Fitting Smooth Manifolds to Point Clouds in a Level Set Formulation

Abstract

Since its inception, curves and surfaces have been the principal means of representation of observed geometry in computer vision. In many practical applications, one’s knowledge of the shapes of real-life objects is obtained through discrete measurements, which are subsequently converted into their continuous counterparts through the process of either curve or surface fitting, depending on the object dimensionality. Unfortunately, the measurement noise due to environmental effects, operator errors and/or hardware limitations makes the fitting problem a challenging one, requiring its solutions to possess a substantial degree of robustness. Moreover, in the case of surface fitting, the use of relatively complex fitting mechanisms might be disadvantageous due to their typically higher computational requirements, which could, in turn, create an implementation bottleneck due to the high dimensionality of the data. Accordingly, in this work, we propose a unified approach to fitting of smooth geometric manifolds, such as curves and surfaces, to point clouds. The proposed method is based on a level-set formulation, which leads to a simple and computationally efficient algorithm, the practical value of which is demonstrated through a series of examples.