The Euler Number of Discretized Sets — On the Choice of Adjacency in Homogeneous Lattices

Ohser, Joachim; Nagel, Werner; Schladitz, Katja

doi:10.1007/3-540-45782-8_12

Cited by 26 publications

(48 citation statements)

References 24 publications

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“…For n = 3 we recall the adjacency systems F 6 , F 14.1 , F 14.2 , and F 26 considered in detail in Ohser et al (2002;.…”

Section: Adjacency and Euler Numbermentioning

confidence: 99%

“…As in Ohser et al (2002;; Schladitz et al (2006) we approximate X by a discretisation X ⊓ F w. r. t. an adjacency system F. Since the set X ⊓F forms a (not necessarily convex) polyhedron, the number #F j (X ⊓ F) of elements of F j (X ⊓ F) is finite and, therefore, the Euler number χ(X ⊓ F) can be computed via the Euler-Poincaré formula,…”

Section: Definitionmentioning

confidence: 99%

“…The Crofton formulae reduce the estimation of intrinsic volumes to the estimation of the Euler number in lower dimensional sections. Therefore, the present paper is based on thorough investigations of digital connectivity and consistent estimation of the Euler number, see Nagel et al (2000); Ohser et al (2002;; Schladitz et al (2006). The consistency results for the Euler number estimates of foreground and background established in Ohser et al (2002) carry over to all intrinsic volumes.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the present paper is based on thorough investigations of digital connectivity and consistent estimation of the Euler number, see Nagel et al (2000); Ohser et al (2002;; Schladitz et al (2006). The consistency results for the Euler number estimates of foreground and background established in Ohser et al (2002) carry over to all intrinsic volumes. In order to give the reader a comprehensive description of our approach making the latter as self-contained as possible, we recall some facts addressed in the papers mentioned above.…”

Section: Introductionmentioning

confidence: 99%

“…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). Based on concepts of section lattices and their translative complements, we derive discrete versions of the Crofton formulae for polyconvex sets and from these, estimators for the densities of the intrinsic volumes of random sets.…”

Section: Introductionmentioning

confidence: 99%

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Miles Formulae for Boolean Models Observed on Lattices

Ohser¹,

Nagel²,

Schladitz³

2011

Image Anal Stereol

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The densities of the intrinsic volumes -in 3D the volume density, surface density, the density of the integral of the mean curvature and the density of the Euler number -are a very useful collection of geometric characteristics of random sets. Combining integral and digital geometry we develop a method for efficient and simultaneous calculation of the intrinsic volumes of random sets observed in binary images in arbitrary dimensions. We consider isotropic and reflection invariant Boolean models sampled on homogeneous lattices and compute the expectations of the estimators of the intrinsic volumes. It turns out that the estimator for the surface density is proved to be asymptotically unbiased and thus multigrid convergent for Boolean models with convex grains. The asymptotic bias of the estimators for the densities of the integral of the mean curvature and of the Euler number is assessed for Boolean models of balls of random diameters. Miles formulae with corresponding correction terms are derived for the 3D case.

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“…For n = 3 we recall the adjacency systems F 6 , F 14.1 , F 14.2 , and F 26 considered in detail in Ohser et al (2002;.…”

Section: Adjacency and Euler Numbermentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%