并更具体地阐述该问题。作为一个应用，我们展示了仅第二个特征函数如何用于管状结构（如人类下颌骨）的复杂形状建模。
