Wavelet
As a representation of a signal

Often, signals can be represented well as a sum of sinusoids. However, consider a non-continuous signal with an abrupt discontinuity; this signal can still be represented as a sum of sinusoids, but requires an infinite number, which is an observation known as Gibbs phenomenon. This, then, requires an infinite number of Fourier coefficients, which is not practical for many applications, such as compression. Wavelets are more useful for describing these signals with discontinuities because of their time-localized behavior (both Fourier and wavelet transforms are frequency-localized, but wavelets have an additional time-localization property). Because of this, many types of signals in practice may be non-sparse in the Fourier domain, but very sparse in the wavelet domain. This is particularly useful in signal reconstruction, especially in the recently popular field of compressed sensing. (Note that the short-time Fourier transform (STFT) is also localized in time and frequency, but there are often problems with the frequency-time resolution trade-off. Wavelets are better signal representations because of multiresolution analysis.)

This motivates why wavelet transforms are now being adopted for a vast number of applications, often replacing the conventional Fourier transform. Many areas of physics have seen this paradigm shift, including molecular dynamics, chaos theory, ab initio calculations, astrophysics, gravitational wave transient data analysis, density-matrix localisation, seismology, optics, turbulence and quantum mechanics. This change has also occurred in image processing, EEG, EMG, ECG analyses, brain rhythms, DNA analysis, protein analysis, climatology, human sexual response analysis, general signal processing, speech recognition, acoustics, vibration signals, computer graphics, multifractal analysis, and sparse coding. In computer vision and image processing, the notion of scale space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation.
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