Study On Parameterization Methods In Spline Fitting

Spline fitting is widely used in geometric modeling and reverse engineering. Conventionally, surfaces in real world are digitized by geometric acquisition methods; the scanned point clouds are cleaned up, fused, and meshed to triangle meshes; the meshes are parameterized onto canonical planar domains; finally, the parameterized meshes are approximated by spline surfaces. Unavoidably, the parameterization step will introduce geometric distortions, which can be classified as angle distortion and area distortion. Most existing spline fitting works are based on parameterization methods minimizing angle distortion, which lead to large area distortion, and severely affect the quality of the splines.In this paper we first propose a spline fitting method based on area-preserving mesh parameterizations, rooted from the Optimal Mass Transportation theory. Then we further propose a brand new mesh parameterization method, which introduces the balance parameter. Thus, we achieve a balance between angle distortion and area distortion. Finally in this paper, we use several samples to present the differences between the spline fitting methods based on the above three types of parameterization. Comparing to the existing methods, the methods proposed in this work have rigorous theoretic foundation, meantime with accuracy and efficiency, greatly improves the quality of the splines.