This paper presents a model of random tessellations that reflect several features of crack patter... more This paper presents a model of random tessellations that reflect several features of crack pattern. There are already several theoretical results derivedwhich indicate that thismodel can be an appropriate referencemodel. Some potential applications are presented in a tentative statistical study.
In order to increase the variety of feasible models for random stationary tessellations (mosaics)... more In order to increase the variety of feasible models for random stationary tessellations (mosaics), two operations acting on tessellations are studied: superposition and iteration (the latter is also referred to as nesting). The superposition of two planar tessellations is the superposition of the edges of the cells of both tessellations. The iteration of tessellations means that one tessellation is chosen as a ‘frame’ tessellation. The single cells of this ‘frame’ are simultaneously and independently subdivided by cut-outs of tessellations of an independent and identically distributed sequence of tessellations. In the present paper, we investigate the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration respectively. Sequences of (renormalised) superpositions of stationary planar tessellations converge weakly to Poisson line tessellations. For consecutive iteration the notion of stability of distributions is adapted and ne...
For random planar stationary tessellations a parameter system of three mean values and two direct... more For random planar stationary tessellations a parameter system of three mean values and two directional distributions of the edges is considered. To investigate the interdependences between these parameters their joint range is described, i.e. the set of values which can be realized by tessellations. The constructive proof is based on mixtures of stationary regular tessellations. To explore the range for the class of stationary ergodic tessellations a procedure is used which we refer to as iteration.
For a class of cell division processes in the Euclidean space ℝd, spatial consistency is investig... more For a class of cell division processes in the Euclidean space ℝd, spatial consistency is investigated. This addresses the problem whether the distribution of the generated structures, restricted to a bounded set V, depends on the choice of a larger region W ⊃ V where the construction of the cell division process is performed. This can also be understood as the problem of boundary effects in the cell division procedure. It is known that the STIT tessellations are spatially consistent. In the present paper it is shown that, within a reasonable wide class of cell division processes, the STIT tessellations are the only ones that are consistent.
Corrigendum of MEAN VALUES FOR HOMOGENEOUS STIT TESSELLATIONS IN 3D (2008 IMAGE ANAL STEREOL 27:2... more Corrigendum of MEAN VALUES FOR HOMOGENEOUS STIT TESSELLATIONS IN 3D (2008 IMAGE ANAL STEREOL 27:29–37)
The intrinsic volumes-in 3d up to constants volume, surface area, integral of mean curvature, and... more The intrinsic volumes-in 3d up to constants volume, surface area, integral of mean curvature, and Euler number-are a very useful set of geometric characteristics. Combining integral and digital geometry we develop a method for efficient simultanous calculation of the intrinsic volumes of sets observed in binary images. In order to achieve consistency in the derived intrinsic volumes for both foreground and background, suitable pairs of discrete connectivities have to be used. To make this rigorous, the concepts discretization w.r.t. an adjacency system and complementarity of adjacency systems are introduced.
The problem of estimating the Euler-Poincare' characteristic (Euler number for short) of a se... more The problem of estimating the Euler-Poincare' characteristic (Euler number for short) of a set in the 3d Euclidean space is considered, given that this set is observed in the points of a lattice. In this situation, which is typical in image analysis, the hoice of an appropriate data-based discretisation of the set is crucial. Four versions of a discretisation method which is based on the notion of adjacency systems are presented; these versions are referred to as (14.1 14.1), (14.2 14.2), (6.26), and (26.6). A comparative assessment of the four approaches is performed with respect to the systematic error occuring in application to Boolean models. It is a surprising result that, except for 26 6 , the estimators yield infinitely large systematic errors when the lattice spacing goes to zero. Furthermore, the measurements of the Euler number from 3d data of autoclaved aerated concrete illustrate the influence of the choice of adjacency and the behaviour of the estimators.
The densities of the intrinsic volumes – in 3D the volume density, surface density, the density o... more 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 thusmultigrid 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 te...
Recently (Nagel and Weiss, 2005), the class of homogeneous random tessellations that are stable u... more Recently (Nagel and Weiss, 2005), the class of homogeneous random tessellations that are stable under the operation of iteration (STIT) was introduced. In the present paper this model is reviewed and new results for the mean values of essential geometric features of STIT tessellations in two and three dimensions are provided and proved. For the isotropic model, these mean values are compared with those ones of the Poisson-Voronoi and of the Poisson plane tessellations, respectively.
We consider homogeneous STIT tessellations Y in the ℓ-dimensional Euclidean space ℝℓ and show the... more We consider homogeneous STIT tessellations Y in the ℓ-dimensional Euclidean space ℝℓ and show the triviality of the tail σ-algebra. This is a sharpening of the mixing result by Lachièze-Rey (2001).
This paper presents a model of random tessellations that reflect several features of crack patter... more This paper presents a model of random tessellations that reflect several features of crack pattern. There are already several theoretical results derivedwhich indicate that thismodel can be an appropriate referencemodel. Some potential applications are presented in a tentative statistical study.
Stochastics An International Journal of Probability and Stochastic Processes, 2008
Homogeneous (i.e. spatially stationary) random tessellations of the Euclidean plane are construct... more Homogeneous (i.e. spatially stationary) random tessellations of the Euclidean plane are constructed which have the characteristic property to be stable under the operation of iteration (or nesting), STIT for short. It is based on a Poisson point process on the space of lines that are endowed with a time of birth. A new approach is presented that describes the tessellation
This paper presents a model of random tessellations that reflect several features of crack patter... more This paper presents a model of random tessellations that reflect several features of crack pattern. There are already several theoretical results derivedwhich indicate that thismodel can be an appropriate referencemodel. Some potential applications are presented in a tentative statistical study.
In order to increase the variety of feasible models for random stationary tessellations (mosaics)... more In order to increase the variety of feasible models for random stationary tessellations (mosaics), two operations acting on tessellations are studied: superposition and iteration (the latter is also referred to as nesting). The superposition of two planar tessellations is the superposition of the edges of the cells of both tessellations. The iteration of tessellations means that one tessellation is chosen as a ‘frame’ tessellation. The single cells of this ‘frame’ are simultaneously and independently subdivided by cut-outs of tessellations of an independent and identically distributed sequence of tessellations. In the present paper, we investigate the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration respectively. Sequences of (renormalised) superpositions of stationary planar tessellations converge weakly to Poisson line tessellations. For consecutive iteration the notion of stability of distributions is adapted and ne...
For random planar stationary tessellations a parameter system of three mean values and two direct... more For random planar stationary tessellations a parameter system of three mean values and two directional distributions of the edges is considered. To investigate the interdependences between these parameters their joint range is described, i.e. the set of values which can be realized by tessellations. The constructive proof is based on mixtures of stationary regular tessellations. To explore the range for the class of stationary ergodic tessellations a procedure is used which we refer to as iteration.
For a class of cell division processes in the Euclidean space ℝd, spatial consistency is investig... more For a class of cell division processes in the Euclidean space ℝd, spatial consistency is investigated. This addresses the problem whether the distribution of the generated structures, restricted to a bounded set V, depends on the choice of a larger region W ⊃ V where the construction of the cell division process is performed. This can also be understood as the problem of boundary effects in the cell division procedure. It is known that the STIT tessellations are spatially consistent. In the present paper it is shown that, within a reasonable wide class of cell division processes, the STIT tessellations are the only ones that are consistent.
Corrigendum of MEAN VALUES FOR HOMOGENEOUS STIT TESSELLATIONS IN 3D (2008 IMAGE ANAL STEREOL 27:2... more Corrigendum of MEAN VALUES FOR HOMOGENEOUS STIT TESSELLATIONS IN 3D (2008 IMAGE ANAL STEREOL 27:29–37)
The intrinsic volumes-in 3d up to constants volume, surface area, integral of mean curvature, and... more The intrinsic volumes-in 3d up to constants volume, surface area, integral of mean curvature, and Euler number-are a very useful set of geometric characteristics. Combining integral and digital geometry we develop a method for efficient simultanous calculation of the intrinsic volumes of sets observed in binary images. In order to achieve consistency in the derived intrinsic volumes for both foreground and background, suitable pairs of discrete connectivities have to be used. To make this rigorous, the concepts discretization w.r.t. an adjacency system and complementarity of adjacency systems are introduced.
The problem of estimating the Euler-Poincare' characteristic (Euler number for short) of a se... more The problem of estimating the Euler-Poincare' characteristic (Euler number for short) of a set in the 3d Euclidean space is considered, given that this set is observed in the points of a lattice. In this situation, which is typical in image analysis, the hoice of an appropriate data-based discretisation of the set is crucial. Four versions of a discretisation method which is based on the notion of adjacency systems are presented; these versions are referred to as (14.1 14.1), (14.2 14.2), (6.26), and (26.6). A comparative assessment of the four approaches is performed with respect to the systematic error occuring in application to Boolean models. It is a surprising result that, except for 26 6 , the estimators yield infinitely large systematic errors when the lattice spacing goes to zero. Furthermore, the measurements of the Euler number from 3d data of autoclaved aerated concrete illustrate the influence of the choice of adjacency and the behaviour of the estimators.
The densities of the intrinsic volumes – in 3D the volume density, surface density, the density o... more 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 thusmultigrid 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 te...
Recently (Nagel and Weiss, 2005), the class of homogeneous random tessellations that are stable u... more Recently (Nagel and Weiss, 2005), the class of homogeneous random tessellations that are stable under the operation of iteration (STIT) was introduced. In the present paper this model is reviewed and new results for the mean values of essential geometric features of STIT tessellations in two and three dimensions are provided and proved. For the isotropic model, these mean values are compared with those ones of the Poisson-Voronoi and of the Poisson plane tessellations, respectively.
We consider homogeneous STIT tessellations Y in the ℓ-dimensional Euclidean space ℝℓ and show the... more We consider homogeneous STIT tessellations Y in the ℓ-dimensional Euclidean space ℝℓ and show the triviality of the tail σ-algebra. This is a sharpening of the mixing result by Lachièze-Rey (2001).
This paper presents a model of random tessellations that reflect several features of crack patter... more This paper presents a model of random tessellations that reflect several features of crack pattern. There are already several theoretical results derivedwhich indicate that thismodel can be an appropriate referencemodel. Some potential applications are presented in a tentative statistical study.
Stochastics An International Journal of Probability and Stochastic Processes, 2008
Homogeneous (i.e. spatially stationary) random tessellations of the Euclidean plane are construct... more Homogeneous (i.e. spatially stationary) random tessellations of the Euclidean plane are constructed which have the characteristic property to be stable under the operation of iteration (or nesting), STIT for short. It is based on a Poisson point process on the space of lines that are endowed with a time of birth. A new approach is presented that describes the tessellation
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Papers by Werner Nagel