Higher-order nonlinear priors for surface reconstruction

Abstract

Abstract-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. An exact solution would require solving a fourth-order partial differential equation (PDE), which can be difficult with conventional numerical techniques. Our approach is to solve a cascade system of two second-order PDEs, which resembles the original fourth-order system. This strategy is based on the observation that the generalization of image processing to surfaces entails filtering the surface normals. We solve one PDE for processing the normals and one for refitting the surface to the normals. Furthermore, we implement the associated surface deformations using level sets. Hence, 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 show results using range and medical data.

Genetic search for optimally-constrained multiple-line fitting of discrete data points

“…A number of studies have been conducted to preserve sharp features during a mesh smoothing process. One technique was anisotropic diffusion that was used in height fields [1], triangle meshes [2][3][4][5][6], level set surfaces [7], surface and function [8], and surface reconstruction [9][10][11]. Another method was bilateral filtering [12,13], which was extended to a trilateral filter [14].…”

“…If one line is perpendicular to the z axis, the z-component in Eqs. (10) and (13a) should be switched with x-component or y-component:…”

Accurate Multiview Stereo Reconstruction With Fast Visibility Integration and Tight Disparity Bounding

“…Some methods search for a minimal surface either imposing to start from a gross initial shape, by smoothing points with high-curvatures (Tasdizen and Whitaker, 2004, Sinha and Pollefeys, 2005 or by imposing planarity Leclerc, 1995, Furukawa et al, 2009). Other methods implicitly search for a maximal surface since they does not impose any surface smoothness (Fromherz and Bichsel, 1995, Seitz and Dyer, 1999, Kutulakos, 2000, Kutulakos and Seitz, 2000, Treuille et al, 2004, Saito and Kanade, 1999.…”

Markov Random Field Surface Reconstruction

“…As will be demonstrated, the priors used in our work are linear. Nonlinear priors for surface normals are demonstrated in a surface reconstruction scheme in [24]. This approach is similar to anisotropic diffusion.…”