Euler's elastica constitute an appealing variational image inpainting model. It minimises an energy that involves the total variation as well as the level line curvature. These components are transparent and make it attractive for shape completion tasks. However, its gradient flow is a singular, anisotropic, and nonlinear PDE of fourth order, which is numerically challenging: It is difficult to find efficient algorithms that offer sharp edges and good rotation invariance. As a remedy, we design the first neural algorithm that simulates inpainting with Euler's Elastica. We use the deep energy concept which employs the variational energy as neural network loss. Furthermore, we pair it with a deep image prior where the network architecture itself acts as a prior. This yields better inpaintings by steering the optimisation trajectory closer to the desired solution. Our results are qualitatively on par with state-of-the-art algorithms on elastica-based shape completion. They combine good rotation invariance with sharp edges. Moreover, we benefit from the high efficiency and effortless parallelisation within a neural framework. Our neural elastica approach only requires 3x3 central difference stencils. It is thus much simpler than other well-performing algorithms for elastica inpainting. Last but not least, it is unsupervised as it requires no ground truth training data.

翻译：欧拉弹性构成了一种吸引人的变分图像修复模型。它最小化一个涉及总变分以及水平线曲率的能量函数，这些成分透明直观，使其在形状补全任务中具有吸引力。然而，其梯度流是一个奇异、各向异性且非线性的四阶偏微分方程，在数值上极具挑战性：难以找到既能提供锐利边缘又具有良好的旋转不变性的高效算法。作为解决方案，我们设计了首个模拟欧拉弹性修复的神经网络算法。我们采用深度能量概念，将变分能量用作神经网络损失函数。此外，我们将其与深度图像先验相结合，使网络架构本身充当先验。通过将优化轨迹引导至更接近理想解，这能够获得更好的修复结果。我们的结果在质量上与基于弹性的形状补全领域的最新算法相当，兼具良好的旋转不变性和锐利边缘。同时，我们还受益于神经框架内的高效性和并行化简易性。我们的神经弹性方法仅需3x3中心差分模板，因此比其它性能良好的弹性修复算法简单得多。最后但同样重要的是，该方法无需监督，因为它不依赖真实训练数据。