A generalization of the inhomogeneity measure for point distributions to the case of finite size objects

Piasecki, R.

doi:10.1016/j.physa.2008.05.034

Cited by 5 publications

(21 citation statements)

References 25 publications

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“…For the sake of clarity a window of width k we identify with the one-dimensional cell of length k while for patterns the cell is of size k × k. In the following we shall simply use the notion "cell" that size be clear from the system's dimension d = 1, 2. To overcome the limitation of standard pattern partitions (into non-overlapping cells) to the scales, for which k is an integer divisor of L here we use a sliding cell-sampling (SCS) approach [7]. To control cell statistics a simple condition for this method is also further given in section 2.4.…”

Section: Actual and Reference Macrostatesmentioning

confidence: 99%

“…1(b) in Ref. [7,13]. Instead of preferred here Greek letter κ, the notation χ a has been used in Ref.…”

Section: Actual and Reference Macrostatesmentioning

confidence: 99%

“…A toy model discussed in sections 2.2 and 2.3 exemplifies those macrostates in context of possible forms of the entropic measure and its maximization. The case (a) we shall neglect here as equivalent one to binary patterns considered previously [6,7]. For the cases (b) and (c), from Eqs.…”

Section: Actual and Reference Macrostatesmentioning

confidence: 99%

“…(4) and (5) On the other hand, to obtain the necessary for our measure the highest possible value of the entropy, Sm ax (k) = ln Ωm ax , we need a reference macrostate RM(k). It is defined by the For case (a) the similar form of the condition but with cell occupation numbers n i has been already applied to binary patterns [6,7]. In the present case (c) the desired formula for the number Ωm ax (c) of microstates for RM(c) can be written as…”

Section: Actual and Reference Macrostatesmentioning

confidence: 99%

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Statistical mechanics characterization of spatio-compositional inhomogeneity

Piasecki

2009

Physica A: Statistical Mechanics and its Applications

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On the basis of a model system of pillars built of unit cubes, a two-component entropic measure for the multiscale analysis of spatio-compositional inhomogeneity is proposed. It quantifies the statistical dissimilarity per cell of the actual configurational macrostate and the theoretical reference one that maximizes entropy. Two kinds of disorder compete: i) the spatial one connected with possible positions of pillars inside a cell (the first component of the measure), ii) the compositional one linked to compositions of each local sum of their integer heights into a number of pillars occupying the cell (the second component). As both the number of pillars and sum of their heights are conserved, the upper limit for a pillar height h max occurs. If due to a further constraint there is the more demanding limit h ≤ h * < h max , the exact number of restricted compositions can be then obtained only through the generating function. However, at least for systems with exclusively composition degrees of freedom, we show that the neglecting of the h * is not destructive yet for a nice correlation of the h * -constrained entropic measure and its less demanding counterpart, which is much easier to compute. Given examples illustrate a broad applicability of the measure and its ability to quantify some of the subtleties of a fractional Brownian motion, time evolution of a quasipattern [28,29] and reconstruction of a laser-speckle pattern [2], which are hardly to discern or even missed.

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Section: Actual and Reference Macrostatesmentioning

confidence: 99%

“…1(b) in Ref. [7,13]. Instead of preferred here Greek letter κ, the notation χ a has been used in Ref.…”

Section: Actual and Reference Macrostatesmentioning

confidence: 99%

Section: Actual and Reference Macrostatesmentioning

confidence: 99%

Section: Actual and Reference Macrostatesmentioning

confidence: 99%

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Statistical mechanics characterization of spatio-compositional inhomogeneity

Piasecki

2009

Physica A: Statistical Mechanics and its Applications

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“…An overview of an aspect of work in that area is given in [16]. For example, Zwicky [17] uses an approach similar to (2).…”

Section: Related Workmentioning

confidence: 99%

Measuring Inhomogeneity in Spatial Distributions

Schilcher

Gyarmati

Bettstetter

et al. 2008

VTC Spring 2008 - IEEE Vehicular Technology Conference

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Abstract-The spatial distribution of nodes in wireless networks has important impact on network performance properties, such as capacity and connectivity. Although random sample models based on a uniform distribution are widely used in the research community, they are inappropriate for scenarios with clustered, inhomogeneous node distribution. This paper proposes a well-defined measure for the level of inhomogeneity of a node distribution. It is based on the local deviation of the actual value of the density of nodes from its expected value. Desired properties of the measure are defined and mathematically proven to be fulfilled. The inhomogeneity measure is also compared to human perception of inhomogeneity gained via an online survey. The results reveal that the measure well fits human perception, although there are notable deviations if linear operations are applied.

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A versatile entropic measure of grey level inhomogeneity

Piasecki

2009

Physica A: Statistical Mechanics and its Applications

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The entropic measure for analysis of grey level inhomogeneity (GLI) is proposed as a function of length scale. It allows us to quantify the statistical dissimilarity of the actual macrostate and the maximizing entropy of the reference one. The maximums (minimums) of the measure indicate those scales at which higher (lower) average grey level inhomogeneity appears compared to neighbour scales. Even a deeply hidden statistical grey level periodicity can be detected by the equally distant minimums of the measure. The striking effect of multiple intersecting curves (MIC) of the measure has been revealed for pairs of simulated patterns, which differ in shades of grey or symmetry properties, only. This indicates for a nontrivial dependence of the GLI on length scale. In turn for evolving photosphere granulation patterns, the stability in time of the first peak position has been found. Interestingly, at initial steps of the evolution clearly dominates the third peak. This indicates for a temporary grouping of granules at length scale that may belong to mesogranulation phenomenon. This behaviour has similarities with that reported by Consolini, Berrilli et al. (2003, 2005 for binarized granulation images of a different data set.

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A generalization of the inhomogeneity measure for point distributions to the case of finite size objects

Cited by 5 publications

References 25 publications

Statistical mechanics characterization of spatio-compositional inhomogeneity

Statistical mechanics characterization of spatio-compositional inhomogeneity

Measuring Inhomogeneity in Spatial Distributions

A versatile entropic measure of grey level inhomogeneity

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