Research On Construction Of Subdivision Schemes And Polynomial Reproduction Based On Generating Functions

Subdivision is one of the most important geometric modeling methods in computer graphics,which has fine properties such as computational efficiency and arbitrary topology.So far,the relevant theoretical research has been relatively mature.H owever,finding the general framework from the relationship between scattered schemes,and constructing subdivision schemes with high polynomial reproduction degree are still two worthy research topics.This thesis mainly studies about these problems based on generating function.The main results are summarized in the following aspects:1.The variant of Lane-Riesenfeld algorithm based on the Chaikin algorithm proposed by Romani in 2015 is generalized by keeping the smoothing operator of LR algorithm unchanged and only changing its refinement operator,and a variant of the LR algorithm with two parameters is obtained,which covers a variety of classic schemes.The parameter range of the new scheme generating C" continuous limit curve is given,and the relation between Romani scheme and the new scheme is presented through graphic illustration,which indicates the other C" continuous new schemes contained in the new scheme different from Romani scheme.The new scheme also contains a subfamily which attains Cn+2-continuity,and polynomial generation degree attains n+2 and n+4.2.Based on the six-point interpolation subdivision scheme,a family of new schemes and a subfamily with cubic polynomial reproduction degree are constructed.The two new families are the variants of the cubic B-spline subdivision and the quintic B-spline subdivision,respectively.And the relationship between the six-point interpolation subdivision and them is similar to the relationship between the four-point interpolation subdivision and the cubic and quintic B-spline subdivision.Analysis shows that the smoothness of the new family attains C3-continuity and the subfamily attains C4-continuity.Compared with the Hormann-Sabin family,which also has the cubic polynomial reproduction degree,the subfamily maintains high smoothness and polynomial generation degree while the support size does not increase as the parameter changes.3.We establish the connection between the push-back method,the finite difference mask and the polynomial reproduction property,and unify the first and second types of methods to study the relationship between the subdivision schemes,providing a simple and intuitive,well-organized perspective and theoretical basis.The concept of elevating factor is proposed,and its geometric meaning is given.A simple method for improving the polynomial reproduction property is obtained.This method does not need to solve the system of equations.It only needs to analyze the generating function of the given subdivision scheme and the elevating factor.The new subdivision obtained can retain the zero condition of the original scheme,and the theoretical analysis is also simple.Finally,the method is used to derive multiple new schemes with higher polynomial reproduction degree.