Sound Analysis of 3D Objects Based on Digitized Data

Abstract

The report reviews selected results in the field of geometrical measurements and reconstructions of 3D objects (i.e. simply-connected compact sets of points) based on gridding techniques. Two soundness properties of approaches are discussed with respect to the selected grid resolution: convergence and convergence towards the "true" value. The existence of sound multigrid approaches is discussed for problems as (1) volume and surface area measurement for Jordan sets (i.e. 3D objects bounded by Jordan surfaces), (2) approximations of planes based on sampled data, (3) surface reconstructions based on gradient information, and (4) surface recovery by solving a (special) linear differential equation system. The paper concludes with a brief discussion of arising digital or computational geometry problems relevant to the discussed subjects.