Fast iterative algorithms for solving the minimization of curvature-related functionals in surface fairing

Abstract

A number of successful variational models for processing planar images have recently been generalized to three-dimensional (3D) surface processing. With this new dimensionality, the amount of numerical computations to solve the minimization of such new 3D formulations naturally grows up dramatically. Though the need of computationally fast and efficient numerical algorithms able to process high resolution surfaces is high, much less work has been done in this area. Recently, a two-step algorithm for the fast solution of the total curvature model was introduced in Tasdizen, Whitaker, Burchard and Osher [Geometric surface processing via normal maps, ACM Trans. Graph. 22(4) (2003), pp. 1012–1033]. In this paper, we generalize and modify this algorithm to the solution of analogues of the mean curvature model of Droske and Martin Rumpf [A level set formulation for Willmore flow, Interfaces Free Bound. 6(3) (2004), pp. 361–378] and the Gaussian curvature model of Elsey and Esedoḡlu [Analogue of the total variation denoising model in the context of geometry processing, Multiscale Model. Simul. 7(4) (2009), pp. 1549–1573]. Numerical experiments are shown to illustrate the good performance of the algorithms and test results.

Keywords:

• surface fairing
• variational models
• fourth-order PDE
• iterative solution techniques
• inverse problem

2010 AMS Subject Classifications :

• 65J15
• 65J20
• 65K10
• 65N21

Acknowledgements

The authors thank Dr Rongjie Lai (University of Southern California, USA) for helpful discussions and sharing his test data.