Models and Methods for One-Dimensional Approximations to Point Cloud Data

In this thesis we investigate the problem of approximating point cloud data, or more generally, measures, by one-dimensional objects. Our approach is variational, as we will study certain functionals and the extent to which their minimizers (of finite length) can provide adequate approximations to data. In the first part of this thesis, the approximating objects we consider are curves, and our main goals are to understand their behavior and provide a robust and efficient algorithm for computing them. Aside from data analysis applications in which we assume data to have a one-dimensional structure, we are also motivated by settings in which the data approximation problem has a physical meaning. Such is the case in urban planning, and in particular the problem of finding optimal networks for transportation. In the second part of the thesis we will propose a new set of functionals that model this problem and establish basic existence properties. We then develop an algorithm for computing local minimizers, and investigate the suitability of the approach through a set of numerical examples that also give a glimpse into how the complexity of low-energy configurations increases with the total mass of the data.