Page - 64 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

2. Modelproblem The initial setting is as follows: The 3D tetrahedral finite element mesh is given in form of a trian- gulation. In particular, the coordinates of points of the triangulation, together with edge information, and a masking of surface points is given. The triangulation is assumed to be regular, in particular, all tetrahedraare non-degenerateand disjoint except for theirboundaries. Since the above-described oscillatory artifacts appear on the surface of the mesh, we will only adapt surfacepointsanduse thepositionof interiorpointsonly todeterminesuitablepointconstraints. This also reduces the computational cost and memory requirements, however, will have some drawbacks asdiscussed inSection6. The triangulation of the surface induces a graphG= (V,E) with verticesV = {v1, . . . ,vN}, where N is the number of vertices, and edge setE such that there is an edge between vi and vj inG if, and only if, {vi,vj} ∈ E. We defineU := R3×N to be the space of point-coordinates of the triangulation, where foru∈U, the jth coordinate of the vertexvi is denoted byuji ∈R. Further, we will use the notationuj ∈RN for the vector containing all jth coordinates ofu andui∈R3 for the coordinates of vi. With this notation, we define the graph-Laplacian operator as the componentwise matrix-multiplicationoperator according to ∆u :=   ∆ˆu1∆ˆu2 ∆ˆu3   ,with thematrix ∆ˆ∈RN×N, givenas(∆ˆ) i,j :=      Deg(vi) if i= j, −1 if{vi,vj}∈E, 0 else, (1) where ∆ˆuj is amatrix-vectormultiplicationandDeg(vi)denotes thedegreeofvi, i.e., thenumberof neighboursofvi inG. In order to smooth the surface, new coordinates of the surface points are computed by minimising the graph-Laplacian under constraints designed to maintain the original mesh structure and to ensure non-degeneracy of themesh. Theminimisationproblemis u+∈argmin u∈U 1 2 ‖∆u‖22, subject tou∈Ω, (2) where the feasible set has the formΩ ={u∈U : ui∈Ωi for i= 1, . . . ,N}, with pointwise feasible sets Ωi as defined in the next section. A solutionu+ corresponds to the coordinates of the nodes of the smoothed surface. Note that the topology of the mesh, and in particular the set of edgesE, does not change and ∆ is linear. A minimisation of‖∆u‖22 results in the node coordinates adapting to the meansof thesurroundingones, and thus, reduces thecurvatureof thesurface. Hence,minimising the graph-Laplacianoperator is expected to implyasmoothingof the surfacemesh. Well-posedness. As we will see in the next section, it is reasonable to choose Ω to be non-empty, bounded and closed. Hence, existence of a solution to (2) follows directly from continuity ofu 7→ ‖∆u‖22 and finitedimensionalityofU. 3. Suitableconstraints Naturally, thesolutionof (2) shouldbeclose to theoriginaldata. Further, thechoiceofΩ isdrivenby two requirements, theconvexity ofΩand themaintenanceofmeshquality: 2 64