Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration

Nagel, Werner; Weiß, V.

doi:10.1017/s0001867800000574

Cited by 70 publications

(237 citation statements)

References 10 publications

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“…They were formally introduced by Nagel and Weiss (2005). In later papers Nagel and Weiss (2006; have shown that many mean values can be obtained from the characteristic stability property of the tessellations by writing and solving certain balance equations.…”

Section: Fig 1 a Realization Of A Homogeneous And Isotropic Stit Tementioning

confidence: 99%

“…4 shows a realization of a homogeneous but anisotropic STIT tessellation in the 3-dimensional space. In Nagel and Weiss (2005) an explicit construction was presented for such tessellations in a bounded convex window in Euclidean spaces of arbitrary dimension d ≥ 2.…”

Section: Stit Tessellationsmentioning

confidence: 99%

“…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%

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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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Section: Fig 1 a Realization Of A Homogeneous And Isotropic Stit Tementioning

confidence: 99%

Section: Stit Tessellationsmentioning

confidence: 99%

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

Thäle

Weiß

2010

Image Anal Stereol

Self Cite

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“…[17,23]). Stationary random tessellations that are stable under the operation of iteration -so-called STIT tessellations -were introduced recently by Mecke, Nagel and Weiß [11,12,14] and they may serve as a new mathematical reference model beside the classical Poisson hyperplane or PoissonVoronoi tessellations. On the other hand, the first-order geometry of these tessellations is already determined by two parameters, the surface density and the so-called directional distribution.…”

Section: Introductionmentioning

confidence: 99%

Shape-driven nested Markov tessellations

Schreiber

Thäle

2012

Stochastics

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A new and rather broad class of stationary (i.e. stochastically translation invariant) random tessellations of the d-dimensional Euclidean space is introduced, which are called shape-driven nested Markov tessellations. Locally, these tessellations are constructed by means of a spatio-temporal random recursive split dynamics governed by a family of Markovian split kernel, generalizing thereby the -by now classical -construction of iteration stable random tessellations. By providing an explicit global construction of the tessellations, it is shown that under suitable assumptions on the split kernels (shape-driven), there exists a unique time-consistent whole-space tessellation-valued Markov process of stationary random tessellations compatible with the given split kernels. Beside the existence and uniqueness result, the typical cell and some aspects of the first-order geometry of these tessellations are in the focus of our discussion.

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“…In a recent work, [18] proposed to use the Delaunay triangles (rather than the Voronoi polygons) and to aggregate them according to linear networks. Furthermore, line-based tessellations, such as proposed by [37], can also be explored to simulate the geometry of agricultural landscapes.…”

Section: Comparing Tessellation Methods Wrt Original Landscapesmentioning

confidence: 99%

Neutral modelling of agricultural landscapes by tessellation methods—Application for gene flow simulation

Ber

Lavigne

Adamczyk³

et al. 2009

Ecological Modelling

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Abstract. Neutral landscape models are not frequently used in the agronomical domain, whereas they would be very useful for studying given agro-ecological or physical processes. Contrary to ecological neutral landscape models, agricultural models have to represent and manage geometrical patches and thus should rely on tessellation methods. We present a three steps approach that aimed at simulating such landscapes. Firstly, we characterized the geometry of three real field patterns; secondly, we generated simulated field patterns with two tessellation methods attempting to control the value of some of the observed characteristics and, thirdly, we evaluated the simulated field patterns. For this evaluation, we considered that good simulated field patterns should capture characteristics of real landscapes that are important for the targeted agro-ecological process. Real landscapes and landscapes simulated using either a Voronoi or a rectangular tessellation were thus compared when used as input data within a gene flow model. The results showed that neither tessellation method captured field shapes correctly, thus leading to over or (small) Voronoi tessellation, though, performed better than the rectangular tessellation. Possible research directions are proposed to improve the simulated patterns, including the use of post-processing, the control of cell orientation or the implementation of other tessellation techniques.

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Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration

Cited by 70 publications

References 10 publications

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

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

Shape-driven nested Markov tessellations

Neutral modelling of agricultural landscapes by tessellation methods—Application for gene flow simulation

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