Abstract :
[en] This work is part of the reconstruction of data from images. Its purpose
is to develop methods to generate a surface of CAO type (B-Spline or NURBS). Indeed, obtaining a mathematical representation of the surface of a solid body from point clouds, images or tetrahedral meshes is a fundamental task of 21st century digital engineering, where simulators interact with real systems. In this thesis, we will develop new algorithms for reconstruction of CAO geometry. First, in order to determine a NURBS surface, a control network, i.e. a quadrangular mesh, is required. Using the eigenfunctions of a Graph Laplacian problem and thanks to the discrete Morse theory, a control network is determined. The surface obtained using this mesh is not a priori optimal, which is why an optimization algorithm is introduced. It allows to adjust the NURBS surface to the triangulation and thus to best approximate the geometry of the object. Then, a model selection is carried out. To do so, a regression model is set up to compare the surfaces obtained with 3D images, and a surface is chosen using a information criterion. These steps being established, we will no longer consider only the noise of the data, but also that of the solution. Thus, using a sampling method, a probabilistic distribution of surface is determined. Finally, in perspective, constraints are applied to the graph Laplacian problem in order to align the NURBS patches along a given curve, for example in the case of an object with a marked edge. The methods developed are robust and do not depend on the topology of the desired 3D object, that is to say that the algorithm works on a wide range of shapes. We apply the developed methodologies in the biomedical field, with examples of vertebrae and femurs. This would make it possible to have the scanned object, a bone, and the implant in the same "format" and thus the adjustment of the implant will be carried out more easily.