Studies On Smoothing Algorithms Of Discrete Models

Representation and processing of discrete models are key problems in computeraided design and computer graphics. They play very important roles in a lot of ap-plications, such as reverse engineering, digital geometry processing, computer gameand film animation. Especially, smoothing is a key factor in the processing of discretemodels. This dissertation focuses on blending and feature-preserving smoothing fortriangle meshes, as well as smoothing, projecting, computation of volume and area forpoint-sample surfaces. The main contributions are summarized as follows:1. To smooth feature edges, this dissertation proposes a new smoothing algorithmcalled mesh blending based on the technique of rolling ball blending. A majorfeature of mesh blending is to move vertices of the blending region to a virtualblending surface by choosing an appropriate parameterization of those vertices.The major advantages are to allow the users to specify different blending radii forcontrolling the blending shape directly, and modify the shape of the profile curvefor complex design. By combining mesh blending with multi-resolution tech-niques, our algorithm can also be used in many applications, including smooth-ing, sharpening, and mesh editing.2. A feature-preserving mesh smoothing algorithm is proposed to prevent shrinkageand drift for corners. First, a new predictor of bilateral filter is presented, whichprevents shrinkage and drift for corners. Then, an algorithm of mesh smoothingcan be obtained by moving every vertex along the direction determined by themean curvature normal with speed defined by the new predictor. This smooth-ing algorithm can quickly remove the noise of the large mesh while preservingits important features, and can also prevent unnatural deformation for irregularmeshes.3. A new algorithm for projecting points onto a point-sampled surface is proposedto smooth the point-sampled surface. This algorithm operates directly on thepoint cloud without any surface reconstruction procedure, and it is simpler and more efficient than those conventional approaches based on surface reconstruc-tion. This proposed algorithm projects points onto the point cloud in a least-squares sense without any specification of projection vector. Furthermore, wedemonstrate the power of this algorithm through a number of application exam-ples including point normal estimation, curves projection and so on.4. In order to implement global volume-preserving and area-preserving smoothing,new quasi-Monte Carlo methods for computing the volume and area of point-sampled surfaces are presented. Our methods operate directly on the point cloudwithout any surface reconstruction procedure. First, an octree of the point cloudis constructed with a volume-based refinement method, and then the volume ofthe point-sampled surface can be estimated by using quasi-Monte Carlo methods.Based on the Cauchy-Crofton formula, the area of the point-sampled surface isalso calculated by counting the number of intersection points between the pointcloud and a set of uniformly distributed lines generated with low-discrepancysequences. By testing on a number of point-based models, experiments suggestthat our methods are more robust and more efficient than those conventional ap-proaches based on surface reconstruction.