Maxima of the continuous wavelet transform of correlated data

The study of continuous wavelet transform (CWT) of signals through the behavior of its local maxima is a well-developed field that has already led to useful applications in signal and image analysis. Meanwhile, the study of level upcrossings of random field is based on expected values of random quantities related to local maxima of the field. Generalizing the notion of level upcrossings from one dimension to higher-dimensional spaces leads to the problem of evaluating the expected value of the Euler characteristic of excursion sets on those fields. This has been done by Adler [The Geometry of Random Fields, Wiley, New York, 198 1] and further extended by Siegmund and Worsley "Testing for a signal with unknown location and scale in a stationary Gaussian random field" [Ann. Stat. 23 (2) (1995) 608-639], who proposed an extension of the method to test for signals not only of unknown location but of unknown scale as well, using an approach quite similar to the CWT. Even for an "irregular" field which does not respect Adler's condition, a proper use of the CWT leads to a representation where the field becomes regular. We first show that this allows us to apply Adler's method to a more general family of irregular random fields as, for instance, a fractional Brownian motion. Then, we introduce a fast implementation based on the discrete dyadic wavelet decomposition that allows us to perform the analysis with fewer operations than the method originally proposed by Siegmund and Worlsey. Finally, we apply this method in order to detect a sharp but continuous signal in a background noise. (C) 2003 Elsevier Inc. All rights reserved.