Variational Graph-based Normal Integration

We present a general optimization-based framework for depth-preserving normal integration. Unlike existing methods that operate on surface orientations defined over regular grids, our approach introduces a unified graph-based formulation capable of integrating semi-differentiable surfaces on unstructured domains. Given a set of points uniformly sampled from a surface, we construct a directed, weighted graph that jointly parameterizes the surface geometry and pairwise point correlations. Surface depth is recovered by minimizing projected point-to-plane distances across the graph, and this objective is optimized through variational inference. In our formulation, estimated surface normals serve as latent variables that encode local geometry via the posterior probabilities of a two-component Gaussian mixture, allowing depth discontinuities to be inferred from sampled triplet configurations. The unknowns are estimated in an alternating fashion, and we provide a geometric interpretation of this inference process by relating it to shape deformation. Experimental results show that the proposed method not only outperforms state-of-the-art techniques on regularly gridded data, but also generalizes effectively to scattered points, which existing approaches do not directly support.