Progressive Interpolation And Smoothing Of Subdivision Surfaces

Subdivision surface has lots of applications in many fields due to its flexibility,multi-resolution structure and the ability to represent arbitrary topological meshes.A subdivision scheme is called an interpolating scheme if the limit surface interpolates vertices of the original mesh.Otherwise,it is called an approximating scheme.In general,interpolating subdivision schemes are sensitive to sharp features & irregular meshes and may produce low quality surfaces comparing with those of the approximating subdivision schemes.In order to make approximating scheme interpolate or fit original vertices,scholars have proposed a series of iterative interpolation algorithms.Among them,progressive interpolating subdivision scheme is an extension of the progressive-iterative approximation(PIA)method on the subdivision surface.This progressive interpolation algorithm has the advantages of local and global methods.That is,it can handle control meshes of any size and any topology while generating smooth subdivision surfaces that faithfully resemble shapes of initial meshes.Traditional progressive interpolation subdivision schemes can interpolate or fit vertices by applying approximating subdivision schemes,and it can achieve desirable results.However,with the increasing number of original vertices,the convergence speeds become slow and computation complexity gets huge.Therefore,how to use interpolation algorithm to deal with large-scale data is a hot issue nowadays.Furthermore,in the field of hydrodynamics and aerodynamics,the interpolation or fitting tangent planes and curvatures of the given surfaces is very important.Consequently,it is necessary to propose an progressive interpolation subdivision schemes which can interpolate or fit tangent planes and curvatures of the given surfaces.In view of the above research status,the following work has been done in this dissertation:1.Base on the local properties of progressive interpolation,an adaptive progressive interpolation subdivision scheme is presented.The vertices of control mesh are classified into two classes: active vertices and fixed ones.When precision is given,two classes vertices are changed dynamically according to result of each iteration.Only the active vertices are adjusted,thus the class of active vertices keep running down while the fixed ones keep rising,which saves computation greatly.Furthermore,an adaptive weight algorithm is also given,which assigns weights to the vertex adaptively,speeds up the convergence of slower vertices,and make them satisfies the given convergence speed threshold,so that the convergence speed can be adaptively controlled.2.One progressive interpolation algorithm of Catmull-Clark subdivision surface with matrix weight is presented.It aims to interpolate the normal vector of subdivision surface which traditional progressive interpolation algorithm can’t make it.First,a 3×3 weight matrix is presented as the weight of the given progressive interpolation algorithm.Different matrix weights have been given in order to not only control convergence speeds & shapes but also interpolate the normal vector so as to smooth the limit surface.Second,the weight matrix can be decomposed into the sum of two matrices,one controls the convergence rate,the other controls the surface shape and smoothness.Finally,this dissertation also presents two different ways to determine the weight matrix.One is designing the diagonal matrix in order to control the shape of the limit surface.The other is designing the rotation matrix in order to iteratively interpolate the normal vector to smooth the limit surface.