Curvature Transport Method For Solving Mesh Singularities

In digital geometry processing,parameterization is widely used in practical appli-cation.And it is always a popular research problem that how to reduce the distortion of parameterization.The main ways to reduce distortion include selecting proper parame-terization method,detection of cone singularities,construction of the seams and so on.For a general 3D mesh,the Gaussian curvature of most vertices on the mesh is not equal to zero,which is the fundamental reason of generating distortion.The distortion will be reduced if Gaussian curvature distributed throughout the vertices can be concentrated on a few cone singularities and the mesh can be cut along these cone singularities.The main research target of this paper is to find a method to construct a set of well cone singularities,which not only solve the distribution and number of cone singularities on the mesh automatically,but also ensure that the seams through these cone singularities can effectively reduce the distortion.In this paper,a method based on curvature transport is proposed to solve the num-ber and positions of cone singularities.This method will solve the positions of cone singularities in two steps.Firstly,the Gaussian curvature of the mesh vertices is used to solve the Yamabe function to obtain the conformal scaling factor of the mesh vertices.The value of the conformal scaling factor reflects the scaling extent of each vertex.Then the conformal scaling persistence is obtained by using the concept of persistence of vertices in computational topology.If the conformal scaling factor is regarded as a scalar field on the mesh,then the vertices with a positive persistence are represented as the local maximum and minimum points of the scalar field.The persistence is larger,the deviation between the extreme point and the points in the neighboring domain is greater.The cone singularity of the first part can be determined by sorting persistence from largest to smallest and setting a persistence threshold.The seams and distortion based on the cone singularity set are calculated.According to the distortion of each triangular face by using the cone singularities in the first part,the second part of cone singularities can be selected hierarchically for triangular faces with larger distortion.In this paper,we choose to ignore the distortion of triangular surface near the seams,since sometimes larger distortion will occur.After the cone singularities of the second part is obtained,its positions need to be optimized.Each mesh edge is given a transport cost,and the curvature is distributed on each vertex in the initial state.The optimal transport cost can be calculated by transporting the curvature of each vertex along each mesh edge to the cone singularities solved above.The positions of the cone singularities are iteratively updated in the 1-ring of the target cone singularity and the transport cost is recalculated to minimize the above optimal transport cost.At this time,the optimiza-tion of the cone singularities positions will stop and the final distribution of the cone singularities will be solved.Experimental results show that the method proposed by this paper can effectively reduce the distortion of parameterization.Compared with state-of-the-art methods,the proposed method can find cone singularities in the region with small curvature,which are frequently ignored by other curvature-based methods,resulting in large local distor-tions.