What is wavelet analysis

Wavelet analysis is a rapidly developing new field in mathematics. It has both profound theoretical and extensive application significance.
The concept of wavelet transform was first proposed by J.Morlet, a French engineer engaged in petroleum signal processing, in 1974. Through physical intuition and the actual needs of signal processing, the inversion formula was established, but it was not recognized by mathematicians at that time. In the 1970s, the discovery of A.Calderon's representation theorem, the atomic decomposition of Hardy space and the in-depth study of unconditional bases made theoretical preparations for the birth of wavelet transform, and J. O. Stromberg also constructed a wavelet basis that was very similar to the present in history; in 1986, the famous mathematician Y.Meyer accidentally constructed a real wavelet basis, and cooperated with S.Mallat to establish a method for constructing wavelet basis (multi-scale analysis). Since then, wavelet analysis has begun to flourish. Among them, Ten Lectures on Wavelets written by Belgian female mathematician I.Daubechies played an important role in promoting the popularization of wavelets. Compared with Fourier transform and windowed Fourier transform (Gabor transform), it is a local transform of time and frequency, so it can effectively extract information from the signal, and perform multi-scale analysis (Multiscale Analysis) on functions or signals through operations such as scaling and translation, solving many difficult problems that Fourier transform cannot solve. Therefore, wavelet transform is known as "mathematical microscope", which is a milestone in the development of harmonic analysis.
The term wavelet, as the name suggests, is a small waveform. The so-called "small" means that it has attenuation; and the term "wave" refers to its volatility, and its amplitude is in the form of alternating positive and negative oscillations. Compared with Fourier transform, wavelet transform is a localized analysis of time (space) frequency. It gradually refines the signal (function) at multiple scales through scaling and translation operations, and finally achieves time subdivision at high frequencies and frequency subdivision at low frequencies. It can automatically adapt to the requirements of time-frequency signal analysis, so that it can focus on any details of the signal, solve the difficult problems of Fourier transform, and become a major breakthrough in scientific methods since Fourier transform.
Wavelet analysis is a rapidly developing new field in applied mathematics and engineering. After nearly 10 years of exploration and research, an important mathematical formalization system has been established, and the theoretical foundation is more solid. Wavelet transform connects multiple disciplines such as applied mathematics, physics, computer science, signal and information processing, image processing, and seismic exploration. Mathematicians believe that wavelet analysis is a new branch of mathematics. It is the perfect crystallization of functional analysis, Fourier analysis, sample analysis, and numerical analysis; signal and information processing experts believe that wavelet analysis is a new technology of time-scale analysis and multi-resolution analysis. It has achieved scientifically significant and application-valued results in signal analysis, speech synthesis, image recognition, computer vision, data compression, seismic exploration, and atmospheric and ocean wave analysis.
In fact, the application field of wavelet analysis is very wide, including: many disciplines in the field of mathematics; signal analysis, image processing; quantum mechanics, theoretical physics; military electronic countermeasures and intelligent weapons; computer classification and recognition; artificial synthesis of music and language; medical imaging and diagnosis; seismic exploration data processing; fault diagnosis of large machinery, etc. For example, in mathematics, it has been used for numerical analysis, construction of fast numerical methods, curve and surface construction, differential equation solution, control theory, etc. In signal analysis, filtering, denoising, compression, transmission, etc. In image processing, image compression, classification, recognition and diagnosis, decontamination, etc. In medical imaging, it reduces the time of B-ultrasound, CT, and MRI imaging, and improves the resolution, etc.