This paper introduces a generative model for 3D surfaces based on a representation of shapes with mean curvature and metric, which are invariant under rigid transformation. Hence, compared with existing 3D machine learning frameworks, our model substantially reduces the influence of translation and rotation. In addition, the local structure of shapes will be more precisely captured, since the curvature is explicitly encoded in our model. Specifically, every surface is first conformally mapped to a canonical domain, such as a unit disk or a unit sphere. Then, it is represented by two functions: the mean curvature half-density and the vertex density, over this canonical domain. Assuming that input shapes follow a certain distribution in a latent space, we use the variational autoencoder to learn the latent space representation. After the learning, we can generate variations of shapes by randomly sampling the distribution in the latent space. Surfaces with triangular meshes can be reconstructed from the generated data by applying isotropic remeshing and spin transformation, which is given by Dirac equation. We demonstrate the effectiveness of our model on datasets of man-made and biological shapes and compare the results with other methods.

中文翻译：

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基于曲率和密度的形状生成表示

本文介绍了一种基于具有平均曲率和度量的形状表示的 3D 表面生成模型，这些形状在刚性变换下是不变的。因此，与现有的 3D 机器学习框架相比，我们的模型大大减少了平移和旋转的影响。此外，形状的局部结构将被更精确地捕获，因为曲率在我们的模型中被明确编码。具体来说，首先将每个表面共形映射到规范域，例如单位圆盘或单位球体。然后，它由两个函数表示：在这个规范域上的平均曲率半密度和顶点密度。假设输入形状在潜在空间中遵循一定的分布，我们使用变分自编码器来学习潜在空间表示。学习之后，我们可以通过随机采样潜在空间中的分布来生成形状的变化。可以通过应用各向同性重新网格划分和自旋变换从生成的数据重建具有三角形网格的表面，这由 Dirac 方程给出。我们证明了我们的模型在人造和生物形状数据集上的有效性，并将结果与​​其他方法进行了比较。