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：我们提出了一种新的见解和范式（StEik），用于学习形状的隐式神经表示（INR）。具体而言，我们深入分析了在INR中施加符号距离函数约束时广泛使用的eikonal损失。我们从理论上证明，随着网络表示能力的增强，优化过程在连续极限下会趋近于一个不稳定的偏微分方程（PDE）。研究表明，这种不稳定性可能出现在现有网络优化中，导致重建表面出现不规则性，或收敛到次优的局部极小值，从而无法捕捉精细的几何与拓扑结构。我们进一步从理论上分析了现有文献中为其他目的而添加的损失项如何消除这些不稳定性。然而，这些项可能导致表面过度正则化，阻碍精细形状细节的表示。基于类似的连续极限PDE理论，我们引入了一个新的正则化项，既能抵消eikonal不稳定性，又不会造成过度正则化。此外，由于连续极限下的稳定性得到了保证，这种稳定化使得我们能够采用新的网络结构来表示更精细的形状细节。我们提出了一种基于二次层（quadratic layers）的结构。在多个基准数据集上的实验表明，与现有最优方法相比，我们的新型正则化方法和网络能够捕捉更精确的形状细节和更准确的拓扑结构。