Euler characteristics for Gaussian fields on manifolds

Taylor, Jonathan; Adler, Robert J.

doi:10.1214/aop/1048516527

Cited by 102 publications

(132 citation statements)

References 13 publications

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“…It is the basis of the work of Taylor (2001); Taylor and Adler (2003); Taylor (2006). It links the statistical information of the distribution used, the hitting set (via the GMFs) and the spatial structure of the correlated RF (via its covariance function second spectral moment) to the geometrical and topological characteristics of the excursion set (via LKCs).…”

Section: Expectation Formulamentioning

confidence: 99%

Meso-scale modeling of concrete: A morphological description based on excursion sets of Random Fields

Roubin

Colliat

Benkemoun

2015

Computational Materials Science

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In view of the significant impact of thin scale heterogeneities in regards with the macroscopic response of concrete (and generally speaking of heterogeneous materials), a particular effort is dedicated to morphological representation and modeling. The development of a model based on spatially correlated random functions (Random Fields) is proposed in this article. It is shown how the stationary ergodic property coupled with the spatial structure of correlated Random Fields can efficiently address the problematic when submitted to a threshold process. Recent mathematical results Adler (2008) give accurate ways to analytically control the resulting morphology, both geometrically and topologically speaking. The generalized aspect of this framework has to be seen here as the ability of the model to represent different kind of morphologies such as matrix/inclusion, opened or closed porosity. With such features, heterogeneous material can be represented at different scales. By expanding a rather abstract formula given by Adler (2008), the authors aim at spreading this point of view with a "ready for use" framework. Furthermore, by addressing common issues such as, reaching high volume fraction with disconnected topologies and representing grain size distributions, solutions to adapt it to cementitious materials are given.

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Section: Expectation Formulamentioning

confidence: 99%

Meso-scale modeling of concrete: A morphological description based on excursion sets of Random Fields

Roubin

Colliat

Benkemoun

2015

Computational Materials Science

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“…Most conventional approaches for signal detection (and enhancement) in brain images are based on analytical formulae that describe the distribution of features in random fields (e.g., SPM; Friston et al, 1995). These are difficult to apply if data lie on curved manifolds such as the cortex (Goebel and Singer, 1999;Jones et al, 2000;Taylor and Adler, 2000), as the computed spatial autocorrelation of the surface data (1) depends on the local parameterization (or metric) of the surface, and (2) it may not be stationary (i.e., the same everywhere on the surface). To overcome this and to apply detection formulae for cortical signals that apply to stationary data (i.e., permutation tests on cluster extent or the Euler characteristics of suprathreshold statistics), a computational grid can be adapted to the roughness tensor of the data using a datadriven PDE (a process called statistical flattening; Worsley et al, 1999).…”

Section: Parameterization Of the Time Axismentioning

confidence: 99%

Mapping cortical change in Alzheimer's disease, brain development, and schizophrenia

et al. 2004

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This paper describes algorithms that can identify patterns of brain structure and function associated with Alzheimer's disease, schizophrenia, normal aging, and abnormal brain development based on imaging data collected in large human populations. Extraordinary information can be discovered with these techniques: dynamic brain maps reveal how the brain grows in childhood, how it changes in disease, and how it responds to medication. Genetic brain maps can reveal genetic influences on brain structure, shedding light on the nature-nurture debate, and the mechanisms underlying inherited neurobehavioral disorders. Recently, we created time-lapse movies of brain structure for a variety of diseases. These identify complex, shifting patterns of brain structural deficits, revealing where, and at what rate, the path of brain deterioration in illness deviates from normal. Statistical criteria can then identify situations in which these changes are abnormally accelerated, or when medication or other interventions slow them. In this paper, we focus on describing our approaches to map structural changes in the cortex. These methods have already been used to reveal the profile of brain anomalies in studies of dementia, epilepsy, depression, childhoodand adult-onset schizophrenia, bipolar disorder, attention-deficit/ hyperactivity disorder, fetal alcohol syndrome, Tourette syndrome, Williams syndrome, and in methamphetamine abusers. Specifically, we describe an image analysis pipeline known as cortical pattern matching that helps compare and pool cortical data over time and across subjects. Statistics are then defined to identify brain structural differences between groups, including localized alterations in cortical thickness, gray matter density (GMD), and asymmetries in cortical organization. Subtle features, not seen in individual brain scans, often emerge when population-based brain data are averaged in this way. Illustrative examples are presented to show the profound effects of development and various diseases on the human cortex. Dynamically spreading waves of gray matter loss are tracked in dementia and schizophrenia, and these sequences are related to normally occurring changes in healthy subjects of various ages. D 2004 Published by Elsevier Inc.

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“…Next, for the random field f defined on the surface, we calculated the expected value of the Euler Characteristic (EC) by appropriately smoothing the random field. The expected EC then determined the points on the surface where the z-statistic differed from zero at a prespecified level of significance (p = .05 − .0001), thereby identifying locations of statistically significant correlations of local morphology with IQ at the surface of each structure (Taylor & Adler, 2003).…”

Section: Surface Analysismentioning

confidence: 99%

Correlates of intellectual ability with morphology of the hippocampus and amygdala in healthy adults

Amat

Bansal

Whiteman

et al. 2008

Brain and Cognition

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Several prior imaging studies of healthy adults have correlated volumes of the hippocampus and amygdala with measures of general intelligence (IQ), with variable results. In this study, we assessed correlations between volumes of the hippocampus and amygdala and full-scale IQ scores (FSIQ) using a method of image analysis that permits detailed regional mapping of this correlation throughout the surface contour of these brain structures. We delineated the hippocampus and amygdala in high-resolution magnetic resonance images of the brain from 34 healthy individuals. We then correlated FSIQ with overall volumes and with the surface morphologies of each of these structures. Hippocampus volumes correlated significantly and inversely with FSIQ independently of gender, age, socioeconomic status, and whole brain volume. Left and right hippocampus volumes correlated respectively with verbal and performance IQ subscales. Higher IQs were significantly associated with large inward deformations of the surface of the anterior hippocampus bilaterally. These findings suggest that a smaller anterior hippocampus contributes to an increased efficiency of neural processing that subserves overall intelligence.

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Euler characteristics for Gaussian fields on manifolds

Cited by 102 publications

References 13 publications

Meso-scale modeling of concrete: A morphological description based on excursion sets of Random Fields

Meso-scale modeling of concrete: A morphological description based on excursion sets of Random Fields

Mapping cortical change in Alzheimer's disease, brain development, and schizophrenia

Correlates of intellectual ability with morphology of the hippocampus and amygdala in healthy adults

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