On the approximation of mean densities of random closed sets

Ambrosio, Luigi; Capasso, Vincenzo; Villa, Elena

doi:10.3150/09-bej186

Cited by 26 publications

(34 citation statements)

References 19 publications

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“…By suitable laws of large numbers, whenever N is sufficiently large, Q N may admit a density given by (A4) [28,29]. Consequently, δ(x − X(t)) in (A10) approaches its mean value [46] λ(t, x) = t 0p (s, x) ds,…”

Section: Discussionmentioning

confidence: 99%

Hybrid modeling of tumor-induced angiogenesis

et al. 2014

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When modeling of tumor-driven angiogenesis, a major source of analytical and computational complexity is the strong coupling between the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network, and the family of interacting underlying fields. To reduce this complexity, we take advantage of the system intrinsic multiscale structure: we describe the stochastic dynamics of the cells at the vessel tip at their natural mesoscale, whereas we describe the deterministic dynamics of the underlying fields at a larger macroscale. Here, we set up a conceptual stochastic model including branching, elongation, and anastomosis of vessels and derive a mean field approximation for their densities. This leads to a deterministic integro-partial differential system that describes the formation of the stochastic vessel network. We discuss the proper capillary injecting boundary conditions and include the results of relevant numerical simulations.

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Section: Discussionmentioning

confidence: 99%

Hybrid modeling of tumor-induced angiogenesis

et al. 2014

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“…point of ∂ A). Nevertheless, we are able to identify two conditions, both stable under finite unions: the first one, see (1), is a kind of quantitative non-degeneracy condition which prevents ∂ A from being too sparse; simple examples (see Example 3) show that SM(A) can be infinite, and H d−1 (∂ A) arbitrarily small, when this condition fails. The second condition, in analogy with the above mentioned Theorem 5, is the existence of the outer Minkowski content and its coincidence with H d−1 (∂ * A).…”

Section: Introductionmentioning

confidence: 97%

Outer Minkowski content for some classes of closed sets

2008

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We find conditions ensuring the existence of the outer Minkowski content for d-dimensional closed sets in R d , in connection with regularity properties of their boundaries. Moreover, we provide a class of sets (including all sufficiently regular sets) stable under finite unions for which the outer Minkowski content exists. It follows, in particular, that finite unions of sets with Lipschitz boundary and a type of sets with positive reach belong to this class.

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“…Within the mathematical framework provided in Ambrosio et al (2009) and in Villa (2014, Theorem 7), based on a stochastic version of the Minkowski content notion, it is proved that if Θ n satisfies (A1), (A2) and (A3), given in the Appendix, then…”

Section: Kernel Estimatormentioning

confidence: 99%

“…The required background regarding the global and local approximation of mean densities of random closed sets has been presented in a series of papers by Capasso and Villa (Capasso and Villa, 2006; 2008; Ambrosio et al, 2009;Villa, 2014); we will report here the basic definitions, while for a detailed mathematical analysis of the proposed estimators we refer to the paper by Camerlenghi et al (2014).…”

Section: Introductionmentioning

confidence: 99%

Numerical Experiments for the Estimation of Mean Densities of Random Sets

Camerlenghi¹,

Capasso²,

Villa³

2014

Image Anal Stereol

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Many real phenomena may be modelled as random closed sets in R d , of different Hausdorff dimensions. The problem of the estimation of pointwise mean densities of absolutely continuous, and spatially inhomogeneous, random sets with Hausdorff dimension n < d, has been the subject of extended mathematical analysis by the authors. In particular, two different kinds of estimators have been recently proposed, the first one is based on the notion of Minkowski content, the second one is a kernel-type estimator generalizing the well-known kernel density estimator for random variables. The specific aim of the present paper is to validate the theoretical results on statistical properties of those estimators by numerical experiments. We provide a set of simulations which illustrates their valuable properties via typical examples of lower dimensional random sets.

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On the approximation of mean densities of random closed sets

Cited by 26 publications

References 19 publications

Hybrid modeling of tumor-induced angiogenesis

Hybrid modeling of tumor-induced angiogenesis

Outer Minkowski content for some classes of closed sets

Numerical Experiments for the Estimation of Mean Densities of Random Sets

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