scholarly journals Local Measures Distribution for the Estimation of the Elongation Ratio of the Typical Grain in Homogeneous Boolean Models

2021 ◽  
Vol 40 (2) ◽  
pp. 95-103
Author(s):  
Tatyana Eremina ◽  
Johan Debayle ◽  
Frederic Gruy ◽  
Jean-Charles Pinoli
Keyword(s):  
Probability Density ◽  
Optimization Algorithm ◽  
Likelihood Function ◽  
Boolean Model ◽  
Spatial Structures ◽  
Elongation Ratio ◽  
Boolean Models ◽  
Minkowski Functionals

We introduce a particular localization of the Minkowski functionals to characterize and discriminate different random spatial structures. The aim of this paper is to present a method estimating the typical grain elongation ratio in a homogeneous Boolean model. The use of this method is demonstrated on a range of Boolean models of rectangles featuring fixed and random elongation ratio. An optimization algorithm is performed to determine the elongation ratio which maximize the likelihood function of the probability density associated with the local perimeter measure. Therefore, the elongation ratio of the typical grain can be deduced.

scholarly journals Exact sampling from conditional Boolean models with applications to maximum likelihood inference

2001 ◽  
Vol 33 (2) ◽  
pp. 339-353 ◽  
Author(s):  
M. N. M. van Lieshout ◽  
E. W. van Zwet
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Function ◽  
Boolean Model ◽  
Boolean Models ◽  
Stochastic Version ◽  
The Past ◽  
Single Realization ◽  
The Em Algorithm ◽  
Monte Carlo Approximation ◽  
Do So

We are interested in estimating the intensity parameter of a Boolean model of discs (the bombing model) from a single realization. To do so, we derive the conditional distribution of the points (germs) of the underlying Poisson process. We demonstrate how to apply coupling from the past to generate samples from this distribution, and use the samples thus obtained to approximate the maximum likelihood estimator of the intensity. We discuss and compare two methods: one based on a Monte Carlo approximation of the likelihood function, the other a stochastic version of the EM algorithm.

Densities of mixed volumes for Boolean models

2001 ◽  
Vol 33 (1) ◽  
pp. 39-60 ◽  
Author(s):  
Wolfgang Weil
Keyword(s):  
Dimensional Space ◽  
Boolean Model ◽  
Boolean Models ◽  
Mixed Volumes ◽  
Explicit Formulae

In generalization of the well-known formulae for quermass densities of stationary and isotropic Boolean models, we prove corresponding results for densities of mixed volumes in the stationary situation and show how they can be used to determine the intensity of non-isotropic Boolean models Z in d-dimensional space for d = 2, 3, 4. We then consider non-stationary Boolean models and extend results of Fallert on quermass densities to densities of mixed volumes. In particular, we present explicit formulae for a planar inhomogeneous Boolean model with circular grains.

scholarly journals A fast FFT method for 3D pore-scale rock-typing of heterogeneous rock samples via Minkowski functionals and hydraulic attributes

2020 ◽  
Vol 146 ◽  
pp. 04002
Author(s):  
Han Jiang ◽  
Christoph H. Arns
Keyword(s):  
Model Systems ◽  
Boolean Models ◽  
Minkowski Functionals ◽  
Hydraulic Unit ◽  
Core Samples ◽  
Physical Measurements ◽  
Classification Framework ◽  
Heterogeneous Rock ◽  
Local Saturation ◽  
Homogeneous Regions

The integration of numerical simulation and physical measurements, e.g. digital and conventional core analysis, requires the consideration of significant sample sizes when heterogeneous core samples are considered. In such case a hierarchical upscaling of properties may be achieved through a workflow of partitioning the sample into homogeneous regions followed by characterization of these homogeneous regions and upscaling of properties. Examples of such heterogeneities are e.g. fine laminations in core samples or different micro-porosity types as consequence of source rock components and diagenesis. In this work we utilize regional measures based on the Minkowski functionals as well as local saturation information derived through a morphological capillary drainage transform as a basis for such a classification/partitioning. An important consideration is the size of the measurement elements utilized, which could be considerable in the case of larger heterogeneities; in such case the calculation of the regional measures can be computationally very expensive. Here we introduce an FFT approach to calculate these measures locally, utilizing their additivity. The algorithms are compared against direct summation techniques and shift-overlap approaches for a selection of different averaging supports to illustrate their speed and practical applicability. We consider a range of artificial Boolean models to illustrate the effect of including hydraulic information on the resulting classifications scheme. This allows the determination of bias, since for these model systems local classes are known ab-initio. The classification framework is tested by comparing to the known initial micro-structure distribution and relative bias quantified in terms of choice of averaging elements (size and shape). Importantly, depending on the actual morphological transition between micro-type partitions, partitions including hydraulic attributes differ from pure morphological partitions with applications to electrofacies and hydraulic unit definitions.

scholarly journals CONTROLLING THE CELL CYCLE RESTRICTION SWITCH ACROSS THE INFORMATION GRADIENT

2019 ◽  
Vol 22 (07n08) ◽  
pp. 1950020 ◽  
Author(s):  
JORDAN C. ROZUM ◽  
RÉKA ALBERT
Keyword(s):  
Cell Cycle ◽  
Control Strategies ◽  
Boolean Model ◽  
Effective Control ◽  
Boolean Models ◽  
Biomolecular Systems ◽  
Regulatory Circuit ◽  
Regulatory Circuits ◽  
System Properties ◽  
Hill Kinetics

Boolean models represent a drastic simplification of complex biomolecular systems, and yet accurately predict system properties, e.g., effective control strategies. Why is this? Parameter robustness has been highlighted as a general feature of biomolecular systems and may play an important role in the accuracy of Boolean models. We argue here that a useful way to view a system’s controllability properties is through its repertoire of self-sustaining positive circuits (stable motifs). We examine attractor control and self-sustaining circuits within the cell cycle restriction switch, a bistable regulatory circuit that allows or prevents entry into the cell cycle. We explore this system using three models: a previously published Boolean model, a Hill kinetics model that we construct from the Boolean model using the HillCube methodology, and a reaction-based model we construct from the literature. We highlight the robustness of stable motifs across these three levels of modeling detail. We also show how consideration of control-robust regulatory circuits can aid in parameter specification.

Co-Existence of the occupied and vacant phase in boolean models in three or more dimensions

1997 ◽  
Vol 29 (4) ◽  
pp. 878-889 ◽  
Author(s):  
Anish Sarkar
Keyword(s):  
Poisson Process ◽  
Boolean Model ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Boolean Models ◽  
Whole Space ◽  
Critical Intensity ◽  
The Way

Consider a continuum percolation model in which, at each point of ad-dimensional Poisson process of rate λ, a ball of radius 1 is centred. We show that, for anyd≧ 3, there exists a phase where both the regions, occupied and vacant, contain unbounded components. The proof uses the concept of enhancement for the Boolean model, and along the way we prove that the critical intensity of a Boolean model defined on a slab is strictly larger than the critical intensity of a Boolean model defined on the whole space.

scholarly journals PARAMETER ESTIMATION IN NON-HOMOGENEOUS BOOLEAN MODELS: AN APPLICATION TO PLANT DEFENSE RESPONSE

2014 ◽  
Vol 33 (2) ◽  
pp. 27
Author(s):  
Maria Angeles Gallego ◽  
Maria Victoria Ibanez ◽  
Amelia Simó
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Plant Immunity ◽  
Simulated Data ◽  
Region Of Interest ◽  
R Package ◽  
Boolean Model ◽  
Boolean Models ◽  
Data Set ◽  
Stationary Point Process

Many medical and biological problems require to extract information from microscopical images. Boolean models have been extensively used to analyze binary images of random clumps in many scientific fields. In this paper, a particular type of Boolean model with an underlying non-stationary point process is considered. The intensity of the underlying point process is formulated as a fixed function of the distance to a region of interest. A method to estimate the parameters of this Boolean model is introduced, and its performance is checked in two different settings. Firstly, a comparative study with other existent methods is done using simulated data. Secondly, the method is applied to analyze the longleaf data set, which is a very popular data set in the context of point processes included in the R package spatstat. Obtained results show that the new method provides as accurate estimates as those obtained with more complex methods developed for the general case. Finally, to illustrate the application of this model and this method, a particular type of phytopathological images are analyzed. These images show callose depositions in leaves of Arabidopsis plants. The analysis of callose depositions, is very popular in the phytopathological literature to quantify activity of plant immunity.

scholarly journals Generalized contact distributions of inhomogeneous Boolean models

2002 ◽  
Vol 34 (01) ◽  
pp. 21-47 ◽  
Author(s):  
Daniel Hug ◽  
Günter Last ◽  
Wolfgang Weil
Keyword(s):  
Convex Body ◽  
Large Class ◽  
Random Vector ◽  
Boundary Point ◽  
Conditional Distribution ◽  
Boolean Model ◽  
Spatial Density ◽  
Boolean Models ◽  
Grain Distribution ◽  
Contact Distribution

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝ d and a gauge body B ⊂ ℝ d , such a generalized contact distribution is the conditional distribution of the random vector (d B (L,Z),u B (L,Z),p B (L,Z),l B (L,Z)) given that Z∩L = ∅, where Z is a Boolean model, d B (L,Z) is the distance of L from Z with respect to B, p B (L,Z) is the boundary point in L realizing this distance (if it exists uniquely), u B (L,Z) is the corresponding boundary point of B (if it exists uniquely) and l B (L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

Critical intensities of Boolean models with different underlying convex shapes

2002 ◽  
Vol 34 (01) ◽  
pp. 48-57
Author(s):  
Rahul Roy ◽  
Hideki Tanemura
Keyword(s):  
Unit Area ◽  
Higher Dimensions ◽  
Boolean Model ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Boolean Models ◽  
Regular Polytope ◽  
Minimum Value ◽  
Critical Intensity ◽  
Poisson Boolean Model

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

scholarly journals Central limit theorems for functionals of stationary germ-grain models

2006 ◽  
Vol 38 (1) ◽  
pp. 76-94 ◽  
Author(s):  
Ursa Pantle ◽  
Volker Schmidt ◽  
Evgueni Spodarev
Keyword(s):  
Limit Theorem ◽  
Central Limit ◽  
Random Fields ◽  
Limit Theorems ◽  
Boolean Model ◽  
Dimensional Euclidean Space ◽  
Boolean Models ◽  
Intrinsic Volumes ◽  
Integrability Conditions ◽  
Vector Valued

Conditions are derived for the asymptotic normality of a general class of vector-valued functionals of stationary Boolean models in the d-dimensional Euclidean space, where a Lindeberg-type central limit theorem for m-dependent random fields, m ∈ N, is applied. These functionals can be used to construct joint estimators for the vector of specific intrinsic volumes of the underlying Boolean model. Extensions to functionals of more general germ–grain models satisfying some mixing and integrability conditions are also discussed.

The surface pair correlation function for stationary Boolean models

2007 ◽  
Vol 39 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Felix Ballani
Keyword(s):  
Pair Correlation ◽  
Boolean Model ◽  
Circular Cylinders ◽  
Necessary Condition ◽  
Order Moment ◽  
Sufficient Condition ◽  
Surface Measure ◽  
Boolean Models ◽  
Random Surface ◽  
Moment Measure

The random surface measure of a stationary Boolean model with grains from the convex ring is considered. A sufficient condition and a necessary condition for the existence of the density of the second-order moment measure of are given and a representation of this density is derived. As applications, the surface pair correlation functions of a Boolean model with spheres and a Boolean model with randomly oriented right circular cylinders in ℝ3 are determined.

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