A Hessian-Based Field Deformer for Real-Time Topology-Aware Shape Editing

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.