ON THE EQUIVALENCE OF THE TUBE AND EULER CHARACTERISTIC METHODS FOR THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS OVER PIECEWISE SMOOTH DOMAINS

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Random fields of multivariate test statistics, with applications to shape analysis (2008, background): “In fact, there is a strong link between this paper and Kuriki and Takemura [11].”
Random Fields and Geometry (2007, methodology): “Takemura and Kuriki [139] had a version of Theorem 13.”
Likelihood ratio tests and singularities (2007, background): “Moreover, as pointed out by a referee, the “tube” and “Euler characteristic” methods may be useful for approximating limiting distributions; see [32, 33] and the references therein.”
NIRS-SPM: statistical parametric mapping for near infrared spectroscopy (2008, unknown): “In the Gaussian SPMs, Sun's tube formula and random field theory give the same solution (Takemura and Kuriki, 2002).”
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…”