Variational surface reconstruction - Wan, Min (2012)
As a fundamental geometry processing task and a typical reverse engineering, surface reconstruction plays a significant role in various areas such as CAD, animation, and medical imaging. In the last several decades, it has drawn a lot of attention and a large number of methods have been proposed. The explicit methods based on Delaunay triangulation are fast, but they lack robustness to noises and outliers. The implicit methods based on level set function have resistance to noises and outliers. However they are not efficient and the grid commonly used in implicit methods causes staircasing in reconstructed results. Furthermore, some topics in surface reconstruction are still unsolved, such as open surface cases, non-orientable cases, and reconstruction with feature preservation. A robust and efficient method to approach most surfaces is still in demand due to the intrinsic ill-posedness and the external reconstruction difficulties such as noises, outliers, non-uniformity, and undersampling, which motivates the studies in this thesis. This thesis includes several works addressing different challenging issues in the surface reconstruction topic. These three works are conducted in a unified framework based on Delaunay triangulation. The proposed mesh framework used in this thesis generalizes the framework used in earlier methods. It not only preserves all theoretic merits of explicit methods, but also adapts to variational methods and numerical solvers in implicit methods. In three studies, novel mathematical models are proposed and validated for the reconstruction purpose; efficient minimization tools are developed; potential applications are suggested through a wide spectrum of numerical examples. The methods not only outperform existing methods on common reconstruction problems, but also first address multi-phase cases, non-orientable cases and reconstruction based on domain decomposition. Firstly, the watertight surface reconstruction problem is addressed. A variational model is proposed with an area regularization term. The energy is minimized via the graph technique. Besides, the multi-phase cases are also approached by applying the approximating multi-way cut algorithm. Secondly, the open surface reconstruction problem is addressed. A “narrow-band” technique and Boolean operation is used to tackle the difficulty in open cases. Furthermore, the surface reconstruction based on domain decomposition is proposed. The reconstruction problem is divided into sub-problems and approached with certain precautious measures for conflicts and cracks. The parallel feasibility and efficiency is discussed. At last, a surface reconstruction problem is proposed for feature preservation. A variational model with curvature term is proposed. Two minimization tools are presented as well, one iterative local swap minimization and one global minimization based on graph-cuts. Advantages and disadvantages are enumerated and analyzed. In summary, three reconstruction methods are included in this thesis regarding three reconstruction topics. They are closely related due to the identical mesh framework and the minimal surface model family. These methods could also be unified into one single approach able to tackle most surfaces encountered, (1) watertight, open, combination of them; (2) multiphase without intersection, multiphase with intersection, combination of them; (3) orientable, non-orientable, combination of them; (4) smooth, sharp, combination of them; and combination of all these types subjecting to all reconstruction difficulties such as noises, outliers, non-uniformity, undersampling, incomplete scanning.