Higher-order nonlinear priors for surface reconstruction

For surface reconstruction problems with noisy and incomplete range data, a Bayesian estimation approach can improve the overall quality of the surfaces. The Bayesian approach to surface estimation relies on a likelihood term, which ties the surface estimate to the input data, and the prior, which ensures surface smoothness or continuity. This paper introduces a new high-order, nonlinear prior for surface reconstruction. The proposed prior can smooth complex, noisy surfaces, while preserving sharp, geometric features, and it is a natural generalization of edge-preserving methods in image processing, such as anisotropic diffusion. The use of these higher-order surface priors requires solving a fourth-order partial differential equation (PDE), which can be difficult with conventional numerical techniques. Our solution is based on the observation that the generalization of image processing to surfaces entails filtering the surface normals. This approach allows us to separate the problem into two second-order PDEs: one for processing the normals and one for refitting the surface. Furthermore, we implement the associated surface deformations using level sets, and thus the algorithm can accommodate very complex shapes with arbitrary and changing topologies. This paper gives the mathematical formulation and describes the numerical algorithms. We also present a quantitative analysis, which demonstrates the effectiveness of the algorithm, and show results using real and synthetic range data.