Approche penalisee en tomographie helicoidale en vue de l'application a la conception d'une prothese personnalisee du genou (French text).

摘要

In order to design an ergonomic knee prosthesis, we present a new reconstruction method that produces significant improvement in the precision of helical tomographic reconstructions. Whereas the standard approach is based on interpolation and convolution backprojection , our technique relies on a penalized approach; in this framework, the 3D image is defined as the minimizer of a penalized least-square criterion, which leads to a very large scale optimization problem. An adequate regularization of the tomographic problem is provided by a convex penalization yielding a precise localization of the edges in the image at a reasonable numerical cost. Experiments carried out on synthetic data show that our method produces a significant improvement in precision over standard reconstruction techniques. However, the very large-scale nature of the numerical reconstruction problem leads to major implementation difficulties. In order to keep the computer cost reasonable, we used a spatial invariance of the observation model and minimized the penalized criterion with a successive over relaxation algorithm. Alternatively, an approximation in the observation model leads to a separable 3D reconstruction problem: as a result, the 3D image can be obtained by successive 2D problems of reduced size. Finally, our work yields an in-depth study of “Half-quadratic” (HQ) algorithms widely used in image or data processing. We were able to point out the connections between these HQ algorithms and already known algorithmic forms; weaker global convergence conditions were provided and faster HQ variants were deduced.