Research Seminar - December 03, 1999

Abstract

Criteria for surface approximation are: multigrid convergence towards the true surface (Hausdorff metric) or towards the true surface contents, support for feature calculations based on the approximated surface (ie. polyhedral approximations support the use of algorithms known in computational geometry), time/space efficiency, robustness etc.

The talk consists of two parts:

A review of local surface fitting techniques with respect to surface area measurement and volume data segmentation (confocal microscopy of chondrons) illustrates the need in developing techniques for convergent surface area measurement. Marching cubes, opaque cubes, marching tetrahedra etc. are local techniques which are not convergent with respect to the surface area measurement problem. Volume data segmentation techniques based on such local isosurface approaches contribute to analysis errors.

The second part informs about the recent situation in surface approximation based on two-dimensional grid continua. There are algorithmic solutions for calculating the length of one-dimensional grid continua in 2D or 3D space. But so far there is no algorithmic solution known for calculating a minimum-area polyhedral surface complete and contained in a closed two-dimensional grid continuum. Global approximations via linear surface patches might be an easier to solve problem, but there are no convergence theorems known for this approach.