Edge Length Based 3D Shape Interpolation

Shape interpolation is a very important and fundamental problem in computer graphics and geometry processing.It can be used to generate natural transitions between two or more geometric shapes.Therefore,it is widely used in computer animation,film and television production,geometric modeling and other fields.The research on shape interpolation in the field of computer graphics has a history of several decades and rich references.In particular,for triangular meshes that discretely represent 3D surfaces,the current mainstream algorithms mostly simultaneously interpolate edge lengths,dihedral angles and other geometric quantities,and then reconstruct the vertex coordinates to obtain the interpolation results.However,the edge lengths and other geometric quantities of reconstructed meshes are generally approximate interpolation quantities.According to Euler’s theorem,for a closed triangular mesh,given all the edge lengths,the continuous degree of freedom is 0 if the global rigid transformation is not considered.If all the edge lengths are known,the number of solutions to the triangular mesh is finite in most cases,and there are no additional degrees of freedom to interpolate other geometric quantities at the same time.In this paper,a triangular mesh interpolation algorithm based on edge lengths is proposed.It is noted that for planar triangular meshes and 3D tetrahedral meshes,interpolating squared edge lengths is equivalent to interpolating pullback metric.So it has the good property that isometric distortion and conformation distortion are bounded simultaneously.We extend this method to triangular meshes.Given the edge lengths,we use Newton’s method to optimize the edge length error energy in the stage of mesh reconstruction.We give the analytic positive definite form of its Hessian matrix,so as to avoid the costly eigenvalue decomposition.Our method directly takes vertex coordinates as optimization variables,which can effectively avoid the accumulation of error compared with the existing algorithm that calculates the dihedral angles first and then reconstructs the vertex coordinates.Therefore it can achieve high-precision mesh reconstruction.Because the edge length error energy is not convex,the convergence result of Newton’s method is highly dependent on the initial value.Note that the result that is obtained by interpolating squared edge lengths of the tetrahedral mesh has very low curvature,which means it can be flattened and embedded in 3D space with only a few modifications.Therefore,this article proposes to convert the triangular meshes into tetrahedral meshes first,and then extract the surface from the interpolation result of the tetrahedral meshes.We use the surface as initialization in the Newton’s iteration of the target edge length error,which makes the convergence result closer to the global optimum.We test our method on a series of input triangular meshes,and the results show that the edge length error of ours is smaller than that of previous edge length based methods.And our results have bounded distortion.