Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics

1995 ◽  
Vol 27 (4) ◽  
pp. 943-959 ◽  
Author(s):  
K. J. Worsley
Keyword(s):  
Euler Characteristic ◽  
Random Fields ◽  
Background Radiation ◽  
Three Dimensions ◽  
Connected Components ◽  
Excursion Sets ◽  
Microwave Background Radiation ◽  
Excursion Set ◽  
Set Of Points ◽  
Fixed Region

Certain images arising in astrophysics and medicine are modelled as smooth random fields inside a fixed region, and experimenters are interested in the number of ‘peaks', or more generally, the topological structure of ‘hot-spots' present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Euler characteristic counts the number of connected components in the excursion set minus the number of ‘holes'. For high thresholds the Euler characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (differential topology) and IG (integral geometry) characteristics, which equal the Euler characteristic provided the excursion set does not touch the boundary of the region. Worsley (1995) finds a boundary correction which gives the expectation of the Euler characteristic itself in two and three dimensions. The proof uses a representation of the Euler characteristic given by Hadwiger (1959). The purpose of this paper is to give a general result for any number of dimensions. The proof takes a different approach and uses a representation from Morse theory. Results are applied to the recently discovered anomalies in the cosmic microwave background radiation, thought to be the remnants of the creation of the universe.

Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics

1995 ◽  
Vol 27 (04) ◽  
pp. 943-959 ◽  
Author(s):  
K. J. Worsley
Keyword(s):  
Euler Characteristic ◽  
Random Fields ◽  
Background Radiation ◽  
Three Dimensions ◽  
Connected Components ◽  
Excursion Sets ◽  
Microwave Background Radiation ◽  
Excursion Set ◽  
Set Of Points ◽  
Fixed Region

Certain images arising in astrophysics and medicine are modelled as smooth random fields inside a fixed region, and experimenters are interested in the number of ‘peaks', or more generally, the topological structure of ‘hot-spots' present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Euler characteristic counts the number of connected components in the excursion set minus the number of ‘holes'. For high thresholds the Euler characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (differential topology) and IG (integral geometry) characteristics, which equal the Euler characteristic provided the excursion set does not touch the boundary of the region. Worsley (1995) finds a boundary correction which gives the expectation of the Euler characteristic itself in two and three dimensions. The proof uses a representation of the Euler characteristic given by Hadwiger (1959). The purpose of this paper is to give a general result for any number of dimensions. The proof takes a different approach and uses a representation from Morse theory. Results are applied to the recently discovered anomalies in the cosmic microwave background radiation, thought to be the remnants of the creation of the universe.

scholarly journals Identification and isotropy characterization of deformed random fields through excursion sets

2018 ◽  
Vol 50 (3) ◽  
pp. 706-725
Author(s):  
Julie Fournier
Keyword(s):  
Random Field ◽  
Euler Characteristic ◽  
Random Fields ◽  
Weak Form ◽  
Invariance Property ◽  
Basic Domain ◽  
Excursion Sets ◽  
The Mean ◽  
Isotropic Random

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

scholarly journals Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure

2017 ◽  
Vol 54 (3) ◽  
pp. 833-851 ◽  
Author(s):  
Anders Rønn-Nielsen ◽  
Eva B. Vedel Jensen
Keyword(s):  
Random Field ◽  
Large Class ◽  
Random Fields ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Excursion Sets ◽  
Excursion Set ◽  
Fixed Radius ◽  
Asymptotic Probability ◽  
The Right

Abstract We consider a continuous, infinitely divisible random field in ℝd, d = 1, 2, 3, given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields, we compute the asymptotic probability that the excursion set at level x contains some rotation of an object with fixed radius as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

The size of the connected components of excursion sets of χ2, t and F fields

1999 ◽  
Vol 31 (03) ◽  
pp. 579-595 ◽  
Author(s):  
J. Cao
Keyword(s):  
Positron Emission Tomography ◽  
Two Dimensions ◽  
Emission Tomography ◽  
Connected Components ◽  
Gaussian Fields ◽  
Connected Component ◽  
Test Statistic ◽  
Excursion Sets ◽  
Positron Emission ◽  
Excursion Set

The distribution of the size of one connected component and the largest connected component of the excursion set is derived for stationary χ2, t and F fields, in the limit of high or low thresholds. This extends previous results for stationary Gaussian fields (Nosko 1969, Adler 1981) and for χ2 fields in one and two dimensions (Aronowich and Adler 1986, 1988). An application of this is to detect regional changes in positron emission tomography (PET) images of blood flow in human brain, using the size of the largest connected component of the excursion set as a test statistic.

scholarly journals RÉNYI FUNCTION FOR MULTIFRACTAL RANDOM FIELDS

Fractals ◽  
2013 ◽  
Vol 21 (02) ◽  
pp. 1350009 ◽  
Author(s):  
NIKOLAI N. LEONENKO ◽  
NARN-RUEIH SHIEH
Keyword(s):  
Cosmic Microwave Background ◽  
Spherical Harmonics ◽  
Random Fields ◽  
Background Radiation ◽  
Microwave Background ◽  
Basic Scheme ◽  
Infinite Products ◽  
Microwave Background Radiation ◽  
Log Normal ◽  
Isotropic Random

This paper presents the basic scheme and the log-normal, log-gamma and log-negative-inverted-gamma scenarios to establish the Rényi function for infinite products of homogeneous isotropic random fields on Rn; in particular for random fields on the sphere in R3. The motivation of this paper is the test of (non-)Gaussianity on the Cosmic Microwave Background Radiation (CMBR) in Cosmology. In the presentation, we need to employ spherical harmonics for some concrete computations.

scholarly journals REVIEW OF RECENT RESULTS ON EXCURSION SET MODELS

2011 ◽  
Vol 20 (2) ◽  
pp. 71 ◽  
Author(s):  
Richard J Wilson ◽  
David J Nott
Keyword(s):  
Boolean Model ◽  
Connected Components ◽  
Random Set ◽  
Binary Images ◽  
Excursion Sets ◽  
Alternative Approach ◽  
Desirable Feature ◽  
Phase Images ◽  
Excursion Set ◽  
Multi Phase

Many images consist of two or more "phases", where a phase is a collection of homogeneous zones. For example, the phases may represent the presence of different sulphides in an ore sample. Frequently, these phases exhibit very little structure, though all connected components of a given phase may be similar in some sense. As a consequence, random set models are commonly used to model such images. The Boolean model and models derived from the Boolean model are often chosen. An alternative approach to modelling such images is to use the excursion sets of random fields to model each phase. In this paper, the properties of excursion sets will be firstly discussed in terms of modelling binary images. Ways of extending these models to multi-phase images will then be explored. A desirable feature of any model is to be able to fit it to data reasonably well. Different methods for fitting random set models based on excursion sets will be presented and some of the difficulties with these methods will be discussed.

The size of the connected components of excursion sets of χ2, t and F fields

1999 ◽  
Vol 31 (3) ◽  
pp. 579-595 ◽  
Author(s):  
J. Cao
Keyword(s):  
Positron Emission Tomography ◽  
Two Dimensions ◽  
Emission Tomography ◽  
Connected Components ◽  
Gaussian Fields ◽  
Connected Component ◽  
Test Statistic ◽  
Excursion Sets ◽  
Positron Emission ◽  
Excursion Set

The distribution of the size of one connected component and the largest connected component of the excursion set is derived for stationary χ2, t and F fields, in the limit of high or low thresholds. This extends previous results for stationary Gaussian fields (Nosko 1969, Adler 1981) and for χ2 fields in one and two dimensions (Aronowich and Adler 1986, 1988). An application of this is to detect regional changes in positron emission tomography (PET) images of blood flow in human brain, using the size of the largest connected component of the excursion set as a test statistic.

On the capacity functional of excursion sets of Gaussian random fields on ℝ2

2016 ◽  
Vol 48 (3) ◽  
pp. 712-725 ◽  
Author(s):  
Marie Kratz ◽  
Werner Nagel
Keyword(s):  
Random Fields ◽  
Stochastic Geometry ◽  
Gaussian Random Fields ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Excursion Sets ◽  
Moment Measure ◽  
Excursion Set ◽  
The One

Abstract When a random field (Xt,t∈ℝ2) is thresholded on a given level u, the excursion set is given by its indicator 1[u, ∞)(Xt). The purpose of this work is to study functionals (as established in stochastic geometry) of these random excursion sets as, e.g. the capacity functional as well as the second moment measure of the boundary length. It extends results obtained for the one-dimensional case to the two-dimensional case, with tools borrowed from crossings theory, in particular, Rice methods, and from integral and stochastic geometry.

A note on the higher moments of the Euler characteristic of the excursion sets of random fields

2009 ◽  
Vol 79 (8) ◽  
pp. 1074-1082 ◽  
Author(s):  
Ali Reza Taheriyoun ◽  
Khalil Shafie ◽  
Mohammad Jafari Jozani
Keyword(s):  
Euler Characteristic ◽  
Random Fields ◽  
Higher Moments ◽  
Excursion Sets

Cosmological Consequences with Randers Conformal of Finsler space

2007 ◽  
Vol 3 (2) ◽  
pp. 203-211
Author(s):  
Arunesh Pandey ◽  
R K Mishra
Keyword(s):  
Background Radiation ◽  
Finsler Space ◽  
Space Time ◽  
Anisotropic Model ◽  
Microwave Background ◽  
Microwave Background Radiation

In this paper we study an anisotropic model of space – time with Finslerian metric. The observed anisotropy of the microwave background radiation is incorporated in the Finslerian metric of space time.

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