Deducing interpolating subdivision schemes from approximating subdivision schemes

Abstract

In this paper we describe a method for directly deducing new interpolating subdivision masks for meshes from corresponding approximating subdivision masks. The purpose is to avoid complex computation for producing interpolating subdivision masks on extraordinary vertices. The method can be applied to produce new interpolating subdivision schemes, solve some limitations in existing interpolating subdivision schemes and satisfy some application needs. As cases, in this paper a new interpolating subdivision scheme for polygonal meshes is produced by deducing from the Catmull-Clark subdivision scheme. It can directly operate on polygonal meshes, which solves the limitation of Kobbelt's interpolating subdivision scheme. A new √3 interpolating subdivision scheme for triangle meshes and a new √2 interpolating subdivision scheme for quadrilateral meshes are also presented in the paper by deducing from √3 subdivision schemes and 4-8 subdivision schemes respectively. They both produce C 1 continuous limit surfaces and avoid the blemish in the existing interpolating √3 and √2 subdivision masks where the weight coefficients on extraordinary vertices can not be described by formulation explicitly. In addition, by adding a parameter to control the transition from approximation to interpolation, they can produce surfaces intervening between approximating and interpolating which can be used to solve the "popping effect" problem when switching between meshes at different levels of resolution. They can also force surfaces to interpolate chosen vertices.

A Method for Deducing Interpolating Subdivision Schemes from Approximating Subdivision Schemes and Blending Subdivisions

“…Maillot et al [17] and Shujin Lin et al [1] discover that there are some connections between the B-spline and the fourpoint curve subdivision methods [4] which can be used to find a solution that directly deduces new interpolating subdivision masks for meshes from their corresponding approximating subdivision masks. However, Maillot's method is rather complex in term of push-back steps needed in each level of refinement and they do not generalize it to extensive applications.…”

“…where (1) Hassan et al [10] proposed an interpolating four-point ternary stationary subdivision scheme with higher order smoothness property compared to most of the existing binary ones. This well-known interpolating ternary scheme is proved to be -continuous when the subdivision parameter is chosen in a certain range.…”

“…We triumphantly find the close connection between the approximating subdivision schemes such as cubic B-spline curve subdivision scheme and interpolating subdivision schemes such as the well-known interpolating four-point C 2 ternary stationary subdivision scheme [10].The basic idea is to make use of the specific linear combination of the displacements of the new vertices which are derived from approximating subdivision scheme to produce associated interpolation displacements for the new vertices therefore generating a corresponding interpolation subdivision scheme. The most importance is that such methods are deduced in premise of keeping the continuity of the original approximating subdivision schemes and easily extended to the case of surface subdivision schemes(see these similar applications in [1]). Our approach is rather simple without the use those rather complex method in term of pushback steps in the existing literatures [17,19], and provides more adjustment parameters for extensive applications compared with the methods in [17,18,19].…”

“…This characteristic is suitable for specific industrial applications. Classical work in interpolating subdivision schemes are presented in [1,3,4,7,10]. Approximating subdivision schemes can be conveniently used for intersection operation, and produce a limiting curve with higher smoothness and continuity.…”

Interpolating Subdivisions Based on Difference Interpolating Functions

“…Maillot and Stam [16] found some relationships between the cubic B-spline and 4-point curve subdivision method. Lin et al [17] proposed a method to directly deduce new interpolating subdivision masks for meshes from the corresponding approximating subdivision masks. The essence comes from Maillot and Stam's idea [16] .…”