Shape manipulation is a central research topic in computer graphics. Topology editing, such as breaking apart connections, joining disconnected ends, and filling/opening a topological hole, is generally more challenging than geometry editing. In this paper, we observe that the saddle points of the signed distance function (SDF) provide useful hints for altering surface topology deliberately. Based on this key observation, we parameterize the SDF into a cubic trivariate tensor-product B-spline function $F$ whose saddle points $\{\boldsymbol{s}_i\}$ can be quickly exhausted based on a subdivision-based root-finding technique coupled with Newton's method. Users can select one of the candidate points, say $\boldsymbol{s}_i$, to edit the topology in real time. In implementation, we add a compactly supported B-spline function rooted at $\boldsymbol{s}_i$, which we call a \textit{deformer} in this paper, to $F$, with its local coordinate system aligning with the three eigenvectors of the Hessian. Combined with ray marching technique, our interactive system operates at 30 FPS. Additionally, our system empowers users to create desired bulges or concavities on the surface. An extensive user study indicates that our system is user-friendly and intuitive to operate. We demonstrate the effectiveness and usefulness of our system in a range of applications, including fixing surface reconstruction errors, artistic work design, 3D medical imaging and simulation, and antiquity restoration. Please refer to the attached video for a demonstration.

翻译：形状操控是计算机图形学的核心研究课题。相较于几何编辑，拓扑编辑（如分离连接、合并断开端点、填充/打开拓扑孔洞）通常更具挑战性。本文发现符号距离函数（SDF）的鞍点为有意识地改变曲面拓扑提供了关键线索。基于此重要发现，我们将SDF参数化为三维三次张量积B样条函数$F$，通过基于细分求根技术结合牛顿法，可快速枚举其所有鞍点$\{\boldsymbol{s}_i\}$。用户可任选候选点（例如$\boldsymbol{s}_i$）实时编辑拓扑。实现中，我们向$F$添加一个以$\boldsymbol{s}_i$为中心的紧支撑B样条函数（本文称为“变形器”），其局部坐标系与Hessian矩阵的三个特征向量对齐。结合光线行进技术，我们的交互系统可达到30 FPS的运算速度。此外，系统支持用户创建所需的表面凸起或凹陷。大量用户研究表明，本系统具有操作友好、直观易用的特点。我们在一系列应用中验证了系统的有效性和实用性，包括曲面重建误差修复、艺术设计、三维医学成像与仿真以及古物修复。演示视频详见附件。