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 often simpler and more 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 that 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 complicated 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 drawbacks. Because the sine wave is infinitely wide, the signal to be analyzed also needs to be meaningful from negative infinity to positive infinity. Therefore, Fourier transformation cannot process some local signals well. For example, if a function with a non-zero value in a local range and all other places are equal to 0, its spectrum will be quite messy. 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 developed several methods for transformation using finite-width basis functions. These basis functions change not only in frequency but also in position. These finite-width waves are called wavelet ). Their transformations are called Wavelet
Transforms ).

The concept of wavelet transform is a French engineer engaged in oil signal processing. morlet first proposed in 1974 that it established an inverse formula based on physical intuition and practical experience required for signal processing, which was not recognized by mathematicians at that time. As French thermal engineer J. b. j. fourier proposes that any function can be expanded into an infinite number of trigonometric functions. l. laplace, p.s. laplace and. m. the same is true for legendh. Fortunately, as early as 1970s,. the discovery of Calderon representation theorem, the atomic decomposition of the hard space, and the in-depth study of the unconditional basis make theoretical preparations for the birth of wavelet transformation. o. stromberg also constructed a wavelet base that is very similar in history. In 1986, the famous mathematician y. meyer accidentally constructs a real wavelet base and corresponds to S. after the Mallat cooperation established the consent method for constructing wavelet basis for multi-scale analysis, wavelet analysis began to flourish. ten lectures on wavelet written by Daubechies
Lectures on Wavelets) plays an important role in promoting the popularization of wavelet. Compared with Fourier transform and window Fourier transform (Gabor Transform), it is a local transformation of time and frequency, and thus can effectively extract information from the signal, perform multi-scale detailed analysis on functions or signals through the scaling and shifting functions (multiscale
Analysis), solves many difficult problems that the Fourier transformation cannot solve, so the wavelet change is known as the "Mathematical microscope", which 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. The purpose of signal processing is: accurate analysis, diagnosis, encoding compression and quantification, fast transmission or storage, and precise reconstruction (or restoration ). From the mathematical point of view, signal and image processing can be regarded as signal processing (images can be regarded as two-dimensional signals). In many applications of wavelet analysis, can all be attributed to signal processing problems. At present, the signal of its nature remains stable with practice, and the ideal processing tool is still Fourier analysis. However, the vast majority of signals in practical applications are unstable, and the tool especially suitable for unstable signals is wavelet analysis.

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. In terms of medical imaging, the time for B-ultrasound, CT, and MRI imaging is reduced, and resolution is improved.

(1) Using Wavelet Analysis for signal and image compression is an important aspect 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) wavelet is also widely used in signal analysis. 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)

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. Generally, 8 directions and 5 frequencies are used for the sake of speed and effect. In 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, a foreigner invented the fast Fourier transform algorithm (FFT) decades ago and reduced the originally N * n calculation to n/2 * log2n, this enables fast signal conversion in the frequency and time domains.