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wavelet_transform.md
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*Good luck trying to understand the corresponding [Wikipedia](wikipedia.md) article.*
Wavelet transform is a [mathematical](math.md) operation, similar to e.g. [Fourier transform](fourier_transform.md), that takes a [signal](signal_processing.md) (e.g. audio or an image) and outputs information about the [frequencies](frequency.md) contained in that signal AS WELL as the locations of those frequencies. This is of course extremely useful when we want to analyze and manipulate frequencies in our signal -- for example [JPEG 2000](jpeg_2000.md) uses wavelet transforms for [compressing](compression.md) images by discarding certain frequencies in them that our eyes are not so sensitive to.
Wavelet transform is a [mathematical](math.md) operation, similar to e.g. [Fourier transform](fourier_transform.md), that takes a [signal](signal_processing.md) (let's say audio or an image) and outputs information about the [frequencies](frequency.md) contained in that signal AS WELL as the locations of those frequencies. This is of course very handy should we want to analyze and manipulate frequencies in the signal -- for example [JPEG 2000](jpeg_2000.md) exploits wavelet transforms for [compressing](compression.md) images by discarding certain frequencies in them that our eyes are not so sensitive to.
The main advantage over [Fourier transform](fourier_transform.md) (and similar transforms such as [cosine transform](cosine_transform.md)) is that wavelet transform shows us not only the frequencies, but ALSO their locations (i.e. for example time at which these frequencies come into play in an audio signal). This allows us for example to locate specific sounds in audio or apply compression only to certain parts of an image. While localizing frequencies is also possible with Fourier transform with tricks such as [spectrograms](spectrogram.md), wavelet transforms are a more elegant, natural and continuous way of doing so. Note that due to [Heisenberg's uncertainty principle](uncertainty_principle.md) it is mathematically IMPOSSIBLE to know both frequencies and their locations exactly, there always has to be a tradeoff -- the input signal itself tells us everything about location but nothing about frequencies, Fourier transform tells us everything about frequencies but nothing about their locations and wavelet transform is a **midway** between the two -- it tells us something about frequencies and their approximate locations.