A Local/Global Approach To Parameterization

As the continuously developing of computer technique,3D geometry data has become the fourth wave of multimedia data,following the revolution of sound,image and video. Digital geometry processing(DGP) is the research of mathematical and computational foundations on how to process 3D geometry data in computer.Since the starting of this century,digital geometry processing has been extensively studied,of which parameterization is a most significant and fundamental problem.Parameterization plays an important role in many applications of geometry processing,such as reconstruction,modeling,denoising, compression,editing and sensor network.There is a popular maxim,"think globally, act locally",which indeed reflects a spirit in methodology,referring to the argument that global environment can be changed by acting on local surroundings.This local/global approach has been successfully applied in economics,sociology,anthropology,and other related fields.In this paper,from such a local/global strategy,we make thorough investigation on the problem of parameterization,and establish different techniques to solve it according to various practical applications.These presented parameterization approaches have been used in other applications in DGP,which show great flexibilities and power.Targeting at the problem of parameterization,this paper provides deep study to develop different parameterization approaches,and contributes the following aspects.1.Based on the local/global methodology,this paper presents a novel approach to planar parameterization.It is observed that the stretch in parameterization can be characterized by the singular values of the Jacobian of parameterization mapping.Hence, locally by analyzing the Jacobian of affine transformation on each triangle and adjusting the corresponding singular values to make the transformation on each triangle selected from a specified matrix set,but globally "stitching" the triangles by solving a linear system,the final parameterization result can be obtained.To minimize angle distortion during parameterization,an ASAP(As-similar-as-possible) approach is proposed;to minimize the area distortion of each triangle,an AAAP(As-authalicas -possible) approach is proposed;to minimize the shape stretch of each triangle, an ARAP(As-rigid-as-possible) approach is proposed.Additionally,a hybrid parameterization approach is also proposed,which allows user to balance the trade-off between angle and area distortion during parameterization,by simply controlling a weight factor,and generate ideal parameterization result.2.A manifold parameterization framework,called ManiPara,is proposed,which can build the one-to-one mapping between two equivalent triangular meshes of arbitrary shape.Technically,it is also based on the local/global strategy.By analyzing the Laplacian operator defined on the source mesh locally,its connectivity can be used to approximate the shape of the target mesh,and finally the parameterization is established by globally solving a linear system.ManiPara can preserve pretty geometry attributes during the parameterization,and also be used conveniently.This approach has been used to solve other problems,like texture transfer,animation,and so on.3.As for parameterizing point cloud,a novel local/global meshless parameterization approach,called NeighborAlign,is presented,which obtains the one-to-one mapping along with the conversion of local neighbors.Locally,the neighbors of each point are mapped onto a reference plane with distance preserving;globally,this reference neighbors are coherently aligned together,which forms the final parameterization. Experimental results show that NeighborAlign approach can preserve the relative distance between neighbor points,which results better parameterization and suits for surface reconstruction and other applications.4.For the parameterization of a general graph network,a novel parameterization technique is proposed by reusing the local/global strategy.From the lengths of edges in the graph,the positions of nodes in the local neighbors are optimized to get the reference patch in the plane.And the final parametedzation is obtained by aligning the reference patches with rigid transformation.This approach is also named "as-rigidas -possible" parameterizaiton,as the rigid transformation is used in the parameter- ization,which can preserve the lengths of edges.Experimental instances show that this parameterization approach can be applied to sensor network localization,and produce more accurate localization,especially for sensor network of sparse connectivity and with noise.