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Constraint-based fairing of surface meshes

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Eurographics Symposium on Geometry Processing (2007)Alexander Belyaev, Michael Garland (Editors)Constraint-based fairing of surface meshesKlaus Hildebrandt Konrad PolthierFreie Univ ersität BerlinAbstractWe propose a constraint-based method for the fairing of surface meshes. The main feature of our approach is thatthe resulting smoothed surface remains within a prescribed distance to the input mesh. For example, specifyingthe maximum distance in the order of the measuring precision of a laser scanner allows noise to be removed whilepreserving the accuracy of the scan.The approach is modeled as an optimization pr oblem where a fairness measure is minimized subject to constraintsthat control the spatial deviation of the surface. The problem is efficiently solved by an active set Newton method.1. IntroductionThe instant availability of high-quality digital models of 3Dsurfaces becomes an essential prerequisite in many researchareas, industrial modeling, e-commerce, medical treatmentplanning, archeology, and restoration, just to name a few.Acquisition technologies such as laser scans for 3D surfacemodels or CT, MRI and other devices for volumetric shapescan measure data with high accuracy. Nevertheless the accu-racy is often lost in the mesh creation pipeline. A crucial stepin this pipeline is the removal of geometric noise containedin the positions of the measured points. Many techniques toeffectively remove the noise have been proposed, but thesemay spoil the accuracy of the data.Our new method provides the guaranty that after noiseremoval the surface still lies within the accuracy of the mea-sured data. This approach is designed for applications whereaccuracy is crucial; for example, technical or medical appli-cations as well as digitalization of cultural heritage.Measuring fairness. Fairness energies are an attempt to es-tablish quantitative measures for fairness of a shape. Findinga commonly-accepted measure of fairness is a delicate task,due to the inherent subjectivity of rating the appearance ofa geometry as well as the specific demands of applications.Nevertheless, one can agree on some general criteria: a fair-ness energy should be independent of the parametrization ofthe surface, in variant under rigid motions and scaling, andspheres should be among the minimizers of the energy.Different measures of fairness have been proposed. Thesecan be classified by the order of the highest derivative of thesurface needed to ev aluate the energy. A measure of first or-der is the area of the surface. Since area is not invariant underscaling, Delingette [Del01] proposed using the isoperimetricratio A3/V2as a scale in variant first-order fairness measurefor closed surfaces. Here A denotes the area and V is thevolume enclosed by the surface. Second-order measures re-late to curvature, such as integrals of squares of curvatureterms. Prominent examples are the bending energyH2dA,the total curvature(κ21+ κ22)dA, and the Willmore energy(κ1− κ2)2dA.Hereκ1and κ2denote the principal curva-tures and H = κ1+ κ2the mean curvature. An example ofa third-order measure is the curvature variation energy pro-posed by Moreton and Sequin [MS92]. The Euler Lagrangeequation of this energy is of sixth order where minimizationbecomes a delicate task; especially on meshes.Improving fairness. Evolution methods are a technique toimprove the fairness of a surface. Such evolving surfacesmonotonically decrease a fairness measure and are describedas the solution of a (usually non-linear) parabolic partial dif-ferential equation. The L2-gradient flow of the area func-tional evolves each point of the surface along its normaldirection with a velocity equal to the mean curvature andis therefore called the mean curvature flow. The underlyingflow equation is of diffusion type ∂tX = ΔX ,whereX is afamily of immersions of the surface and Δ is the Laplace-Beltrami operator of X. Smoothing algorithms based on thisequation are often referred to as Laplace smoothing.Taubin [Tau95] applied this approach to surface smooth-ing using a linearization of the flow equation, which keepsthe Laplace operator unchanged during the evolution. Basedc The Eurographics Association 2007.K. Hildebrandt and K. Polthier / Constraint-based fairing of surface meshesFigure 1: An original noisy scan of a Chinese lion with 1.3 m triangles and height about 10 cm (left).The smoothed mesh (right)stays within a 0.1 mm distance from the initial mesh. The surfaces are colored by mean curvature, with color range from white(negative curvature) t o red (positive curvature).on this linear equation, he constructed a low-pass filter formeshes in analogy to filters used in signal processing. Des-brun et al. [DMSB99] applied the mean curvature flow tomesh smoothing and used the more faithful cotan discretiza-tion of the Laplace-Beltrami operator. Furthermore, they em-ployed an implicit scheme for the time integration to over-come the step-size restrictions of explicit schemes.Feature preserving schemes. A side effect of Laplacesmoothing is that it quickly smoothes out geometricfeatures such as sharp corners. Anisotropic diffusionschemes [CDR00, TWBO02, BX03, HP04] preserve or evenenhance sharp corners by suppressing diffusion in direc-tions of high curvature. Anisotropic geometric diffusion de-rives from t he Perona Malik filter [PM87] in image process-ing. A related technique - the bilateral filter [TM98] - hasbeen transferred to meshes by Fleishman et al. [FDCO03]and Jones et al. [JDD03]. Recently, an extension employ-ing non-local means has been presented by Yoshizawa etal. [YBS06]. Comparable results can be achie ved by meth-ods based on Wiener filtering [PSZ01, GP01, Ale02].Second order fairness measures. While minimizers of thearea functional are determined by values at the boundary,second order fairness measures such as bending energy orWillmore energy allow for C1boundary conditions; i.e. po-sitions and normals can be prescribed. This makes second-order functionals attractive for the construction of fair splinesurfaces in geometric design [WN01]. By specifying consis-tent data at the boundaries of the individual patches, glob-ally G1surfaces can be constructed. Kobbelt and Schnei-der [SK00] construct fair meshes with G1boundary condi-tions as the solution of a fourth-order differential equation.For meshes, stable discretizations of the Willmore energyand its fourth-order gradient flow have been developed inrecent years. Yoshizawa


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