Range Surface Fitting

Transforming a discrete representation of a surface (i.e. a finite set of 3D points) into a smooth and analytical form is a problem of wide interest. Such a representation can then be used for complex computations such as the estimation of geodesic distances. Finding a smooth surface that approximates a set of 3D points has other advantages. In particular, it allows one to reduce the amount of noise inherent to datasets acquired with real sensors such as Time-of-Flight cameras. A large part of our work is dedicated to this problem in the case of range data.

In this chapter, we tackle the problem of fitting an analytic surface to range data. We start by giving the first definitions and concepts related to range data. In particular, we give a brief review of the sensors and techniques that allows one to acquire range data. We also give an introductory and practical example that allows us to explain the basic concepts of range data fitting. This example can also be considered as a practical implementation of the concepts related to parameters and hyperparameters estimation that were reviewed in the previous two chapters.

We then present our contributions related to range data fitting. We propose two main contributions which will be presented in separate sections. First, we propose a new criterion, the L-Tangent Norm, that enables one to automatically select the regularization trade-off in range surface fitting (31). Second, we propose a method for fitting a surface to range data with heteroskedastic noise, i.e. noise whose variance is not the same for all the data points (32). By exploiting the fact that the data are arranged on a regular grid, our method features a low computational complexity.