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.

应用可微分曲面演化的理论框架，通过利用拓扑导数来实现离散拓扑变化，以对图像函数进行变分优化。通过导出拓扑导数与消失孔和相位引入与图像强度变化的关系，我们实现了形状扰动的可微变化方法，如孔或相位起源，从而解决了现有方法的局限性并改进了图像矢量化、基于文本提示的矢量图生成、单图形重建和多视角三维重建等应用。