Seite - 63 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

Graph-Laplacian minimisation for surfacesmoothing in 3D finiteelement tetrahedralmeshes RichardHuber, Martin Holler and KristianBredies∗ UniversityofGraz, Institute for Mathematicsand ScientificComputing Abstract We propose a new method to improve surface regularity of 3D tetrahedral meshes associated with finite element simulations of the heart. Our approach is to minimise the graph-Laplacian subject to suitablepointconstraints. Theseconstraintsarecomputedfromthewholetriangulationandpreventa worseningofmeshquality thatwouldotherwisebecausedby thesmoothing. Theresultingminimisa- tion problem is solved via a primal-dual algorithm, leading to a method that globally updates vertex coordinates in each iteration. Experiments confirm that our method reduces surface oscillations of the mesh whilepreventingdegeneration of the triangulationas indicatedbymeshqualitymetrics. 1. Introduction In biomedical engineering, the development of a realistic 3D simulation framework for the human heart is currently an active research topic. Such a framework would allow, for example, patient- specific models and more individualised treatment [6]. In order to carry out such simulations, 3D meshes are typically created from segmented magnetic resonance (MR) images, using mesh-genera- tionsoftwaresuchasdescribedin[12]. Inviewofthesubsequentsimulations, theseproceduresensure a sufficient quality of the triangulation, as indicated by quality metrics, and prevent the creation of degenerateelements. However,duetophysical limitationsintheimageacquisitionand,consequently, a lowresolutionof the imagedata, suchmeshesoftensuffer fromartifacts. Thoseappear inparticular in formof oscillations on theotherwise smoothsurface (seeSection5.). It is thegoalof thiswork toprovideamethodthat reduces theseoscillations,butmaintainshighmesh quality. To this aim, we minimise the graph-Laplacian under suitable constraints and adapt the mesh coordinatesaccordingly. Theconstraintsarecomputedfromthewhole initial triangulationandensure non-degeneracyof theresulting triangulationandmaintenanceofahighmeshquality, the latterbeing indicated byquality metrics. As the computation of meshes from segmented image data and a subsequent reduction of mesh arti- facts isachallenge thatcommonlyappears inmeshgeneration forfiniteelement simulations inmany differentcontexts,a lotof researchhasalreadybeencarriedout in thatdirection. Different toclassical mesh improvementdedicated toenhancing thequalityof the triangluation, thatoften focusona local adaption of nodes [8, 9, 11], our method aims at reducing mesh artifacts and hence is more related to mesh denoising approaches. For the latter, we exemplary refer to [10, 13, 14] and the references therein for recentmethods. For ageneraloverviewonmesh related topics see [2,3]. ∗University of Graz, Institute for Mathematics and Scientific Computing, Heinrichstrasse 36, A-8010 Graz, Austria. Email: richard.huber@edu.uni-graz.at, martin.holler@uni-graz.at, kristian.bredies@uni-graz.at 1 63