From the perspective of modern mathematics, Fourier transform is a special integral transformation. It can represent a function that satisfies a certain condition as a linear combination or integral of the sine basis function. In different fields of study, Fourier transform has many different forms, such as continuous Fourier transform and discrete Fourier transform.
Fourier transform belongs to the content of harmonic analysis. The word "analysis" can be interpreted as an indepth study. Literally, the word "analysis" is actually "truthfully". It achieves the deep understanding and research to the complex function by the "truthfully" of the function. Philosophically, "Analytic doctrine" and "reductionism", is to improve the understanding of the essence of the object through the proper analysis of the internal things. For example, modern atomic theory attempts to analyze the origin of all the material in the world as atoms, but only hundreds of of atoms, relative to the infinite richness of the material world, this kind of analysis and classification undoubtedly provides a good means to recognize the various natures of things.
In the field of mathematics, it is the same, although the initial Fourier analysis as an analytical tool for thermal processes, but its methods of thinking still has a typical reductionism and analytical features. The "arbitrary" function can be expressed as a linear combination of sine function by a certain decomposition, and the sine function is fully studied and relatively simple function class, which is similar to the idea of atomic theory in chemistry. Amazingly, modern mathematics finds that Fourier transforms have a very good character, making it so useful and helpful that people have to lament the wonders of creation:
1. Fourier transform is a linear operator, and if given the proper norm, it is also a unitary operator;
2. The inverse transformation of Fourier transform is easy to find, and the form is very similar to the positive transformation;
3. The sine basis function is the intrinsic function of the differential operation, thus, the solution of linear differential equation can be transformed into algebraic equation of constant coefficient. In a linear time invariant physical system, the frequency is an invariant property, so the response of the system to the complex excitation can be obtained by combining its response to the different frequency sinusoidal signals;
4. The famous convolution theorem points out that Fourier transforms can transform complex convolution operations into simple product operations, thus providing a simple means of calculating convolution;
5. The discrete form of Fourier transforms can be computed quickly using a digital computer (the algorithm is called a fast Fourier transform (FFT) algorithm).
Because of the good nature of the above, Fourier transform has been widely used in physics, number theory, combinatorial mathematics, signal processing, probability, statistics, cryptography, acoustics, optics and other fields.
Fourier transform plays a very important role in image processing.
Fourier transform plays a very important role in image processing. Because not only Fourier analysis involves many aspects of image processing, Fourier's improved algorithm, For example, the discrete cosine transform, Gabor and wavelet in the image processing also has important components. In the impression, the Fourier transform has an important effect on the following topics in image processing: 1. Image enhancement and image denoising Most of the noise is the highfrequency component of the image, through Lowpass filter to remove highfrequencynoise; The edge is also the high frequency component of the image, which can enhance the edge of the original image by adding high frequency components. 2. Edge Detection of image segmentation Extracting high frequency component of image 3. Image feature Extraction: Shape characteristics: Fourier descriptor Texture features: Calculating texture features directly by Fourier coefficients Other features: A Fourier transform of the extracted eigenvalues to enable translation, scaling, and rotation invariance of the feature 4. Image compression The data can be compressed directly by Fourier coefficients, and the common discrete cosine transform is the real transformation of Fourier transform. Fourier transform Fourier transform is to decompose the time domain signal into the sum of the sinusoidal signal or cosine function superposition of different frequencies. In continuous case, the original signal is required to satisfy the absolute integrable condition in one cycle. In discrete cases, the Fourier transform must exist. Gonzalez < image processing > The explanation is very vivid: an appropriate analogy is to compare the Fourier transform to a glass prism. A prism is a physical instrument that decomposes light into different colours, and the color of each component is determined by the wavelength (or frequency). The Fourier transform can be regarded as a mathematical prism, and the function is decomposed into different components based on frequency. When we consider the light, we discuss its spectral or frequency spectrum. Similarly, Fourier transforms enable us to analyze a function by the frequency component. Fourier transforms have many excellent properties. such as linearity, symmetry (which can be used in the Fourier transform of the computed signal); Timeshift: The time shift of function in time domain corresponds to the phase shift in the frequency domain, while the amplitude spectrum is unchanged. Frequency Shift: The function is multiplied by the E^JWT in the time domain, which enables the entire spectrum to move W. This is also called modulation theorem, communication inside the signal frequency division multiplexing needs to use this feature (modulation of different signals to different frequencies at the same time transmission); Convolution theorem: Time domain convolution equals frequency domain product, time domain product equals frequency domain convolution (attach a coefficient). (Image processing This is an important point) The performance of signal in frequency domain In frequency domain, the greater the frequency, the faster the original signal changes; the smaller the frequency, the smoother the original signal. When the frequency is 0 o'clock, the DC signal is indicated, unchanged. Therefore, the frequency of the size of the signal to reflect the speed of change. The Highfrequency component interprets the abrupt part of the signal, and the Lowfrequency component determines the overall image of the signal. In the image processing, the frequency domain reflects the intensity of the image in the spatial level, that is, the change speed of the image grayscale, i.e. the gradient size of the image. For the image, the edge of the image is a mutation part, the change is fast, so the response in the frequency domain is high frequency component; the noise of the image is mostly high frequency, and the part of the image is low frequency component. In other words, the Fourier transform provides another angle to observe the image, it can transform the image from the gray distribution to the frequency distribution to observe the characteristics of the image. In a written note, Fourier transforms provide a way to free conversion from airspace to frequency. For image processing, the following concepts are very important: Image High Frequency component: The image mutation part, in some cases refers to the image edge information, in some cases refers to the noise, more is the mixture of the two; Lowfrequency component: the part of the image that changes gently, namely the image contour information Highpass filter: Allows the image to suppress lowfrequency components, highfrequency components through Lowpass filter: Contrary to Qualcomm, let the image of the Highfrequency component suppression, Lowfrequency components through Band pass filter: Make the image in a certain part of the frequency of information through the other too low or too high are suppressed There is also a bandstop filter, is a bandpass reverse. Template operation and convolution theorem In the time domain to do the template operation, is actually to the image convolution. Template operation is an important process of image processing, many image processing processes, such as enhancement/denoising (these two are not clear), edge detection is commonly used. According to convolution theorem, the time domain convolution equivalence and frequency domain product. Therefore, in the time domain for the image template operation is equivalent to the image in the frequency domain to do filter processing. For example, a mean template, its frequency domain response to a lowpass filter, in the time domain to the image of the average filter is equivalent to the image in the frequency domain of the average template response to the frequency domain response to the image of a lowpass filter.
Image denoising Image denoising is to suppress the noise part of the image. Therefore, if the noise is high frequency, from the point of view of frequency domain, it is necessary to use a lowpass filter to deal with the image. The high frequency component of the image can be suppressed by Lowpass filter. But in this case, it often leads to the suppression of edge information. Common denoising templates have meanvalue templates, Gaussian templates and so on. Both of these filters suppress the high frequency component of the image in the local area, and suppress the noise while blurring the edge of the image. There is also a nonlinear filtermedian filter. The median filter is well removed from the pulse type noise. Because the pulse point is the point of mutation, the order of the output after the value, then the maximum and minimum points can be removed. The median filter has a poor effect on Gaussian noise.
Salt and pepper noise: for salt and pepper using median filter can be very good removal. The mean value can also achieve a certain effect, but it will cause blurred edges. Gaussian white Noise: white noise in the entire frequency domain are distributed, it seems more difficult. Gonzalez version image processing P185: Arithmetic mean filter and geometric mean filter (especially the latter) are more suitable for dealing with Gaussian or even random noises. Harmonic mean filter is more suitable for the processing of impulse noise. Image enhancement Sometimes feel image enhancement and image denoising is a contradictory process, image enhancement is often needed to enhance the edge of the image to obtain better display effect, this need to increase the high frequency components of the image. And the image denoising is to eliminate the noise of the image, that is, the need to suppress highfrequency components. Sometimes these two refer to something similar. For example, when the noise is eliminated and the display of the image is significantly improved, then this is the same meaning. The common image enhancement methods are contrast stretching, histogram equalization, image sharpening and so on. The first two are pixel based transformations in the airspace, and the latter one is processed in the frequency domain. I understand the sharpening is directly on the image plus the image of high pass filter after the component, that is, the edge of the image effect. Contrast stretching and histogram equalization are in order to improve the contrast of the image, that is, to make the image look more obvious difference, I think, after such processing, the image should also enhance the high frequency component of the image, making the image details of the difference is greater. At the same time, some noise is introduced.

The physical meaning of Fourier transform in image
The frequency of the image is the index of the intensity of the gray change in the image, and is the gradient of the gray level in the plane space. For example: A large area of the desert in the image is a slow change in gray area, the corresponding frequency is very low, but for the Surface property transformation of the edge region is a drastic change in the image of the region, the corresponding frequency value is higher.
Fourier transform has a very obvious physical meaning in practice, and the f is an analog signal with finite energy, and its Fourier transform represents F spectrum. From a purely mathematical point of view, Fourier transforms are processed by converting a function into a series of periodic functions. From the physical effect, Fourier transform is to transform the image from the space domain to the frequency domain, and its inverse transformation is to transform the image from the frequency domain to the space domain. In other words, the physical meaning of the Fourier transform is to transform the gray distribution function of the image into the frequency distribution function of the image, and the Fourier inverse transform is to transform the frequency distribution function of the image into the gray distribution function.
Before Fourier transforms, an image (an uncompressed bitmap) is a collection of points from the sampling on the continuous space (the real space), and we are accustomed to representing each point in the space with a twodimensional matrix, the image can be represented by z=f (x,y). Because the space is threedimensional, the image is twodimensional, so the relationship of the object in the other dimension is represented by the gradient, so that we can see the corresponding relation of the object in the threedimensional space by observing the image.
Why do you want to raise the gradient. Because in fact, the twodimensional Fourier transform of the image to get the spectrum map, is the image gradient distribution, of course, the spectrum of points and the image of the various points does not exist one by one of the corresponding relationship, even in the absence of frequency shift is not. On the Fourier spectrum, we see the bright and dark dots, in fact, the difference between a point on the image and the neighborhood points, that is, the size of the gradient, that is, the size of the frequency of the point (it can be understood that the lowfrequency part of the image refers to the lower gradient of the point, Generally speaking, the gradient is large, the brightness of the point is strong, otherwise the point brightness is weak. So by looking at the Fourier transform spectrum, also called power graphs, we can first see that the energy distribution of the image, if the spectrum of dark points more, then the actual image is relatively soft (because each point and neighborhood difference is not small, gradient relatively small), on the contrary, if the spectrum of the number of bright points, Then the actual image must be sharp, the boundary is distinct and the pixels on both sides of the boundary are big difference. After frequency spectrum shift to the origin, we can see that the frequency distribution of the image is centered on the origin and symmetrically distributed. In addition to clearly seeing the frequency distribution of the spectrum, it is possible to shift frequency to the center there is also a benefit, it can be separated from the periodic rules of interference signals, such as sinusoidal interference, a sinusoidal interference, frequency shift to the origin of the spectrum can be seen in addition to the center of a certain point as the center, symmetrical distribution of bright points, This set is to interfere with the noise generated, at this point can be very intuitive by placing a bandstop filter in this position to eliminate interference.
In addition, I would like to explain the following points:
1, after the image after the twodimensional Fourier transform, the transformation coefficient matrix shows:
If the FN origin of the transformation matrix is located in the center, its spectral energy is concentrated near the center of the transformation coefficient short array (shaded area in the figure). If the original point of FN of the twodimensional Fourier transform matrix is set in the upperleft corner, then the image signal energy will be concentrated on the four corners of the coefficient matrix. This is determined by the nature of the twodimensional Fourier transform itself. At the same time, it also shows that the low frequency region of the image energy concentration.
2, after the transformation of the image in the original point before the four Corners are lowfrequency, the brightest, the middle part of the translation after the lowfrequency, the brightest, the brightness of the large low frequency of energy (amplitude angle is relatively large)
The general idea may be that the analytic function is part of a harmonic function, while the harmonic function in the disk can be expressed as an integral of the limit on its circumference, and this place seems to have some connection with the rich transformation. You have a very good problem, and I have been thinking about this kind of problem.
The derivation way of Laplace transformation:
1, from the mathematical point of view: through the integral transformation function to function transformation, the differential equation into algebraic equations.
2, from the physical meaning of derivation: the essence is still the signal decomposition into a number of orthogonal signal and (integral), or can be generalized from the FT.
By deriving the Laplace transform from Fourier transform, the physical meaning can be explained more clearly, and the two transformations can be closely connected.
The Laplace transform provides a way to transform domain definitions, mapping a signal (function) defined on a time domain to a complex frequency domain (to understand this sentence, you need to understand the concept of the function spacewe know that the function defines a relationship between an element of a set and an element of another set. The combination of two or more functions is the function space, that is, the function space is also a set; the "definition domain" of Laplace transform, which is the function space, can be said that Laplace transform is a function of processing functions. Because the Laplace transform is so cleverly defined, it has some peculiar qualities, and it is a corresponding relationship (as long as the convergent domain of the complex frequency domain is given), so long as a given time domain function (signal), It can transform to a complex frequency domain signal (whether this signal is a real signal or a complex signal) via Laplace transform, thus, as long as we deal with this complex frequency domain signal, it is equivalent to processing the time domain signal (for example, F (t) ←→f (s), re[s]>a, if we are to F (s) For delay processing, to get the signal f (sz), re[s]>a+re[z], then we multiply the time domain function by a rotational factor e^zt, that is, F (t) e^zt←→f (SZ),re[s]>a+re[z]; as long as the F (sz) is reversed , you can get an F (t) e^zt).
Laplace transform is used to solve the differential equation, mainly by applying the properties of Laplace transform to solve the differential equation to solve the algebraic equations (because it is much easier to solve the equation than to solve the differential equations). Furthermore, the solution of the original differential equation can be obtained by Laplace inverse transformation of the result (easily).
We can always easily draw images of realvariable functions (most functions do), but it is difficult to draw the image of a complex function, which may be one of the reasons why Laplace transforms are more abstract, and the other is that the complex frequency s in Laplace transform has no definite physical meaning.
As for the feature root and complex number, it should be easy to understand that the questioner should take another look at the definition in the book.