Wavelets, Fourier Transform, and Fractals

Wavelets and Fourier analysis in digital signal processing are comparatively discussed. In data processing, the fundamental idea behind wavelets is to analyze according to scale, with the advantage over Fourier methods in terms of optimally processing signals that contain discontinuities and sharp spikes. Apart from the two major characteristics of wavelet analysis multiresolution and adaptivity to nonstationarity or local features in data there is a related application in functional neuroimaging on the basis of the fractal or scale invariant properties demonstrated by the brain imaging data. Wavelets provide an orthonormal basis for multiscale analysis and decorrelation of nonstionary time series and spatial processes. The discrete wavelet transform (DWT) has applications to statistical analysis in functional magnetic resonance imaging (fMRI) like time series resampling by wavestrapping, linear model estimation, and methods for multiple hypothesis testing in the wavelet domain. Key–Words: Wavelets, Fourier transform, fractals, multiresolution analysis, functional magnetic resonance imaging.