2008, Gloger, Oliver, Ehrhardt, Matthias, Dietrich, Thore, Hellwich, Olaf, Graf, Kristof, Nagel, Eike

In this work we propose an adapted level set segmentation technique for the recognition of atherosclerotic plaque tissue on magnetic resonance images. The images are 2dimensional crosssectional images and show different profiles from ex-vivo human vessels with high variability in vessel shape. We used a curvature based anisotropic diffusion technique to denoise the magnetic resonance images. The segmentation technique is subdivided into three level set steps. Hence, the result of every phase serves as constructive knowledge for the next level set step. By analyzing and combining carefully all available channel information during the first and second step we are capable to delineate exactly the vessel walls by using and adapting two well-known level set segmentation techniques. The third step controls an enclosing level set which separates the plaque patterns from healthy media tissue. In this step we introduce a local weighting concept to consider intensity information for conspicuous plaque patterns. Furthermore, we propose the introduction of a maximal shrinking distance for the third level set in the vessel wall and compare the results of the local weighting algorithm with and without the concept of the maximal shrinking distance. The incorporation of locally weighted intensity information into the level set method allows the algorithm to automatically distinguish plaque from healthy media tissue. The knowledge of the maximal shrinking distance can improve the segmentation results and enables to delineate tissue areas where plaque is most likely.

2013, Kamenski, Lennard, Huang, Weizhang

The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix is investigated using a density function approach proposed by Fried in 1973. It is shown that the approach can be made mathematically rigorous for general domains and used to develop bounds on the smallest eigenvalue and the condition number that are sharper than existing estimates in one and two dimensions and comparable in three and higher dimensions. The new results reveal that the mesh concentration near the boundary has less influence on the condition number than the mesh concentration in the interior of the domain. This is especially true for the Jacobi preconditioned system where the former has little or almost no influence on the condition number. Numerical examples are presented.