This year's Abel Prize was awarded to Yves Meyer in recognition of his great role in the development of wavelet theory. Wavelet theory allows us to decompose different types of information into simpler components, thus making information analysis, processing and storage easier. Therefore, wavelet theory has been applied in a wide range of fields, including harmonic analysis and calculation, data compression, noise reduction, medical imaging, archiving, digital cinema and gravitational wave detection.
In 20 16, LIGO detected gravitational wave events radiated by two black holes, and its signal analysis applied wavelet theory.
Interestingly, Meyer's work was inspired not by mathematics, but by the oil industry. In the1980s, French engineer Jean Morlet wanted to know how to make better use of seismic data to find oil. Morlet analyzed the reflection data collected from oil exploration. Transmit vibration to the ground and collect echoes. This is the same principle that bats use sonar. The problem is how to analyze the reflected data and extract valuable information about the oil layer. Morlet and physicist Alex Grossmann thought of a method to analyze signals and introduced a new function category called "wavelet", which was obtained by stretching and translating fixed functions. However, the oil industry is not interested in this. Morlet's method was not adopted, but their paper was still published in the scientific journal 1984 in the spring.
A year later, while Meyer was copying something at the Paris Institute of Technology, his colleagues copied the paper on Morlet for him. On the train to Marseille, he discovered the great potential of Xiaobo.
Mathematicians and engineers have long known a powerful tool for analyzing and processing specific types of information: Fourier analysis. Sound is the best example to explain Fourier analysis. For example, the sound of center A emitted by a tuning fork is represented by a perfect sine wave, as shown below:
This is a sine wave. It extends infinitely to the left and right. Because sine wave is related to cosine wave, it can also be regarded as the representation of cosine wave.
Other sounds, such as the same notes played by the violin, are more complicated. But we later found that any periodic sound, in fact, any type of periodic signal, can be decomposed into the sum of sine waves and cosine waves with different frequencies.
The function f varies with time and represents sound waves. The Fourier transform process decomposes the function f into a sine wave with a specific frequency and amplitude. Fourier transform is represented as a peak in frequency domain, and the height of the peak represents the amplitude of the wave at this frequency.
Fourier analysis is a very useful tool. It can also be used to analyze and process images and other types of information. But it also has defects: because the basic components-sine wave and cosine wave-are periodic, Fourier analysis can only play the greatest role in repeated signals. But it is not so useful for aperiodic signals with irregular characteristics (such as peaks, etc.). Unfortunately, most phenomena in real life, from sound to seismic data, belong to the non-periodic category.
This waveform comes from the human voice. It is regular, but not periodic.
This is also the time when wavelet theory appeared. As the name implies, wavelet is a "very small wave". The theory is based on "mother wavelet", which is a small part of the oscillation function. The frequency of oscillation is different, so is the width of wavelet. But there is a close relationship between them: the higher the frequency, the narrower the width.
Meyer wavelet. The compressed wavelet (top) has a higher frequency, while the stretched wavelet (bottom) has a lower frequency.
The wavelet can be generated by changing the scale of the mother wavelet, such as narrowing (increasing frequency), amplifying (decreasing frequency) or moving. A signal, such as our voice, can be represented by the combination of this set of wavelets. This decomposition allows us to capture the repeated information in the signal and use a series of decreasing mother wavelet versions to amplify local irregularities (such as peaks).
In order to store such a signal decomposition, you only need to describe the information of the original mother wavelet and the contributions of different wavelet. They are enough to reconstruct the original signal.
Fourier transform (i) and wavelet transform (II). The former has only frequency ω, while the latter has two variables: scale A (controlling the expansion and contraction of wavelet function) and translation τ (controlling the translation of wavelet function).
The original idea of wavelet theory can be traced back to a long time ago. A hundred years ago, mathematician Alfred Hall constructed a kind of wavelet. Haar wavelet has some good characteristics, but it also has some shortcomings. Meyer played a key role in the development of wavelet theory, and he established a strong and solid mathematical foundation for wavelet theory.
Some examples of wavelet types: (a) COIF1; Db2 Meyer; Sym3(e)mor let； Mexico. (Source: Krishna B)
Meyer's first major contribution is to construct smooth orthogonal wavelet bases. In the wavelet analysis constructed by Morlet, all the functions in Meyer wavelet base are generated by a single smooth "mother wavelet", which can be clearly specified by translation and expansion. Although the wavelet constructed by Morlet is very basic in nature, it is quite incredible. Subsequently, Stéphane Mallat and Yves Meyer systematically developed the theory of multi-resolution analysis, which is a general framework for constructing wavelet bases.
Yves Meyer.
1late 1980s1early 1990s, signal processing ushered in the "wavelet revolution", and wavelet transform was also applied to many basic signal processing tasks. Such as compression (such as JPEG2000 image compression format) and denoising, as well as more modern applications (such as compressed sensing). The FBI also uses wavelet to store fingerprint information, otherwise it will take up a lot of storage space. In addition, Meyer's work has promoted the important theoretical development in the field of harmonic analysis and partial differential equations, from proving the boundedness of Cauchy integral on Lipschitz curve (solved by Coifman, McIntosh and Meyer) to developing new tools that are indispensable for understanding the nonlinear effects of partial differential equations (such as compensation compactness). In addition, Meyer has made important contributions to quasicrystals, singular integral operators and Naville-Stokes equations. It can be said that Meyer's work and opinions not only promote the development of the application of pure mathematics and mathematical analysis, but also build a fruitful communication bridge between them.
Stefan marat called him a "dreamer". His work does not belong to any field (such as pure mathematics, applied mathematics or computer science) and can only be labeled as "magic".