Wavelets, statistics, and biomedical applications
This paper emphasizes the statistical properties of the wavelet transform (WT) and discusses some examples of applications in medicine and biology. The redundant forms of the transform (continuous wavelet transform (CWT) and wavelet frames) are well suited for detection tasks (e.g., spikes in EEG, or microcalcifications in mammograms). The CWT, in particular, can be interpreted as a prewhitening multi-scale matched filter. Redundant wavelet decompositions are also very useful for the characterization of singularities, as well as for the time-frequency analysis of non-stationary signals. We discuss some examples of applications in phonocardiography, electrocardiography (EGG), and electroencephalography (EEG). Wavelet bases (WB) provide a similar, non-redundant decomposition of a signal in terms of the shifts and dilations of a wavelet. This makes WB well suited for any of the tasks for which block transforms have been used traditionally. Wavelets, however, may present certain advantages because they can improve the signal-to-noise ratio, while retaining a certain degree of localization in the time (or space) domain. We present three illustrative examples. The first is a denoising technique that applies a soft threshold in the wavelet domain. The second is a more refined version that uses generalized Wiener filtering. The third is a statistical method for detecting and locating patterns of brain activity in functional images acquired using magnetic resonance imaging (MRI). We conclude by describing a wavelet generalization of the classical Karhunen-Loeve transform.