scholarly journals EULER INTEGRATION OF GAUSSIAN RANDOM FIELDS AND PERSISTENT HOMOLOGY

2012 ◽  
Vol 04 (01) ◽  
pp. 49-70 ◽  
Author(s):  
OMER BOBROWSKI ◽  
MATTHEW STROM BORMAN
Keyword(s):  
Random Field ◽  
Euler Characteristic ◽  
Persistent Homology ◽  
Gaussian Random Field ◽  
Gaussian Random Fields ◽  
Closed Form Expression ◽  
Form Expression ◽  
Kinematic Formula ◽  
Euler Integral ◽  
The Relationship

In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to compute the expected Euler integral of a Gaussian random field using the Gaussian kinematic formula and obtain a simple closed form expression. This results in the first explicitly computable mean of a quantitative descriptor for the persistent homology of a Gaussian random field.

scholarly journals Independence of the increments of Gaussian random fields

1982 ◽  
Vol 85 ◽  
pp. 251-268 ◽  
Author(s):  
Kazuyuki Inoue ◽  
Akio Noda
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Euclidean Distance ◽  
Gaussian Random Field ◽  
Gaussian Random Fields

Let be a mean zero Gaussian random field (n ⋜ 2). We call X Euclidean if the probability law of the increments X(A) − X(B) is invariant under the Euclidean motions. For such an X, the variance of X(A) − X(B) can be expressed in the form r(|A − B|) with a function r(t) on [0, ∞) and the Euclidean distance |A − B|.

A Solution of Fractional Laplace's Equation by Modified Separation of Variables

2014 ◽  
Vol 4 (1) ◽  
pp. 397-403
Author(s):  
Amir Pishkoo ◽  
Maslina Darus ◽  
Fatemeh Tamizi
Keyword(s):  
Closed Form ◽  
Separation Of Variables ◽  
Closed Form Expression ◽  
Fractional Dimension ◽  
Laplace’S Equation ◽  
Form Expression ◽  
Laplace's Equation ◽  
Radial Functions ◽  
Separation Of Variables Method ◽  
The Relationship

This paper applies the Modified separation of variables method (MSV) suggested by Pishkoo and Darus towards obtaining a solution for fractional Laplace's equation. The closed form expression for potential function is formulated in terms of Meijer's G-functions (MGFs). Moreover, the relationship between fractional dimension and parameters of Meijer’s G-function for spherical Laplacian in R, for radial functions is derived.

On the asymptotic form of convex hulls of Gaussian random fields

2014 ◽  
Vol 12 (5) ◽  
Author(s):  
Youri Davydov ◽  
Vygantas Paulauskas
Keyword(s):  
Banach Space ◽  
Asymptotic Behavior ◽  
Random Field ◽  
Hausdorff Distance ◽  
Random Fields ◽  
Asymptotic Form ◽  
Gaussian Random Field ◽  
Gaussian Random Fields ◽  
Convex Hulls ◽  
Limit Set

AbstractWe consider a centered Gaussian random field X = {X t : t ∈ T} with values in a Banach space $$\mathbb{B}$$ defined on a parametric set T equal to ℝm or ℤm. It is supposed that the distribution of X t is independent of t. We consider the asymptotic behavior of closed convex hulls W n = conv{X t : t ∈ T n}, where (T n) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b n)n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(W n), where f is some function, is also studied.

Local Maxima and the Expected Euler Characteristic of Excursion Sets of χ 2, F and t Fields

1994 ◽  
Vol 26 (1) ◽  
pp. 13-42 ◽  
Author(s):  
K. J. Worsley
Keyword(s):  
Blood Flow ◽  
Euler Characteristic ◽  
Pain Perception ◽  
Gaussian Random Field ◽  
Three Dimensional ◽  
Gaussian Random Fields ◽  
Local Maxima ◽  
Excursion Sets ◽  
Positron Emission ◽  
Unknown Point

The maximum of a Gaussian random field was used by Worsley et al. (1992) to test for activation at an unknown point in positron emission tomography images of blood flow in the human brain. The Euler characteristic of excursion sets was used as an estimator of the number of regions of activation. The expected Euler characteristic of excursion sets of stationary Gaussian random fields has been derived by Adler and Hasofer (1976) and Adler (1981). In this paper we extend the results of Adler (1981) to χ2, F and t fields. The theory is applied to some three-dimensional images of cerebral blood flow from a study on pain perception.

Gaussian Random Fields on Undirected Graphs

2006 ◽  
Author(s):  
Ulf Grenander ◽  
Michael I. Miller
Keyword(s):  
Partition Function ◽  
Random Field ◽  
Random Fields ◽  
Gaussian Random Field ◽  
Gaussian Random Fields ◽  
Difference Operators ◽  
Neighborhood Structure ◽  
Undirected Graphs ◽  
State Spaces ◽  
Discrete Lattices

This chapter focuses on state spaces of the continuum studying Gaussian random fields on discrete lattices. Covariances are induced via difference operators and the associated neighborhood structure of the resulting random field is explored. Texture representation and segmentation are studied via the general Gaussian random field structures. For dealing with the partition function determined by the log-determinant of the covariance asymptotics are derived connecting the eigenvalues of the finite covariance fields to the spectrum of the infinite stationary extension.

Local Maxima and the Expected Euler Characteristic of Excursion Sets of χ 2, F and t Fields

1994 ◽  
Vol 26 (01) ◽  
pp. 13-42 ◽  
Author(s):  
K. J. Worsley
Keyword(s):  
Blood Flow ◽  
Euler Characteristic ◽  
Pain Perception ◽  
Gaussian Random Field ◽  
Three Dimensional ◽  
Gaussian Random Fields ◽  
Local Maxima ◽  
Excursion Sets ◽  
Positron Emission ◽  
Unknown Point

The maximum of a Gaussian random field was used by Worsley et al. (1992) to test for activation at an unknown point in positron emission tomography images of blood flow in the human brain. The Euler characteristic of excursion sets was used as an estimator of the number of regions of activation. The expected Euler characteristic of excursion sets of stationary Gaussian random fields has been derived by Adler and Hasofer (1976) and Adler (1981). In this paper we extend the results of Adler (1981) toχ2,Fandtfields. The theory is applied to some three-dimensional images of cerebral blood flow from a study on pain perception.

scholarly journals Polar Functions for Anisotropic Gaussian Random Fields

2014 ◽  
Vol 2014 ◽  
pp. 1-18
Author(s):  
Zhenlong Chen
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Gaussian Random Field ◽  
Continuous Functions ◽  
Gaussian Random Fields ◽  
Upper And Lower Bounds ◽  
Packing Dimensions ◽  
Anisotropic Fields ◽  
The Times ◽  
Polar Functions

LetXbe an (N,d)-anisotropic Gaussian random field. Under some general conditions onX, we establish a relationship between a class of continuous functions satisfying the Lipschitz condition and a class of polar functions ofX. We prove upper and lower bounds for the intersection probability for a nonpolar function andXin terms of Hausdorff measure and capacity, respectively. We also determine the Hausdorff and packing dimensions of the times set for a nonpolar function intersectingX. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise, and the operator-scaling Gaussian random field with stationary increments.

scholarly journals Synthesizing of Planar and Conformal Double-Sided Parallel-Cross Dipoles FSS Based on Closed-Form Expression

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Yassine Zouaoui ◽  
Larbi Talbi ◽  
Khelifa Hettak ◽  
Naresh K. Darimireddy
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Form Expression

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

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