A wavelet is a kind of mathematical function used to divide a given function into different frequency components and study each component with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are
scaled and
translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional
Fourier transforms for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-
periodic and/or non-
stationary signals.
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<
mathematics> A waveform that is bounded in both
frequency and duration. Wavelet tranforms provide an alternative to more traditional
Fourier transforms used for analysing waveforms, e.g. sound.
The
Fourier transform converts a signal into a continuous series of
sine waves, each of which is of constant frequency and
amplitude and of infinite duration. In contrast, most real-world signals (such as music or images) have a finite duration and abrupt changes in frequency.
Wavelet transforms convert a signal into a series of wavelets. In theory, signals processed by the wavelet transform can be stored more efficiently than ones processed by Fourier transform. Wavelets can also be constructed with rough edges, to better approximate real-world signals.
For example, the United States Federal Bureau of Investigation found that Fourier transforms proved inefficient for approximating the whorls of fingerprints but a wavelet transform resulted in crisper reconstructed images.
SBG Austria.
["Ten Lectures on Wavelets", Ingrid Daubechies].
(1994-11-09)
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