Non-uniform Subdivision Surfaces Via Eigen Polyhedron

Non-uniform rational B-splines(NURBS)and subdivision are two main free-form surface representation methods.NURBS is the standard for industry and subdivision is the standard representation for animations.In order to generalize the NURBS to arbitrary topology,the non-uniform subdivision is introduced.This thesis is focusing on the non-uniform subdivision and includes the following three main contributions.In the first work,we construct a new non-uniform Doo-Sabin subdivision scheme via eigen polygon.We proved that the limit surface is always convergent and is G1 con-tinuous for any valence and any positive knot intervals under a minor assumption,where we assume that ? is the second and third eigenvalues of the subdivision matrix.And then,a million of numerical experiments are tested with randomly selecting positive knot intervals,which verify that our new subdivision scheme satisfies the assumption.However this is not true for the other two existing non-uniform Doo-Sabin schemes in[1-2].In additional,numerical experiments indicate that the quality of the new limit surface can be improved.In the second work,we define a non-uniform interpolatory subdivision scheme("NUISS" for short hereinafter)for arbitrary topological control mesh via eigen poly-hedron.We first split the edge and face point rules into the average of the contribu-tions from the adjacent vertices.And then we define the eigen-polyhedron for the non-uniform extraordinary points and solves the contributions of the extraordinary points to the adjacent face and edge points guided by the eigen-polyhedron.This construction can guarantee that the subdivision matrix can have two identical eigenvalues 1/2 which is the necessary condition for the subdivision surface to be G1.Numerical experiment results demonstrate that our algorithm can be improved subdivision surfaces with comparable or even better high-quality.In the third work,we give an evaluation algorithm for non-uniform subdivision sur-face at arbitrary parameters.Our evaluation algorithm calculates the position of the non-uniform subdivision surface inside the patch around the extraordinary vertex,and the remaining patches are evaluated through the non-uniform B-splines.The non-uniform subdivision surface obtained high-quality surface under the non-uniform parameteriza-tion.The subdivision basis function is global linearly independent,unit subdivision,and the subdivision space is nested,so that the geometric mapping can be kept unchanged.