The Euler Number of Discretised Sets – Surprising Results in Three Dimensions

Ohser, Joachim; Nagel, Werner; Schladitz, Katja

doi:10.5566/ias.v22.p11-19

Cited by 29 publications

(17 citation statements)

References 24 publications

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“…For each intersection of this image with the markers, we calculate the Euler number χ using the algorithm of Ohser et al . (). This topological characteristic identifies markers referring to pillars by their toroidal shape.…”

Section: An Algorithm To Classify Pillarsmentioning

confidence: 97%

Three‐dimensional image analytical detection of intussusceptive pillars in murine lung

Föhst

Wagner

Ackermann

et al. 2015

Journal of Microscopy

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A variety of diseases can lead to loss of lung tissue. Currently, this can be treated only symptomatically. In mice, a complete compensatory lung growth within 21 days after resection of the left lung can be observed. Understanding and transferring this concept of compensatory lung growth to humans would greatly improve therapeutic options. Lung growth is always accompanied by a process called angiogenesis forming new capillary blood vessels from preexisting ones. Among the processes during lung growth, the formation of transluminal tissue pillars within the capillary vessels (intussusceptive pillars) is observed. Therefore, pillars can be understood as an indicator for active angiogenesis and microvascular remodelling. Thus, their detection is very valuable when aiming at characterization of compensatory lung growth. In a vascular corrosion cast, these pillars appear as small holes that pierce the vessels. So far, pillars were detected visually only based on 2D images. Our approach relies on high-resolution synchrotron microcomputed tomographic images. With a voxel size of 370 nm we exploit the spatial information provided by this imaging technique and present the first algorithm to semiautomatically detect intussusceptive pillars. An at least semiautomatic detection is essential in lung research, as manual pillar detection is not feasible due to the complexity and size of the 3D structure. Using our algorithm, several thousands of pillars can be detected and subsequently analysed, e.g. regarding their spatial arrangement, size and shape with an acceptable amount of human interaction. In this paper, we apply our novel pillar detection algorithm to compute pillar densities of different specimens. These are prepared such that they show different growing states. Comparing the corresponding pillar densities allows to investigate lung growth over time.

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Section: An Algorithm To Classify Pillarsmentioning

confidence: 97%

Three‐dimensional image analytical detection of intussusceptive pillars in murine lung

Föhst

Wagner

Ackermann

et al. 2015

Journal of Microscopy

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“…Notice that the values for c 1 , c 2 , and c 3 correspond to those ones published in Ohser et al (2003) for a Boolean model with balls of constant diameter.…”

Section: Boolean Models In Rmentioning

confidence: 99%

“…It is important to realise that these weights depend on F, but they are independent of X . On the other hand, the vector h = (h ℓ ) is independent of F. The components of w for the adjacency systems F 6 , F 14.1 , F 14.2 and F 26 on L 3 are given in Ohser et al (2003). Now following Jernot et al (2004), we consider a refinement of Eq.…”

Section: Definitionmentioning

confidence: 99%

“…Here we follow Ohser et al (2003); Schladitz et al (2006) where the definition of adjacency is based on a consistency relation for the Euler number.…”

Section: Adjacency and Euler Numbermentioning

confidence: 99%

“…Finally, we present Miles' formulae for Boolean models in R n sampled on homogeneous lattices. In principle, we follow Ohser et al (2003) where formulae are given for the density of the Euler number of Boolean models in R 3 with balls of constant diameter. These results are generalised as follows: We derive formulae for the expectations of the densities of the intrinsic volumes for Boolean models with convex grains.…”

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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3D Images of Materials Structures

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The Euler Number of Discretised Sets – Surprising Results in Three Dimensions

Cited by 29 publications

References 24 publications

Three‐dimensional image analytical detection of intussusceptive pillars in murine lung

Three‐dimensional image analytical detection of intussusceptive pillars in murine lung

Miles Formulae for Boolean Models Observed on Lattices

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