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 (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.

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.

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.

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 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.

Characterization of Multi-Phase and Multi-Component Polymer Systems Using the Atomic Force Microscope

1999 ◽  
Vol 5 (S2) ◽  
pp. 962-963
Author(s):  
M. VanLandingham ◽  
X. Gu ◽  
D. Raghavan ◽  
T. Nguyen
Keyword(s):  
Energy Dissipation ◽  
Atomic Force Microscope ◽  
Mechanical Response ◽  
Sample Surface ◽  
Phase Changes ◽  
Phase Imaging ◽  
Polymer Systems ◽  
Atomic Force ◽  
Phase Images ◽  
Multi Phase

Recent advances have been made on two fronts regarding the capability of the atomic force microscope (AFM) to characterize the mechanical response of polymers. Phase imaging with the AFM has emerged as a powerful technique, providing contrast enhancement of topographic features in some cases and, in other cases, revealing heterogeneities in the polymer microstructure that are not apparent from the topographic image. The enhanced contrast provided by phase images often allows for identification of different material constituents. However, while the phase changes of the oscillating probe are associated with energy dissipation between the probe tip and the sample surface, the relationship between this energy dissipation and the sample properties is not well understood.As the popularity of phase imaging has grown, the capability of the AFM to measure nanoscale indentation response of polymers has also been explored. Both techniques are ideal for the evaluation of multi-phase and multi-component polymer systems.

A PARALLEL ALGORITHM FOR ANALYZING CONNECTED COMPONENTS IN BINARY IMAGES

1992 ◽  
Vol 06 (02n03) ◽  
pp. 315-333 ◽  
Author(s):  
FRANCO CHIAVETTA ◽  
VITO DI GESÙ ◽  
ROSALIA RENDA
Keyword(s):  
Parallel Algorithm ◽  
Parallel Implementation ◽  
Discrete Space ◽  
Space Complexity ◽  
Connected Components ◽  
Two Dimensional ◽  
Binary Images ◽  
Cylindrical Algebraic Decomposition ◽  
Time Space ◽  
Connectivity Degree

In this paper, a parallel algorithm for analyzing connected components in binary images is described. It is based on the extension of the Cylindrical Algebraic Decomposition (CAD) to a two-dimensional (2D) discrete space. This extension allows us to find the number of connected components, to determine their connectivity degree, and to solve the visibility problem. The parallel implementation of the algorithm is outlined and its time/space complexity is given.

scholarly journals UNBIASED ESTIMATORS OF SPECIFIC CONNECTIVITY

2011 ◽  
Vol 26 (3) ◽  
pp. 129
Author(s):  
Jean-Paul Jernot ◽  
Patricia Jouannot ◽  
Christian Lantuéjoul
Keyword(s):  
Boolean Model ◽  
Quantitative Metallography ◽  
Random Set ◽  
Unbiased Estimators ◽  
Sintered Bronze

This paper deals with the estimation of the specific connectivity of a stationary random set in IRd. It turns out that the "natural" estimator is only asymptotically unbiased. The example of a boolean model of hypercubes illustrates the amplitude of the bias produced when the measurement field is relatively small with respect to the range of the random set. For that reason unbiased estimators are desired. Such an estimator can be found in the literature in the case where the measurement field is a right parallelotope. In this paper, this estimator is extended to apply to measurement fields of various shapes, and to possess a smaller variance. Finally an example from quantitative metallography (specific connectivity of a population of sintered bronze particles) is given.

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