Mathematics has filled our lives for as long as we can remember. When I was a child, my concept of mathematics was that mathematics was the main course and should be taken seriously. However, primary school mathematics has been very poor, and I dare not do math problems when I encounter problems and give up. Later, I went to middle school and got a better understanding of the importance of mathematics. Like Chinese and English, the math score is 150. I met a good teacher and gradually learned some thinking about mathematics. I began to understand that mathematics is to do more problems and be well informed. In high school, the knowledge system of mathematics has gradually formed, and it is gradually understood that the tactics of questioning the sea are not necessarily a good method. Everyone has his own learning method. In order to prepare for the college entrance examination, I insist on making a set of math papers every day, reviewing old ones and learning new ones. I think this method is very useful, and I think it is fate. When I was in college, I was assigned to the math major, and I thought math was pretty good at that time. Normal students and girls will be good careers in the future, but in universities. Moreover, abstract things are boring, and my feelings about mathematics have gradually changed. Sometimes I think it's useless to learn these things. It's all theoretical stuff But later, the teacher told us that what the university learned was not knowledge, but the ability to think about mathematical problems. Four years' study in college has made me understand some basic frameworks of mathematics. Many students are going to continue their postgraduate entrance examination and continue to study mathematics. It feels like math is getting narrower and narrower. In the past, they all did some basic knowledge reserves. Now I feel a little directional after graduate school, but basic mathematics is a very theoretical thing. I think it is to give a definition, give a theorem, then prove this theorem, and then prove some propositions with this theorem. Teachers often say that mathematics can't survive without definition, and now they are getting used to this idea.
We arranged a lecture on the frontier knowledge of mathematics in the first class on Monday. The teacher invited many professors and doctors to give us lectures, mainly explaining some hot topics today. The teachers' work is excellent. We not only learned a lot of technical terms of mathematics, but also got a general concept of all directions of mathematics to prepare for future research. Several of them talked about wavelet analysis, which is a rapidly developing new field in mathematics and profound in theory.
To this end, I read some information about wavelet analysis.
The concept of wavelet transform was first put forward in 1974 by J.Morlet, a French engineer engaged in petroleum signal processing. The inversion formula was established through physical intuition and the actual needs of signal processing, but it was not recognized by mathematicians at that time. Just as the French thermal engineer J.B.J Fourier put forward the innovative concept that any function can be expanded into infinite series of trigonometric functions in 1807, it was not recognized by famous mathematicians J.L. Lagrange, P.S. Laplace and A.M. Legendre. Fortunately, as early as the 1970s, The discovery of A.Calderon's representation theorem, the atomic decomposition of Hardy space and the in-depth study of unconditional basis made theoretical preparations for the birth of wavelet transform, and J.O.Stromberg also constructed 1986. The famous mathematician Y.Meyer accidentally constructed a real wavelet basis, and established a protocol method for constructing wavelet basis in cooperation with S.Mallat. After multi-scale analysis, wavelet analysis began to flourish, among which Ten Lectures on Wavelet written by Belgian female mathematician I.Daubechies played an important role in promoting the popularization of wavelet. Compared with Fourier transform and windowed Fourier transform (Gabor transform), it is a local transformation of time and frequency, so it can effectively extract information from the signal, and solve many problems that Fourier transform can't solve by expanding and translating the multi-scale analysis of the function or signal. Therefore, wavelet transform is known as the "mathematical microscope" and is a landmark progress in the development history of harmonic analysis.
The term wavelet, as its name implies, is a small waveform. The so-called "small" means that it has attenuation; Calling it "wave" refers to its volatility, and its amplitude is in the form of alternating positive and negative shocks. Compared with Fourier transform, wavelet transform is a localized analysis of time (space) frequency. It refines the signal (function) step by step through telescopic translation operation, and finally realizes time subdivision at high frequency and frequency subdivision at low frequency, which can automatically adapt to the requirements of time-frequency signal analysis, thus focusing on any details of the signal and solving the problem of Fourier transform, which has become a major breakthrough in scientific methods since Fourier transform. Some people call wavelet transform "mathematical microscope".
The application of wavelet analysis is closely combined with the theoretical research of wavelet analysis. Now, it has made remarkable achievements in the field of science and technology information industry. Electronic information technology is an important field among the six high technologies, and its important aspects are image and signal processing. Nowadays, signal processing has become an important part of contemporary scientific and technological work. The purpose of signal processing is: accurate analysis, diagnosis, coding compression and quantization, rapid transmission or storage, accurate reconstruction (or recovery). From a mathematical point of view, signal and image processing can be regarded as signal processing (images can be regarded as two-dimensional signals), and in many applications of wavelet analysis, it can be attributed to signal processing. At present, Fourier analysis is still the ideal tool for processing signals with stable properties in practice. However, most of the signals in practical applications are unstable, and the tool especially suitable for unstable signals is wavelet analysis.
Wavelet analysis is a rapidly developing new field in applied mathematics and engineering. After nearly 10 years of exploration and research, an important mathematical formal system has been established, and the theoretical foundation is more solid. Compared with Fourier transform, wavelet transform is a local transformation of space (time) and frequency, so it can effectively extract information from signals. Through expansion and translation, functions or signals can be analyzed in detail at multiple scales, which solves many problems that Fourier transform cannot solve. Wavelet transform involves applied mathematics, physics, computer science, signal and information processing, image processing, seismic exploration and many other disciplines. Mathematicians believe that wavelet analysis is a new branch of mathematics and the perfect crystallization of functional analysis, Fourier analysis, sample modulation analysis and numerical analysis. Experts in signal and information processing believe that wavelet analysis is a new technology of time scale analysis and multi-resolution analysis, and has made achievements in signal analysis, speech synthesis, image recognition, computer vision, data compression, seismic exploration, atmosphere and ocean wave analysis, etc.
In fact, wavelet analysis has a wide range of applications, including: many disciplines in the field of mathematics; Signal analysis and image processing; Quantum mechanics, theoretical physics; Military electronic countermeasures and weapon intelligence: computer classification and identification; 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 in numerical analysis, constructing fast numerical methods, constructing curves and surfaces, solving differential equations, cybernetics and so on. Filtering, denoising, compression and transmission in signal analysis. Image compression, classification, identification and diagnosis, decontamination, etc. in image processing. In medical imaging, the imaging time of B-ultrasound, CT and MRI is reduced and the resolution is improved.
The application of (1) wavelet analysis in signal and image compression is an important aspect of wavelet analysis. Its characteristics are high compression ratio and high compression speed, which can keep the characteristics of signals and images unchanged after compression and can resist interference in transmission. There are many compression methods based on wavelet analysis, such as wavelet packet optimal basis method, wavelet domain texture model method, wavelet transform zerotree compression, wavelet transform vector compression and so on.
(2) Wavelet is also widely used in signal analysis. It can be used in boundary processing and filtering, time-frequency analysis, signal-noise separation and weak signal extraction, fractal index calculation, signal identification and diagnosis, multi-scale edge detection and so on.
(3) Application in engineering technology. Including computer vision, computer graphics, curve design, turbulence, remote universe research and biomedicine.
I don't know much about wavelet analysis The teacher showed us some applications of wavelet analysis. For image processing, we can restore the damaged image. I'm impressed. Two women's images are restored, but the process of wavelet analysis applied to these two images is different. I understand the importance of mathematics as a basic subject and the charm of combining mathematics with computer programs. I suddenly remembered some historical documentaries I had seen before. There will be many ancient damaged cultural relics on it. Scientists put these cultural relics on the computer through some program operations to get the restoration map of cultural relics. At that time, it was amazing. Now think about this "magic" feeling so close to yourself, very happy.
The lecture on frontier knowledge of mathematics not only brought me thinking, but also played a guiding role in my future study, which made me deeply aware of many unsolved problems in mathematics, contacted excellent professors and set an example for myself. I hope I can be as good as a teacher in my future study and make some contributions to the study of mathematics.