Remeshing Of Surfaces Of Triangular Mesh

In this paper we present a remeshing scheme of triangular mesheswhich are gotten from the 3D digital camera .It includes simplificationof triangular mesh, optimization of vertices'positions of triangularmesh and optimization of topological connectivity of triangular mesh .The final mesh generated by our remeshing scheme approximates theoriginal mesh well. It also reflects the curvature of the originaltriangular mesh, and every triangle of the final mesh approximatesequilateral triangle .It not only guarantees the visual quality of thegeometrical model but also reduces the redundant vertices.Mesh simplification is an important content in remeshing oftriangular mesh, which is deeply meaningful to storage, transmissi-on ,process, especially rendering timely of geometrical models .In thispaper we present an error measure function about collapsing an edge toguide mesh simplification .We define it as follows:tijvvfnormalfnormalWhere ?,? is the dot product of two vectors, f · normal and f~1· normalare unit normals of the triangular face f and the triangular face f~1respectively. T (i ) is the set of triangular faces that contain thevertex i and T (i , j) is the set of triangular faces that contain the edge(i , j).In our algorithm about mesh simplification we get the orders ofthe collapsing edges by the above function and collapse the edge withminimal error measure edge at each step in order to reduce the numberof the vertices of the mesh. We mustn't collapse the boundary edge inorder not to add the approximating error during the collapsing edge.Mesh optimization is a core content in remeshing of triangular mesh,which mainly includes optimization of vertices'positions andoptimization of topological connectivity. We reduce the numbers ofvertices after mesh simplification. But distribution of vertices isvery irregular and it has many skinny triangles .So we must optimizethe mesh in order to get the regular distribution of vertices and letevery triangle approximate the equilateral triangle.In this paper, we optimize vertices'positions of triangular meshby the method based area equalization, and then we optimize topologicalconnectivity of triangular mesh by a serials of operations of edge swap.We apply optimizations on the varying mesh M which is initializedto M 0.Firstly, we map the 1-ring neighborhood of a vertex of M into itstangential plane by the conformal map .Let i be a vertex of M andv is the corresponding point in R 3. N (i ) is the 1-ring neighbor-hood , N (i ) = {k 1 ≤k≤D}, and D is the degree of i .Here we supposethe vertices of N (i ) have an order .That is to say, the triangular face(k ? 1,i,k) is a triangle of M ,where j0 = j D .The definition of theconformal map is as follows:where ( )11, , , , 2 /kk i k k l i l Dlr v v θ v ?v vα π θ== ? = ∑ ∠ = , Φ ( v ) = p= ( 0,0 ),Φ ( vk ) = p k = ( xk ,yk).Then we map the sub-mesh of M containing the1-ring neighborhood of i into a planar mesh whose vertices are p andp k.Secondly, we optimize the position of the vertex by the method basedarea equalization in the tangential plane ,whose target is to find anew point p 1 = ( x,y) in the tangential plane such that the ratios of theareas of the triangles which are adjacent to p are as close as possibleto μ1 , μ1 , L , μD.All μ k are positive and sum to unity. Denote by Α k ( x,y)the area of the triangle ( )1p , pk ,pk +1 , we compute it as follows:( )1112, 111xyxyxyx ykkkkΑ k =++Let Α be the area of polygon ( )p1 , p2,LpD,which may be computed as∑( )=Α =DΑkk10, 0. Now the location of p 1 = ( x,y) is defined as follows:( ) ( ) ( ( ))2, argm,i n∑1,== DΑ?Αkx yx ykxyμ kThis is an classical least square question which can be computed bysolving a system of two linear equations in x and y.Lastly we can get the new position v new after optimization by theinverse of the conformal map which map the new point p1 into the varyingmesh M . Then we can get the final position v 1 of i by projecting v newinto the triangular cubic Bezier patches for every face of M 0. Afteroptimizing the position of every vertex of M we can optimize thetopological connectivity of triangular mesh by a serials of operationsof edge swap.In this paper we further optimize the mesh by the method based onweighted angle-based technique which improves angles of the trianglesincidents on i by moving its position. Let α k be the angle adjacentto v k in the polygon v1 , v2,L ,vD.We define v 1k to be the point lying onthe bisector of α k such that v k ? v=vk?v1k, namely, the edge (v k, v)is rotated around v k to coincide with the bisector of α k. The newposition of i is defined as the average of all v 1k for all the neighbors,namely:∑== Dkvnew kvk1μ 1Where μ k is equal to ∑=Dk kk12211αα.In this paper we use two error measures to evaluate the distancebetween the new mesh and the original mesh .When we optimize the mesh ,some new triangular faces are generated .We don't modify the mesh ifthe error measures of the new face is much larger than the thresholdset beforehand. By this means we can guarantee that the new meshapproximates the original mesh well.An important factor of measuring the quality of a remeshed modelis fidelity to the original mesh, which is the difference between theprocessed mesh and original mesh. The error is the Hausdorff distancenormalized by the bounding box diagonal, obtained using the Metro tool[13]. Another important factor of measuring the quality of a remeshedmodel is by measuring the geometric properties of the resultingtriangles and the combinational properties of the mesh, such as thedegree of vertices. Statistics are usually collected on the minimalangle and the average angle of all triangles. Obviously this value isanywhere between 0 o and 60 o .For a high-quality mesh, the minimum ofthese values should be no less than 10 o, and the average angle shouldbe no less than 45 o.In this paper we test our remeshing scheme to some 3D modelsdownloaded in the internet and some 3D human face models gotten fromour own 3D digital camera .The testing results demonstrates that ourremeshing scheme satisfy needs about accuracy and speed and can be usedby the software system of the 3D digital camera .