Implementation of MPEG-4s Subdivision Surfaces Tools

This work is about the implementation of a MPEG-4 decoder for subdivision surfaces, which are powerful 3D paradigms allowing to compactly represent piecewise smooth surfaces. This study will take place in the framework of MPEG-4 AFX, the extension of the MPEG-4 standard including the subdivision surfaces. This document will introduce, with some details, the theory of subdivision surfaces in the two forms present in MPEG-4: plain and detailed/ wavelet subdivision surfaces. It will particularly concentrate on wavelet subdivision surfaces, which permit progressive 3D mesh compression

Generalized Debye Sources Based EFIE Solver on Subdivision Surfaces

The electric field integral equation is a well known workhorse for obtaining fields scattered by a perfect electric conducting (PEC) object. As a result, the nuances and challenges of solving this equation have been examined for a while. Two recent papers motivate the effort presented in this paper. Unlike traditional work that uses equivalent currents defined on surfaces, recent research proposes a technique that results in well conditioned systems by employing generalized Debye sources (GDS) as unknowns. In a complementary effort, some of us developed a method that exploits the same representation for both the geometry (subdivision surface representations) and functions defined on the geometry, also known as isogeometric analysis (IGA). The challenge in generalizing GDS method to a discretized geometry is the complexity of the intermediate operators. However, thanks to our earlier work on subdivision surfaces, the additional smoothness of geometric representation permits discretizing these intermediate operations. In this paper, we employ both ideas to present a well conditioned GDS-EFIE. Here, the intermediate surface Laplacian is well discretized by using subdivision basis. Likewise, using subdivision basis to represent the sources, results in an efficient and accurate IGA framework. Numerous results are presented to demonstrate the efficacy of the approach

Ck Continuity of Subdivision Surfaces

Stationary subdivision is an important tool for generating smooth free-form surfaces for CAGD and computer graphics. One of the challenges in construction of subdivision schemes for arbitrary meshes is to guarantee that the limit surface will have smooth regular parameterization in a neighborhood of any point. First results in this direction were obtained only recently. In this paper we derive necessary and sufficient criteria for Ck -continuity that generalize and extend most known conditions. We create a general mathematical framework that can be used for analysis of more general types of schemes. Finally, we prove a degree estimate for Ck -continuous polynomial schemes generalizing an estimate of Reif [20] and give a practical sufficient condition for smoothness