Maxima of discretely sampled random fields, with an application to 'bubbles'

A smooth Gaussian random field with zero mean and unit varianceis sampled on a discrete lattice, and we are interested in theexceedance probability or P-value of the maximum in a finiteregion. If the random field is smooth relative to the mesh size,then the P-value can be well approximated by results for thecontinuously sampled smooth random field (Adler, 1981; Worsley,1995a; Taylor & Adler, 2003; Adler & Taylor, 2007).If the random field is not smooth, so that adjacent latticevalues are nearly independent, then the usual Bonferroni boundis very accurate. The purpose of this paper is to bridge thegap between the two, and derive a simple, accurate upper boundfor intermediate mesh sizes. The result uses a new improvedBonferroni-type bound based on discrete local maxima. We givean application to the ‘bubbles’ technique for detectingareas of the face used to discriminate fear from happiness.