The recent surge of utilizing deep neural networks for geometric processing and shape modeling has opened up exciting avenues. However, there is a conspicuous lack of research efforts on using powerful neural representations to extend the capabilities of parametric surfaces, which are the prevalent surface representations in product design, CAD/CAM, and computer animation. We present Neural Parametric Surfaces, the first piecewise neural surface representation that allows coarse patch layouts of arbitrary $n$-sided surface patches to model complex surface geometries with high precision, offering greater flexibility over traditional parametric surfaces. By construction, this new surface representation guarantees $G^0$ continuity between adjacent patches and empirically achieves $G^1$ continuity, which cannot be attained by existing neural patch-based methods. The key ingredient of our neural parametric surface is a learnable feature complex $\mathcal{C}$ that is embedded in a high-dimensional space $\mathbb{R}^D$ and topologically equivalent to the patch layout of the surface; each face cell of the complex is defined by interpolating feature vectors at its vertices. The learned feature complex is mapped by an MLP-encoded function $f:\mathcal{C} \rightarrow \mathcal{S}$ to produce the neural parametric surface $\mathcal{S}$. We present a surface fitting algorithm that optimizes the feature complex $\mathcal{C}$ and trains the neural mapping $f$ to reconstruct given target shapes with high accuracy. We further show that the proposed representation along with a compact-size neural net can learn a plausible shape space from a shape collection, which can be used for shape interpolation or shape completion from noisy and incomplete input data. Extensive experiments show that neural parametric surfaces offer greater modeling capabilities than traditional parametric surfaces.

翻译：近期，利用深度神经网络进行几何处理与形状建模的研究热潮开辟了令人振奋的新途径。然而，在利用强大神经表示扩展参数曲面能力方面，研究明显不足——参数曲面是产品设计、CAD/CAM及计算机动画中广泛采用的表面表示形式。我们提出神经参数曲面（Neural Parametric Surfaces），这是首个分段神经表面表示方法，允许任意$n$边形曲面片的粗略面片布局以高精度建模复杂曲面几何形状，相比传统参数曲面具有更强灵活性。通过构造设计，这种新型表面表示方法可保证相邻面片间的$G^0$连续性，并能经验性地实现现有神经面片方法无法达到的$G^1$连续性。其核心要素是一个可学习的特征复形$\mathcal{C}$，嵌入高维空间$\mathbb{R}^D$中，且拓扑等价于曲面的面片布局；复形的每个面单元通过插值其顶点处的特征向量定义。学习得到的特征复形由MLP编码函数$f:\mathcal{C} \rightarrow \mathcal{S}$映射，生成神经参数曲面$\mathcal{S}$。我们提出一种曲面拟合算法，通过优化特征复形$\mathcal{C}$并训练神经映射$f$，以高精度重建给定目标形状。进一步证明，该表示方法配合紧凑型神经网络可从形状集合中学习合理的形状空间，用于形状插值或对含噪声、不完整输入数据进行形状补全。大量实验表明，神经参数曲面相比传统参数曲面具有更强的建模能力。