Length distributions of edges in planar stationary and isotropic STIT tessellations

Mecke, Joseph; Nagel, Werner; Weiß, V.

doi:10.3103/s1068362307010025

Cited by 21 publications

(55 citation statements)

References 8 publications

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“…We will furthermore obtain a general moment relationship for the conditional and unconditional length distribution of the segments. Our results are demonstrated on several concrete examples, in particular we consider the isotropic case and confirm again the results of Mecke et al (2007). Also the new rectangular case and a case with unequal weights is discussed.…”

Section: Introductionsupporting

confidence: 82%

“…More results about STIT tessellations can be found in Nagel and Weiss (2003), Nagel and Weiss (2005), Nagel and Weiss (2006), Mecke et al (2007), Mecke et al (2008a), Mecke et al (2008b), Thäle (2008), Thäle (2009). Mackisack and Miles (1996) introduced the notion of I-, K-and J-segments and showed that their analysis can be fruitful, especially in the case of tessellations which are not face-to-face.…”

Section: Fig 1 Realization Of An Isotropic Stit Tessellationmentioning

confidence: 99%

“…This includes mean value formulas in 2D and 3D and length distributions of several linear segments. For the latter Mecke et al (2007) calculated also first and second moments, but only for the isotropic case. They especially observed that the variance of the length of the so-called typical Isegment is infinite.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Moments of the Length of Line Segments in Homogeneous Planar Stit Tessellations

Thäle¹

2011

Image Anal Stereol

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Homogeneous planar tessellations stable under iteration (STIT tessellations) are considered. Using recent results about the joint distribution of direction and length of the typical I-, K-and J-segment we prove closed formulas for the first, second and higher moments of the length of these segments given their direction. This especially leads to the mean values and variances of these quantities and mean value relations as well as general moment relationships. Moreover, the relation between these mean values and certain conditional mean values (and also higher moments) is discussed. The results are also illustrated for several examples.

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Section: Introductionsupporting

confidence: 82%

Section: Fig 1 Realization Of An Isotropic Stit Tessellationmentioning

confidence: 99%

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Moments of the Length of Line Segments in Homogeneous Planar Stit Tessellations

Thäle¹

2011

Image Anal Stereol

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“…For details on STIT tessellations we refer to Nagel and Weiss (2005), Nagel and Weiss (2006), Mecke et al (2007), Nagel and Weiss (2008), Mecke et al (2008a) and Mecke et al (2008b), Mecke (2009), Thäle (2009 or Schreiber and Thäle (2010). Also Schneider and Weil (2008), pp.…”

Section: Fig 4 a Realization Of A Homogeneous And Anisotropic Stit mentioning

confidence: 99%

New Mean Values for Homogeneous Spatial Tessellations That Are Stable Under Iteration

Thäle

Weiß

2010

Image Anal Stereol

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Homogeneous random tessellations in the 3-dimensional Euclidean space are considered that are stable under iteration -STIT tessellations. A classification of vertices, segments and flats is introduced and a couple of new metric and topological mean values for them and for the typical cell are calculated. They are illustrated by two examples, the isotropic and the cuboid case. Several extremum problems for these mean values are solved with the help of techniques from convex geometry by introducing an associated zonoid for STIT tessellations.

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“…The authors first learned from R. V. Ambartzumian the idea of this operation in the 1980s; cf. [6], [8], and [9]. He also posed the problem of the existence of a limit for repeated iteration.…”

Section: Introductionmentioning

confidence: 99%

The iteration of random tessellations and a construction of a homogeneous process of cell divisions

2008

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A random tessellation of R d is said to be homogeneous if its distribution is invariant under all shifts of R d . The iteration of homogeneous random tessellations is described in a new manner that makes it evident that the resulting random tessellation is homogeneous again. Furthermore, a tessellation-valued process is constructed, the random states of which are homogeneous random tessellations stable under iteration (STIT). It can be interpreted as a process of subsequent division of cells.

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Length distributions of edges in planar stationary and isotropic STIT tessellations

Cited by 21 publications

References 8 publications

Moments of the Length of Line Segments in Homogeneous Planar Stit Tessellations

Moments of the Length of Line Segments in Homogeneous Planar Stit Tessellations

New Mean Values for Homogeneous Spatial Tessellations That Are Stable Under Iteration

The iteration of random tessellations and a construction of a homogeneous process of cell divisions

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