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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 meshes Klaus Hildebrandt Konrad Polthier Freie Universit t Berlin Abstract We propose a constraint based method for the fairing of surface meshes The main feature of our approach is that the resulting smoothed surface remains within a prescribed distance to the input mesh For example specifying the maximum distance in the order of the measuring precision of a laser scanner allows noise to be removed while preserving the accuracy of the scan The approach is modeled as an optimization problem where a fairness measure is minimized subject to constraints that control the spatial deviation of the surface The problem is efficiently solved by an active set Newton method 1 Introduction The instant availability of high quality digital models of 3D surfaces becomes an essential prerequisite in many research areas industrial modeling e commerce medical treatment planning archeology and restoration just to name a few Acquisition technologies such as laser scans for 3D surface models or CT MRI and other devices for volumetric shapes can measure data with high accuracy Nevertheless the accuracy is often lost in the mesh creation pipeline A crucial step in this pipeline is the removal of geometric noise contained in the positions of the measured points Many techniques to effectively remove the noise have been proposed but these may spoil the accuracy of the data Our new method provides the guaranty that after noise removal the surface still lies within the accuracy of the measured data This approach is designed for applications where accuracy is crucial for example technical or medical applications as well as digitalization of cultural heritage Measuring fairness Fairness energies are an attempt to establish quantitative measures for fairness of a shape Finding a commonly accepted measure of fairness is a delicate task due to the inherent subjectivity of rating the appearance of a geometry as well as the specific demands of applications Nevertheless one can agree on some general criteria a fairness energy should be independent of the parametrization of the surface invariant under rigid motions and scaling and spheres should be among the minimizers of the energy Different measures of fairness have been proposed These can be classified by the order of the highest derivative of the c The Eurographics Association 2007 surface needed to evaluate the energy A measure of first order is the area of the surface Since area is not invariant under scaling Delingette Del01 proposed using the isoperimetric ratio A3 V 2 as a scale invariant first order fairness measure for closed surfaces Here A denotes the area and V is the volume enclosed by the surface Second order measures relate to curvature such as integrals of squares of curvature terms Prominent examples are the bending energy H 2 dA 2 the total curvature 1 22 dA and the Willmore energy 1 2



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