Researches And Applications On The Local/Global Approach To Surface Parameterization

Surface parameterization has been a hot topic in computer graphics field,and it has a very wide application in this field.In this paper,we do some research around the two core issues about planar parameterization and spherical parameterization,and has achieved some innovative results.The main work is shown as follows:(1)A novel local/global parameterization approach was proposed for single and multi-boundary triangular meshes.It is an extension of the ARAP(As-Rigid-As-Possible)approach,which stitches together 1-ring patches instead of individual triangles.To optimize the spring energy,we introduce a linear iterative scheme which employs convex combination weights and a fitting Jacobian matrix corresponding to a prescribed family of transformations.Our algo-rithm is simple,efficient,and robust.The geometric properties(angle and area)of the original model can also be preserved by appropriately prescribing the singular values of the fitting ma-trix.To reduce the area and stretch distortions for high-curvature models,a stretch operator is introduced.Numerical results demonstrate that ARAP++ outperforms several state-of-the-art methods in terms of controlling the distortions of angle,area,and stretch.Furthermore,it achieves a better visualization performance for several applications,such as texture mapping and surface remeshing.(2)A novel local/global spherical parameterization was proposed for the genus-zero trian-gular mesh,which naturally extends the planar approach to the spherical case.In our method,we derive two fitting matrices(conformal and isometric)in 3D space.By optimizing the so-called spring energy,the spherical results are achieved by solving a nonlinear system with spherical constraints.Intuitively,it represents the stitching together of the 1-ring patches to form a unit sphere.Moreover,the derivation of the 3D fitting matrices can also be applied to planar trian-gles directly,so that we can obtain a class of novel planar approaches(conformal,isometric,authalic)to the problem of flattening triangular meshes.In order to enhance robustness of the proposed spherical method,a stretch operator is introduced for dealing with high-curvature mod-els.Numerical results demonstrate that our method is simple,efficient and convergent,and it outperforms several state-of-the-art methods in terms of trading-off the distortions of angle,area and stretch.Furthermore,it achieves better visualization in texture mapping.(3)A novel spherical parameterization was proposed to process the genus-zero triangular mesh.The method is based on the planar ARAP++ method and further research of the lo-cal/global parameterization.It consists of two main steps,local phase and global phase.In the local phase,we employ the planar ARAP++ method to optimize the Spring energy,and achieve the displacement of planar vertex around its 1-ring neighborfiood,then map the new vertex to the 3D 1-ring neighborhood on the sphere.In the global phase,we add the spherical constrains to the planar ARAP++ method,and then obtain the final result iteratively according to the New?ton method.Numerical results demonstrate that our method is efficient and convergent,and outperfonns several popular methods in term of controlling the distortion measures(angle,area,rigidity).Furthermore,it achieves a better visualization performance in texture mapping.