NURBS-based surface generation from 3D images: spectral construction and data-driven model selection

In this paper, we present a set of improved algorithms for recovering computer aided design (CAD-type) surface models from three-dimensional (3D) images. The goal of the proposed framework is to generate B-spline or non-uniform rational B-spline (NURBS) surfaces, which are standard mathematical representations of solid objects in digital engineering. To create a NURBS surface, we first compute a control network (a quadrilateral mesh) from a triangular mesh using the Marching Cubes algorithm and Discrete Morse theory. To create a NURBS surface, we first compute a triangular mesh using the Marching Cubes algorithm, then the control network (a quadrilateral mesh) is determined from the triangular mesh by using Discrete Morse theory. Discrete Morse theory uses the critical points of a specific scalar field defined over the triangulation to generate a quad mesh. Such a scalar field is obtained by solving a graph Laplacian eigenproblem over the triangulation. However, the resulting surface is not optimal. We therefore introduce an optimization algorithm to better approximate the geometry of the object. In addition, we propose a statistical method for selecting the most appropriate eigenfunction of the graph Laplacian to generate a control network that is neither too coarse nor too fine, given the precision of the 3D image. To do this, we set up a regression model and use an information criterion to choose the best surface. Finally, we extend our approach by taking into account both model and data uncertainty using probabilistic regression and sampling the posterior distribution with Hamiltonian Markov Chain Monte Carlo. Graphical Abstract