Local Digital Estimators of Intrinsic Volumes for Boolean Models and in the Design-Based Setting

Svane, Anne Marie

doi:10.1017/s0001867800006923

Cited by 6 publications

(16 citation statements)

References 15 publications

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“…The resulting formulae for the mean digital estimators generalize the formulae of Ohser et al (2009) and Svane (2014a) to non-isotropic grain distributions. They have a resemblance to the Miles formulae (Miles, 1976) for specific intrinsic volumes, but contain a bias depending on how the lattice is rotated relatively to Z.…”

Section: Introductionmentioning

confidence: 79%

“…Comparing Theorem with the Miles formulae , we see that an estimator for

{\overset{V}{true}}_{d - 1} (Z)

{\overset{V}{true}}_{d - 2} (Z)

is asymptotically unbiased exactly if the weights satisfy a set of linear equations involving the constants c k ( B l , W l ), k = 1,2,3. In 2D, these equations were determined and the full solution was given in Svane (). In the following sections, we determine the constants and the corresponding equations in 3D.…”

Section: Optimal Estimators For Isotropic Boolean Modelsmentioning

confidence: 99%

“…Because Z is isotropic, the constants c ( B l , W l ) depend only on the motion equivalence class of W l . Thus, we may as well let the weights be motion independent, that is, for all configurations ( B l , W l ) with W l ∈ η j , we choose the weight

w_{l}^{(q)}

equal to some weight

{\overset{w}{true}}_{j}^{(q)}

depending only on j (see Svane, , for a justification). Thus simplifies to

{\hat{V}}_{q} (Z \cap A) = a^{q - d} \sum_{j = 1}^{22} {\tilde{w}}_{j}^{(q)} \sum_{l : W_{l} \in η_{j}} \frac{N_{l} (Z \cap A \cap a L)}{N (A)} .

By and Propositions and , we must set

{\overset{w}{true}}_{1}^{(q)} = {\overset{w}{true}}_{22}^{(q)} = 0

for all q < d in order for the asymptotic mean to be finite.…”

Section: Optimal Estimators For Isotropic Boolean Modelsmentioning

confidence: 99%

“…Because Z is isotropic, the constants c.B l ; W l / depend only on the motion equivalence class of W l . Thus, we may as well let the weights be motion independent, that is, for all configurations .B l ; W l / with W l 2 Á j , we choose the weight w .q/ l equal to some weight Q w .q/ j depending only on j (see Svane, 2014a, for a justification). Thus (16) simplifies to…”

Section: The 3d Situationmentioning

confidence: 99%

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Local Digital Algorithms Applied to Boolean Models

Hörrmann

Svane

2016

Scandinavian J Statistics

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We investigate the estimation of specific intrinsic volumes of stationary Boolean models by local digital algorithms; that is, by weighted sums of n × . . . × n configuration counts. We show that asymptotically unbiased estimators for the specific surface area or integrated mean curvature do not exist if the dimension is at least two or three, respectively. For 3-dimensional stationary, isotropic Boolean models, we derive asymptotically unbiased estimators for the specific surface area and integrated mean curvature. For a Boolean model with balls as grains we even obtain an asymptotically unbiased estimator for the specific Euler characteristic. This solves an open problem from [18].

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Section: Introductionmentioning

confidence: 79%

“…Comparing Theorem with the Miles formulae , we see that an estimator for

{\overset{V}{true}}_{d - 1} (Z)

{\overset{V}{true}}_{d - 2} (Z)

Section: Optimal Estimators For Isotropic Boolean Modelsmentioning

confidence: 99%

w_{l}^{(q)}

equal to some weight

{\overset{w}{true}}_{j}^{(q)}

depending only on j (see Svane, , for a justification). Thus simplifies to

{\hat{V}}_{q} (Z \cap A) = a^{q - d} \sum_{j = 1}^{22} {\tilde{w}}_{j}^{(q)} \sum_{l : W_{l} \in η_{j}} \frac{N_{l} (Z \cap A \cap a L)}{N (A)} .

By and Propositions and , we must set

{\overset{w}{true}}_{1}^{(q)} = {\overset{w}{true}}_{22}^{(q)} = 0

for all q < d in order for the asymptotic mean to be finite.…”

Section: Optimal Estimators For Isotropic Boolean Modelsmentioning

confidence: 99%

Section: The 3d Situationmentioning

confidence: 99%

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Local Digital Algorithms Applied to Boolean Models

Hörrmann

Svane

2016

Scandinavian J Statistics

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“…If the algorithms are applied to a stationary isotropic Boolean model with a fixed lattice, a similar result seems to hold: There exists asymptotically unbiased estimators for V n , V n−1 , and V n−2 . At least, this has been shown in both 2D [21] and 3D [5]. Again, isotropy is essential.…”

Section: Isotropic Latticesmentioning

confidence: 91%

Valuations in Image Analysis

Svane

2017

Lecture Notes in Mathematics

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When intrinsic volumes and Minkowski tensors of a real world structure are computed, this is often based on a digital image. The digitization causes some estimation problems due to the anisotropic nature of the digital grid. Even the most natural and frequently used algorithms based on counting the local pixel/voxel configurations are often biased. In this chapter, we survey the known results on convergence of these local algorithms with a focus on estimation of intrinsic volumes. Moreover, we present some of the latest attempts to define convergent algorithms.

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Bicovariograms and Euler characteristic of regular sets

Lachièze‐Rey

2017

Mathematische Nachrichten

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We establish an expression of the Euler characteristic of a -regular planar set in function of some variographic quantities. The usual  2 framework is relaxed to a  1,1 regularity assumption, generalising existing local formulas for the Euler characteristic. We give also general bounds on the number of connected components of a measurable set of ℝ 2 in terms of local quantities. These results are then combined to yield a new expression of the mean Euler characteristic of a random regular set, depending solely on the third order marginals for arbitrarily close arguments. We derive results for level sets of some moving average processes and for the boolean model with non-connected polyrectangular grains in ℝ 2 . Applications to excursions of smooth bivariate random fields are derived in the companion paper [25], and applied for instance to  1,1 Gaussian fields, generalising standard results. K E Y W O R D S Euler characteristic, intrinsic volumes, shot noise processes, boolean model M S C ( 2 0 1 0 ) 28A75, 52A22, 60D05, 60G10It is more generally an indicator of the regularity of the set, as an irregular structure is more likely to be shredded in many small pieces, or pierced by many holes, which results in a large value for | ( )|.As an integer-valued quantity, the Euler characteristic can be easily measured and used in estimation and modelisation procedures. It is an important indicator of the porosity of a random media [7,19,33], it is used in brain imagery [23,37], astronomy, [15,27,31], and many other disciplines. See also [1] for a general review of applied algebraic topology. In the study of parametric random media or graphs, a small value of | ( )| indicates the proximity of the percolation threshold, when that makes sense. See [30], or [13] in the discrete setting.The mathematical additivity property is expressed, for suitable sets and , by the formula ( ∪ ) = ( ) + ( ) − ( ∩ ), which applies recursively to finite such unions. In the validity domain of this formula, the Euler characteristic of a set can therefore be computed by summing local contributions. The Gauss-Bonnet theorem formalises this notion for  2 manifolds, stating that the Euler characteristic of a smooth set is the integral along the boundary of its Gaussian curvature. Exploiting the local nature of the curvature in applications seems to be a geometric challenge, in the sense that it is not always 398

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Local Digital Estimators of Intrinsic Volumes for Boolean Models and in the Design-Based Setting

Cited by 6 publications

References 15 publications

Local Digital Algorithms Applied to Boolean Models

Local Digital Algorithms Applied to Boolean Models

Valuations in Image Analysis

Bicovariograms and Euler characteristic of regular sets

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