Adaptive spline approximation: data-driven parameterization and CAD model (re-)construction

In this thesis, we combine Computer Aided Geometric Design methods with Deep Learning technologies. The final objective is to develop and apply advanced geometric reverse engineering methods for the design of complex data-driven free-form spline geometries. In particular, we address the (re-)construction of highly accurate CAD models from point clouds, which is both a fundamental and challenging problem. Depending on the acquisition process, the nature of the data can strongly vary, from uniformly distributed to scattered and affected by noise; yet the reconstructed geometric models are required to be compact, highly accurate, and smooth, while simultaneously capturing key geometric features. An important and crucial step of any parametric model reconstruction scheme consists in solving the parameterization problem, namely to suitably map the input data to a parametric domain. We propose data-driven parameterization methods based on (geometric) deep learning to address this problem both in the univariate and multivariate cases, considering either structured or unstructured point cloud configurations. The accuracy of the model obtained with the proposed schemes is higher than the one usually achieved by standard methods. It should be noted that, to the best of our knowledge, data-driven models to address the parameterization problem of scattered data sets in a general multivariate setting were not previously proposed. In addition, we introduce \emph{novel adaptive fitting schemes} with moving parameterization and truncated hierarchical B-splines, based on the optimization of different error metrics. In particular, we propose adaptive alternating and joint optimization methods to optimize the parameter locations and the control points of the (hierarchical) spline geometric model. The alternating methods optimize the parameters separately from the control points computation, whereas the joint approach optimizes the parameters and control points simultaneously. The use of moving parameterization instead of fixed parameter values, when suitably combined with adaptive spline approximation, can significantly improve the resulting geometric model, thus outperforming state-of-the-art hierarchical spline model reconstruction schemes.