We present new insights and a novel paradigm (StEik) for learning implicit neural representations (INR) of shapes. In particular, we shed light on the popular eikonal loss used for imposing a signed distance function constraint in INR. We show analytically that as the representation power of the network increases, the optimization approaches a partial differential equation (PDE) in the continuum limit that is unstable. We show that this instability can manifest in existing network optimization, leading to irregularities in the reconstructed surface and/or convergence to sub-optimal local minima, and thus fails to capture fine geometric and topological structure. We show analytically how other terms added to the loss, currently used in the literature for other purposes, can actually eliminate these instabilities. However, such terms can over-regularize the surface, preventing the representation of fine shape detail. Based on a similar PDE theory for the continuum limit, we introduce a new regularization term that still counteracts the eikonal instability but without over-regularizing. Furthermore, since stability is now guaranteed in the continuum limit, this stabilization also allows for considering new network structures that are able to represent finer shape detail. We introduce such a structure based on quadratic layers. Experiments on multiple benchmark data sets show that our new regularization and network are able to capture more precise shape details and more accurate topology than existing state-of-the-art.

翻译：我们提出了关于学习形状隐式神经表示（INR）的新见解与新范式（StEik）。具体而言，我们深入分析了用于在INR中施加符号距离函数约束的常用Eikonal损失函数。理论分析表明，随着网络表示能力的增强，优化过程在连续极限下趋近于一个不稳定的偏微分方程（PDE）。我们证明这种不稳定性会在现有网络优化中显现，导致重建表面出现不规则性，或收敛至次优局部极小值，从而无法捕获精细的几何与拓扑结构。我们从理论上揭示了文献中为其他目的引入的损失附加项如何消除这些不稳定性，但此类项可能导致表面过度正则化，阻碍精细形状细节的表征。基于连续极限的类似PDE理论，我们引入一项新型正则化项，既能抵消Eikonal不稳定性，又避免过度正则化。此外，由于连续极限下的稳定性得到保证，这种稳定化方法还允许我们采用能够表征更精细形状细节的新型网络结构——我们提出了基于二次层的网络架构。多个基准数据集上的实验表明，与现有最先进方法相比，我们的新型正则化与网络能够捕获更精准的形状细节与更准确的拓扑结构。