Surface Area Estimation of Digitized 3D Objects Using Local Computations

Abstract

We describe surface area measurements based on local estimates of isosurfaces originating from a marching cubes representation. We show how improved precision and accuracy are obtained by optimizing the area contribution for one of the cases in this representation. The computations are performed on large sets (approximately 200,000 3D objects) of computer generated spheres, cubes, and cylinders. The synthetic objects are generated over a continuous range of sizes with randomized alignment in the digitization grid. Sphericity, a scale invariant measure of compactness, allows us, in combination with the improved surface area estimate, to distinguish among the test sets.

Miles Formulae for Boolean Models Observed on Lattices

“…It also allows a comparison of the various methods for surface estimation. As shown in Ziegel and Kiderlen (2009), the maximum asymptotic relative error for general sets X is 12.8 % for the weights suggested in Lindblad and Nyström (2002) The approach based on the Crofton formulae has a number of advantages. First, the method of surface area estimation can simply be extended to arbitrary dimensions and to most of the other intrinsic volumes, in particular the integral of the mean curvature which measures e.g.…”

“…The probably most intuitive method for measuring the surface area is based on rendering data, see, e.g., Lindblad and Nyström (2002), where the areas of the surface patches serve as weights for computing the surface area from local knowledge, i.e., from the numbers of pixel configurations. However, this type of estimator is not multigrid convergent.…”

“…We are starting from the local knowledge represented by the numbers of 2 × 2 × 2 pixel configurations on cubic primitive lattices and ask for the best choice of surface weights. The weights suggested by Lindblad (2005) minimise the estimation variance of the surface area of a plane with random normal direction uniformly distributed on the unit sphere. This idea goes back to Mullikin and Verbeek (1993), see also the discussion in Windreich et al (2003).…”

“…Moreover, compared to empirical marching cubes methods like in Windreich et al (2003), only a small number of pixel configurations has to be taken into account. In Schladitz et al (2006) the weights for the surface are compared with those of Lindblad and Nyström (2002), see also Ziegel and Kiderlen (2009) for a sound investigation of surface area estimation. This paper is organised as follows: First we repeat some facts about homogeneous lattices and summarise results on adjacency systems and Euler number estimation from Nagel et al (2000); Ohser et al (2002;; Schladitz et al (2006).…”

Surface Area Estimation in Practice

“…For an alternative voxel-based surface area estimation method, see e.g. [6]. Table 1 shows the frequency of the nine surface voxel types in a 160 × 200 × 160 segmented white matter MR brain image (in the grey-white matter interface).…”