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的隐式神经表征（INR）的新见解和新范式（StEik）。具体而言，我们揭示了在INR中用于施加有符号距离函数约束的常用Eikonal损失函数。分析表明，随着网络表征能力的增强，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可能出现在现有网络优化中，导致重建表面不规则和/或收敛到次优局部极小值，从而无法捕捉精细的几何与拓扑结构。我们通过分析展示了文献中目前用于其他目的的损失附加项如何实际消除这些不稳定性。然而，此类项可能过度正则化表面，阻碍精细形状细节的表征。基于类似连续极限下的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，又不会过度正则化。此外，由于连续极限下稳定性得到保证，这种稳定化还允许采用能够表征更精细形状细节的新网络结构。我们提出了基于二次层（quadratic layers）的此类结构。在多个基准数据集上的实验表明，我们的新正则化与网络能够比现有最先进方法捕捉更精确的形状细节和更准确的拓扑结构。