Random mosaics with cells of general topology

1996 ◽  
Vol 28 (02) ◽  
pp. 332
Author(s):  
Richard Cowan ◽  
Albert K. L. Tsang
Keyword(s):  
Euler Characteristic ◽  
Ergodic Theory ◽  
General Topology ◽  
Traditional Theory ◽  
Random Partition ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Partition Process

This paper considers a structure, named a ‘random partition process’, which is a generalisation of a random tessellation. The cells, possibly multi-part and with holes, have a general topology summarised by the Euler characteristic. Vertices of all orders are allowed. Using the tools of ergodic theory, all of the formulae, from the traditional theory of random tessellations with convex cells, are generalised. Some motivating examples are given.

Random mosaics with cells of general topology

1996 ◽  
Vol 28 (2) ◽  
pp. 332-332
Author(s):  
Richard Cowan ◽  
Albert K. L. Tsang
Keyword(s):  
Euler Characteristic ◽  
Ergodic Theory ◽  
General Topology ◽  
Traditional Theory ◽  
Random Partition ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Partition Process

This paper considers a structure, named a ‘random partition process’, which is a generalisation of a random tessellation. The cells, possibly multi-part and with holes, have a general topology summarised by the Euler characteristic. Vertices of all orders are allowed. Using the tools of ergodic theory, all of the formulae, from the traditional theory of random tessellations with convex cells, are generalised. Some motivating examples are given.

Further random tessellations with the classic Poisson polygon distributions

1996 ◽  
Vol 28 (02) ◽  
pp. 338-339 ◽  
Author(s):  
Roger E. Miles ◽  
Margaret S. Mackisack
Keyword(s):  
Orientation Distribution ◽  
Convex Polygons ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Random Convex

It is well-known that Poisson lines in the plane, with orientation distribution Θ on [0, π), generate a (random) tessellation M 0 of (random) convex polygons whose characteristics (area, perimeter, etc.) conform, in an ergodic sense, to a certain class {D Θ} of distributions.

New Classes of Random Tessellations Arising from Iterative Division of Cells

2010 ◽  
Vol 42 (1) ◽  
pp. 26-47 ◽  
Author(s):  
Richard Cowan
Keyword(s):  
Probability Distribution ◽  
Mathematical Formalism ◽  
Simulation Methods ◽  
Vast Number ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Typical Cell ◽  
New Ideas ◽  
Successive Division

We present new ideas about the type of random tessellation which evolves through successive division of its cells. These ideas are developed in an intuitive way, with many pictures and only a modicum of mathematical formalism–so that the wide application of the ideas is clearly apparent to all readers. A vast number of new tessellation models, with known probability distribution for the volume of the typical cell, follow from the concepts in this paper. There are other interesting models for which results are not presented (or presented only through simulation methods), but these models have illustrative value. A large agenda of further research is opened up by the ideas in this paper.

scholarly journals EIKONAL-BASED MODELS OF RANDOM TESSELLATIONS

2019 ◽  
Vol 38 (1) ◽  
pp. 15 ◽  
Author(s):  
Bruno Figliuzzi
Keyword(s):  
Efficient Method ◽  
Eikonal Equation ◽  
Velocity Model ◽  
Local Velocity ◽  
Implicit Functions ◽  
Multi Scale ◽  
Random Tessellation ◽  
Straight Line ◽  
Random Tessellations ◽  
Roughness Amplitude

In this article, we propose a novel, efficient method for computing a random tessellation from its vectorial representation at each voxel of a discretized domain. This method is based upon the resolution of the Eikonal equation and has a complexity in O(N log N), N being the number of voxels used to discretize the domain. By contrast, evaluating the implicit functions of the vectorial representation at each voxel location has a complexity of O(N²) in the general case. The method also enables us to consider the generation of tessellations with rough interfaces between cells by simulating the growth of the germs on a domain where the velocity varies locally. This aspect constitutes the main contribution of the article. A final contribution is the development of an algorithm for estimating the multi-scale tortuosity of the boundaries of the tessellation cells. The algorithm computes the tortuosity of the boundary at several scales by iteratively deforming the boundary until it becomes a straight line. Using this algorithm, we demonstrate that depending on the local velocity model, it is possible to control the roughness amplitude of the cells boundaries.

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

2008 ◽  
Vol 40 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Joseph Mecke ◽  
Werner Nagel ◽  
Viola Weiss
Keyword(s):  
Homogeneous Process ◽  
Cell Divisions ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Random States

A random tessellation of ℝd is said to be homogeneous if its distribution is invariant under all shifts of ℝd. 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.

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

2008 ◽  
Vol 40 (01) ◽  
pp. 49-59 ◽  
Author(s):  
Joseph Mecke ◽  
Werner Nagel ◽  
Viola Weiss
Keyword(s):  
Homogeneous Process ◽  
Cell Divisions ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Random States

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.

scholarly journals New Classes of Random Tessellations Arising from Iterative Division of Cells

2010 ◽  
Vol 42 (01) ◽  
pp. 26-47 ◽  
Author(s):  
Richard Cowan
Keyword(s):  
Probability Distribution ◽  
Mathematical Formalism ◽  
Simulation Methods ◽  
Vast Number ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Typical Cell ◽  
New Ideas ◽  
Successive Division

We present new ideas about the type of random tessellation which evolves through successive division of its cells. These ideas are developed in an intuitive way, with many pictures and only a modicum of mathematical formalism–so that the wide application of the ideas is clearly apparent to all readers. A vast number of new tessellation models, with known probability distribution for the volume of the typical cell, follow from the concepts in this paper. There are other interesting models for which results are not presented (or presented only through simulation methods), but these models have illustrative value. A large agenda of further research is opened up by the ideas in this paper.

Further random tessellations with the classic Poisson polygon distributions

1996 ◽  
Vol 28 (2) ◽  
pp. 338-339
Author(s):  
Roger E. Miles ◽  
Margaret S. Mackisack
Keyword(s):  
Orientation Distribution ◽  
Convex Polygons ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Random Convex

It is well-known that Poisson lines in the plane, with orientation distribution Θ on [0, π), generate a (random) tessellation M0 of (random) convex polygons whose characteristics (area, perimeter, etc.) conform, in an ergodic sense, to a certain class {DΘ} of distributions.

Ergodic Theory

1983 ◽  
Author(s):  
Karl E. Petersen
Keyword(s):  
Ergodic Theory
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