Fourier transform and Wavelet Analysis

Both Fourier transform and wavelet transform are essentially the same. They convert signals between time and frequency domains and find some intuitive information from seemingly complex data, then analyze it. Because the signal is more simple and intuitive in the frequency domain than in the time domain,Most of the signal analysis work is carried out in the frequency domain.. Music is an excellent example of time/frequency analysis. Music score is the signal distribution of music in the frequency domain, and music is a function after the music score is transformed to the time domain. From music to music, it is a Fourier or wavelet transformation; from music to music, it is a Fourier or wavelet inverse transformation.

1, Fourier Transformation

It can be understood:Any meaningful curve in the real number field can be decomposed into the superposition of several sine curves.The principles of Fourier transformation and fragment are actually the same source. Do not underestimate this transformation. It may be the most basic principle of the universe. In other words:No matter how complex a substance is, several simple basic substances can be combined in a certain way.This is actually the case when it comes to molecular atoms, topography, and the Galaxy universe.

However,Fourier transformation also has its defects.. Because the sine wave is infinitely wide, the signal to be analyzed also needs to be meaningful from negative infinity to positive infinity.Fourier transform cannot process some local signals very well.. For example, a non-0And all other places are equal0Function, its spectrum will present a very chaotic situation. At this time, the signal in the frequency domain is not as intuitive as the time domain, and the spectrum analysis becomes very difficult.

2, Wavelet analysis

To overcome these defects of Fourier transformation, scientists and engineers have developedSeveral methods of using the finite-width Basis Function for Transformation. These basis functions change not only in frequency but also in position. These finite-width waves are called"Wavelet"(Wavelet). Their transformations are called"Wavelet Transform"(Wavelet
Transforms).

The concept of wavelet transform is a French engineer engaged in oil signal processing. J. Morlet In 1974 First proposed by the year, an inverse formula was established through the physical intuition and the practical experience required for signal processing, which was not recognized by mathematicians at that time. As 1807 French thermal Engineer J. B. J. Fourier Proposal Any function can be expanded into an infinite series of trigonometric functions. The concept of innovation cannot be obtained by famous mathematicians. J. L. Laplace , P.s. Laplace And A. M. legendh . Fortunately, as early as 1970s, A. Calderon Represents theorem discovery, Hard The atomic decomposition and unconditional basis of space have made theoretical preparations for the birth of wavelet transform. J. O. Stromberg It also constructs a wavelet base that is very similar to the current one in history; 1986 Famous mathematician of the year Y. Meyer Construct a real wavelet base by accident and S. Mallat After the establishment of the consent method for constructing wavelet basis, the wavelet analysis began to flourish. I. Daubechies Written 《 Wavelet 10 lecture ( Ten
Lectures on Wavelets ) It has played an important role in promoting the popularization of wavelet. It corresponds Fourier Transformation, window Fourier Transform ( Gabor In contrast, this is a local transformation of Time and Frequency. Therefore, it can extract information from the signal effectively, and perform multi-scale refined analysis on functions or signals through scaling and shifting operations ( Multiscale
Analysis ), Solved Fourier Transformation cannot solve many difficult problems, so the wavelet change is called " Mathematical Microscope " It is a milestone in the history of harmonic analysis.

The Application of Wavelet Analysis is closely integrated 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 in the six major high-tech fields, and its important aspect is image and signal processing. Nowadays, signal processing has become an important part of contemporary scientific and technological work,Purpose of Signal ProcessingIs: accurate analysis, diagnosis, encoding compression and quantification, fast transmission or storage, precise reconstruction (or recovery ). From the mathematical point of view, signal and image processing can be regarded as signal processing (The image can be viewed as a two-dimensional signal.In many applications of wavelet analysis, it can be attributed to signal processing problems. Now,The ideal processing tool is still Fourier analysis.. HoweverThe vast majority of signals in practical applications are unstable., AndWavelet Analysis is a tool especially suitable for non-stable signals..

In fact, wavelet analysis is widely used in many fields, including mathematics, signal analysis, image processing, quantum mechanics, theoretical physics, and intelligence of Military Electronic Confrontation and weapons; computer classification and recognition; artificial synthesis of music and language; Medical Imaging and diagnosis; seismic exploration data processing; Fault Diagnosis of large machinery; for example, in mathematics, it has been used for numerical analysis, Fast Numerical Method construction, curve and surface construction, solving differential equations, and control theory. Filtering, de-noise, compression, transmission, and so on in signal analysis. Image compression, classification, recognition and diagnosis, and decontamination in image processing. Reduction in medical imagingBChao,CT, MRI imaging time, improve resolution, etc.

(1)Wavelet Analysis for signal and Image CompressionIt is an important aspect of the application of wavelet analysis. It features a high compression ratio and high compression speed. After compression, it can keep the signal and image features unchanged, and it can resist interference during transmission. There are many compression methods based on wavelet analysis. The most successful methods include the best base method of wavelet packet, the wavelet domain texture model method, the wavelet transform zero tree compression, and the wavelet transform vector compression.

(2)Application of Wavelet in Signal AnalysisIt is also very extensive. It can be used for border processing and filtering, time-frequency analysis, signal-to-noise separation and extraction of weak signals, determination of fragment index, signal recognition and diagnosis, and multi-scale edge detection.

(3)Applications in engineering and other aspects. Including computer vision, computer graphics, curve design, turbulence, remote universe research and biomedicine.

 

4,Gabor Filter(GaborFilter)

Gabor wavelet is a type of wavelet set. For the formula, see (the following link ). In terms of shape, Gabor wavelet is encapsulated in a gaussion distribution shape and has zero points. If you want to display the Gabor filter in three dimensions, it should look like the following. The left side is the real part (even function), and the right side is the imaginary part (odd function ).Http://cs.ccnu.edu.cn/wjylwjyl/kjqy/paper/Recognition.htm

A set of different Gabor filters can be obtained by changing the phase and wavelength of K.Generally8Direction and5FrequencyIn this way, a total of 5x8 = 40 different Gabor filters can be generated. For an image with a size of X, the minimum and maximum frequencies are 16 and 4 pixels respectively.

4. Jet

Convolution of the original image and each Gabor filter will produce 40 results (note that the real and virtual parts must be separated before convolution ). For each input pixel, 40 output complex numbers are generated. We sort these 40 complex numbers in the filter order, which is a jet.

How to perform convolution? For each pixel, the two images are staggered for a certain distance, and the overlapping pixels are multiplied and then accumulated. Of course, don't forget to have a classic formula: Time Domain convolution = frequency-domain multiplication. You only need to first convert the image to the frequency domain, multiply it, and then reverse transform it back to the time domain. Fortunately, foreigners invented the fast Fourier transform decades ago.Algorithm(FFT) reduces the original N * n calculation to n/2 * log2n, making the signal conversion in the frequency and time domain very fast.