Wavelet theory offers a new approach to the analysis of functions, focusing not on the values of a function or its derivatives at a point, but on averages over intervals. French mathematician Yves Meyer is considered to have been one of its founders in 1986, which earned him the 2017 Abel Prize. Wavelet theory developed extraordinarily quickly, opening up unexpected possibilities and applications such as file compression, notably the JPEG 2000 format.
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*How to compress a file*
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The most common application is file compression. A file containing many 0s is easy to compress: for example, instead of storing the 0s, we need only store the non-zero values and their positions in the file. Thus, a file of N numbers containing only n non-zero values can be stored using just n real values (the non-zero values) and n integer values (their positions), making 2n values in all. To compress a file, then, we seek to create 0s, even if this means setting values "close to 0" to 0.
Let's take a file to compress. Its signal is the collection of numbers in the file; it may represent a sound, an image, a video, or any sequence of measured values. The simplest files are one-dimensional, such as sound files (twenty thousand values per second are needed to transmit frequencies up to 20 kHz, the limit of human hearing). If we want to transmit only one value, it seems natural to send the mean m00 of all the values in the file. To transmit two values, we can take the mean m10 of the first half of the file, followed by the mean m11 of the second half. But we can instead send m00 and d10 = m10 – m00 (d10 is called a detail coefficient). It is then easy to reconstruct m10 and m11 (since m10 + m11 = 2m00). This amounts to the same thing… except that if m10 = m11 (and both are non-zero), then d10 = 0. We have therefore turned a non-zero value (m10) into a zero (d10).
