Topology-controllable Implicit Surface Reconstruction Based on Persistent Homology

Abstract

In geometric design, reconstructing an implicit surface with high-quality geometry and expected topology from point clouds has been a challenging problem. However, traditional topological invariants, such as Betti numbers, are not easily used in controlling topology in implicit surface reconstruction. Persistent homology provides a quantitative measurement, i.e. , persistence diagram (PD), and allows tracking of pairs of key points that generate and destroy topological holes. In this study, we propose a topology-controllable implicit surface reconstruction method from point clouds based on persistent homology. Specifically, given a point cloud with normals, the signed distance field is first constructed, and then a B-spline function represented distance function is generated by fitting the signed distance values through progressive iterative approximation. By designing a topological target function using the persistent pairs in PDs according to topological priors, the control coefficients of the B-spline function are optimized to extract an implicit surface, i.e. , an iso-surface of the B-spline function, with expected topology. Experiments show that the proposed method can reconstruct surfaces with higher topological quality than other reconstruction methods. Moreover, the proposed method can be used to edit the topology of an implicit surface. • The topology-controllable framework for implicit B-spline surface reconstruction improves geometric quality and ensures the desired topology of the extracted surface. • Principles of designing the topological target function for implicit surface reconstruction are proposed. • Reconstruction on sparse point clouds and the application of topology editing are explored.