Page - 68 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

Note that this is a global method, i.e., it updates the positions of all surface vertices in each iteration, unlike many other surface smoothing algorithmswhichoperatepointwise. Reiteration. In some situations, the proposed constraints are too restrictive, and hence, the smooth- ing results are not satisfactory. To overcome that, the point-constraints for each single point would need to be updated iteratively with the position of all other points. This would, however, result in a non-convexproblem, preventing thecomputationofglobaloptima. Aheuristicapproachtostillachievesomeimprovement,withoutre-designingtheoverallmethod, is to restart Algorithm 1 after convergence. To this aim, new constraints are computed from the outputu+ and thegraph-Laplacian isoptimisedagainsubject to theseupdatedconstraints. Thiscanberepeated a few times, e.g., 4 times, to allow some more flexibility in the constraint set. In practice, it can be reasonable to reduce thenumberof iterationsperformedinAlgorithm1,anddoafewouter iterations inorder toallowformoremovement,while still guaranteeing thatnoself-intersectionoccursand the meshquality remains high. Independent of such heuristics, the point-constraints of our method always ensure a non-degenerate triangulation. Also, the inner points of the mesh are not moved by our methods and hence limit the effect of the re-iteration. This, together with the point-constraints, in particular prevents a strong decreaseof thevolumeof theshape,as frequentlyobservedwithunconstrainedLaplaciansmoothing. 5. Experimental results Theproposedmethod,althoughrathersimple, isquiteeffective. Itallowstosmooththesurfaceandto reduce artifacts significantly while maintaining the original level of mesh quality. Figure 2 illustrates the effects of smoothing, with the original model on the left side, and the smoothed version on the right. The figure shows a mesh of a human heart, where the smoothed version was computed with 3 outer and1000 inner iterationsandwith theconstraintparameterα= 2/5. The effect of the proposed method on mesh quality can be evaluated quantitatively by measuring ρ, the skewness of a tetrahedron, i.e., the ratio of a tetrahedron’s volume to its circumscribed ball’s volume. Additionally,wequantify thechangeof thevolumeofeachtetrahedronandidentifychanged orientations. This is done for each tetrahedron in the mesh by measuring the ratio of det(A) in the original and the smoothed mesh, denoted by θ, whereA is a parallelepiped induced by a the tetrahedron. Furthermore, one can observe maximal and minimal angles in the tetrahedra in order to find very flat tetrahedra. Table 1 depicts a quantitative evaluation of the effect of our method on mesh quality by comparingρ for the original and the smoothed mesh and computing θ. As one can see, the number of flat structures does not increase significantly due to smoothing and for only 1% of the tetrahedra thevolumereducedbymore thanonehalf. Further,weobserved thatnosign-flipsof thedeterminant occurred,hence therearenoself-intersections. Percentiles ofP 1% 5% 10% Originalmesh 0.0900 0.2350 0.3348 Smoothed mesh 0.0934 0.2047 0.2812 PercentilesofΘ 1% 5% 0.5473 0.6648 Table 1: Mesh quality corresponding to mesh considered in Figure 2. Percentiles ofP andΘ, whereP is avector ofρ forall tetrahedra andΘ is avectorofθ forall tetrahedra. 6 68