Surface Area Estimation for Digitized Regular Solids

Regularly gridded data in Euclidean 3-space are assumed to be digitizations of regular solids with respect to a chosen grid resolution. Gauss and Jordan introduced different digitization schemes, and the Gauss center point scheme is used in this paper. The surface area of regular solids can be expressed finitely in terms of standard functions for specific sets only, but it is well defined by triangulations for any regular solid. We consider surface approximations of regularly gridded data characterized to be polyhedrizations of boundaries of these data. The surface area of such a polyhedron is well defined, and it is parameterized by the chosen grid resolution. A surface area measurement technique is multigrid convergent for a class of regular solids iff it holds that for any set in this class the surface areas of approximating polyhedra of the digitized regular solid converge towards the surface area of the regular solid if the grid resolution goes to infinity. Multigrid convergent volume measurements have been studied in mathematics for more than one hundred years, and surface area measurements had been discussed for smooth surfaces. The problem of multigrid convergent surface area measurement came with the advent of computer-based image analysis. The paper proposes a classification scheme of local and global polyhedrization approaches which allows us to classify different surface area measurement techniques with respect to the underlying polyhedrization scheme. It is shown that a local polyhedrization technique such as marching cubes is not multigrid convergent towards the true value even for elementary convex regular solids such as cubes, spheres or cylinders. The paper summarizes work on global polyhedrization techniques with experimental results pointing towards correct multigrid convergence. The class of general ellipsoids is suggested to be a test set for such multigrid convergence studies. 1 JAIST, Japan 2 The University of Auckland, Tamaki Campus, Centre for Image Technology and Robotics, Computer Vision Unit, Auckland, New Zealand Surface Area Estimation for Digitized Regular Solids Yukiko Kenmochi a and Reinhard Klette b a School of Information Science, Japan Advanced Institute of Science & Technology 1-1 Asahidai Tatsunokuchi, Ishikawa 923-1292, Japan b CITR Tamaki, University of Auckland Tamaki Campus, Building 731, Auckland, New Zealand ABSTRACT Regularly gridded data in Euclidean 3-space are assumed to be digitizations of regular solids with respect to a chosen grid resolution. Gauss and Jordan introduced di erent digitization schemes, and the Gauss center point scheme is used in this paper. The surface area of regular solids can be expressed nitely in terms of standard functions for speci c sets only, but it is well de ned by triangulations for any regular solid. We consider surface approximations of regularly gridded data characterized to be polyhedrizations of boundaries of these data. The surface area of such a polyhedron is well de ned, and it is parameterized by the chosen grid resolution. A surface area measurement technique is multigrid convergent for a class of regular solids i it holds that for any set in this class the surface areas of approximating polyhedra of the digitized regular solid converge towards the surface area of the regular solid if the grid resolution goes to in nity. Multigrid convergent volume measurements have been studied in mathematics for more than one hundred years, and surface area measurements had been discussed for smooth surfaces. The problem of multigrid convergent surface area measurement came with the advent of computer-based image analysis. The paper proposes a classi cation scheme of local and global polyhedrization approaches which allows us to classify di erent surface area measurement techniques with respect to the underlying polyhedrization scheme. It is shown that a local polyhedrization technique such as marching cubes is not multigrid convergent towards the true value even for elementary convex regular solids such as cubes, spheres or cylinders. The paper summarizes work on global polyhedrization techniques with experimental results pointing towards correct multigrid convergence. The class of general ellipsoids is suggested to be a test set for such multigrid convergence studies.