We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse rendering of geometry rely on silhouette gradients for topology changes, such signals are sparse. In contrast, our theory derives topological derivatives that relate the introduction of vanishing holes and phases to changes in image intensity. As a result, we enable differentiable shape perturbations in the form of hole or phase nucleation. We validate the proposed theory with optimization of closed curves in 2D and surfaces in 3D to lend insights into limitations of current methods and enable improved applications such as image vectorization, vector-graphics generation from text prompts, single-image reconstruction of shape ambigrams and multi-view 3D reconstruction.

翻译：我们提出了一种可微曲面演化的理论框架，该框架通过使用拓扑导数对图像泛函进行变分优化，允许离散拓扑变化。虽然先前的几何逆渲染方法依赖轮廓梯度实现拓扑变化，但此类信号是稀疏的。相比之下，我们的理论推导出了将消失孔洞和相的引入与图像强度变化相关联的拓扑导数。由此我们实现了以孔洞或相成核形式进行的可微形状扰动。我们通过二维闭合曲线和三维表面的优化验证了所提出的理论，以揭示当前方法的局限性，并推动改进应用，包括图像矢量化、基于文本提示的矢量图形生成、形状双关图的单图像重建以及多视图三维重建。