Euler characteristics for Gaussian fields on manifolds

By Jonathan E. Taylor, Robert J. Adler. 2003

131 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/eb41fd33ba9079add41e538a7a710f6e06299039

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

NIRS-SPM: statistical parametric mapping for near infrared spectroscopy (2008, unknown): “We then use random field theory results, which are in fact available for practically any non-homogeneous randomfield, even onmanifolds, provided that it is smooth (Worsley et al.,1999; Taylor and Adler, 2003; Taylor and Worsley, 2008; Ye et al., 2006).”
Random fields of multivariate test statistics, with applications to shape analysis (2008, methodology): “However, in Taylor and Adler [21] the actual derivation of the expected EC (2) in the nonisotropic case proceeds more naturally from first principles, namely, the variogram metric (14), with isotropic random fields in Euclidean space following as a special case.”
Persistent homology for random fields and complexes (2010, background): “The Gaussian kinematic formula (hereafter GKF) was due originally to Taylor in [34] (but for the form below see [2, 35]) and states that”
Some new random field tools for spatial analysis (2008, background): “We now come to the main result of the new theory, the fundamental starting point of which was in Jonathan Taylor’s 2001 McGill PhD thesis (cf. [ 16 , 19]).”
A gaussian kinematic formula (2006, methodology): “an approximation to the Dirac delta, as in [18].”
Topological complexity of smooth random functions (2011, background): “While special cases appeared as early as the mid 1970s, the elegant geometric form in which it can be found in these notes first appeared in Taylor’s 2001 McGill PhD thesis [79] and the paper [80].”
Level crossings and other level functionals of stationary Gaussian processes (2006, background): “(see [157] and [2]), and by Azäıs and Wschebor (see [14]).”
Validity of the expected Euler characteristic heuristic (2005, unknown): “Piecewise smooth manifolds are required to have the additional property that they are locally approximated by SxM in the sense that for every x ∈ M there exists a diffeomorphism ϕx : TxM̂ → M̂ which, when restricted to SxM maps any sufficiently small neighbourhood of the origin to a neighbourhood…”
Unified univariate and multivariate random field theory. (2004, methodology): “To use this result, we need EC densities beyond the d = 3 dimensions given in Worsley et al. (1996). A general expression can be found in Worsley (1994), and for convenience, it is reproduced in Appendix B.”
On the distribution of the maximum of a Gaussian field with d parameters (2005, methodology): “When such a Riemannian metric is not given, it is always possible to use the metric g induced by the process itself (see Taylor and Adler, 2002 ) by setting”
Nonstationary cluster-size inference with random field and permutation methods (2004, background): “Furthermore, a new result by Taylor and Adler (2003) shows that the existence of the warp is no longer necessary and that the random field results work under almost any form of nonstationarity.”