The physical meaning of Fourier transform

1, why the Fourier transform, what is the physical meaning of it?

Fourier transform is a very important algorithm in the field of digital signal processing. To know the meaning of Fourier transform algorithm, we must first understand the meaning of Fourier principle. Fourier principle shows that any continuous measurement of the timing or signal, can be expressed as a different frequency sine wave signal infinite superposition. The Fourier transform algorithm based on this principle is used to calculate the frequency, amplitude and phase of different sine wave signals in this signal by using the original signals directly measured.

and Fourier transform algorithm is corresponding to the inverse Fourier transform algorithm. The inverse transformation is essentially an additive process, so that a single changed sine wave signal can be converted into a single signal.

Therefore, it can be said that Fourier transform the original difficult to process the time-domain signal into an easy to analyze the frequency domain signal (signal spectrum), can use some tools to these frequency domain signal processing, processing. Finally, Fourier inverse transform can be used to convert these frequency domain signals into time domain signals.

From the perspective of modern mathematics, Fourier transform is a special integral transformation. It can represent a function that satisfies certain conditions as a linear combination or integral of a sinusoidal basis function. In different research fields, Fourier transforms have many different variants, such as continuous Fourier transform and discrete Fourier transform.

In the field of mathematics, although the original Fourier analysis is a tool for analytical analysis of the thermal process, its method of thinking still has the characteristics of typical reductionism and analysis. The function of "arbitrary" can be expressed as a linear combination of the sine function by certain decomposition, while the sine function is physically fully researched and relatively simple function class:

1. Fourier transform is a linear operator, if the appropriate norm is given, it is also unitary operator;

2. The inverse transformation of Fourier transform is easy to find, and the form and positive transformation are very similar;

3. Sine basis function is the Eigen function of differential operation, so that the solution of the linear differential equation can be transformed into the solution of the algebraic equation with constant coefficients. The convolution operation with constant clutter is a simple product operation, thus providing a simple method for calculating convolution.

4. The well-known convolution theorem points out that the Fourier transform can be transformed by a digital computer to quickly calculate (its algorithm is called Fast Fourier transform algorithm (FFT)).

5. In the discrete form of Fourier's physical system, the frequency is a constant property, so the response of the system to the complex excitation can be obtained by combining its response to the sinusoidal signals of different frequencies;

Because of the good properties mentioned above, Fourier transform has been widely used in physics, number theory, combinatorial mathematics, signal processing, probability, statistics, cryptography, acoustics, optics and other fields. http://hovertree.com/

2. The physical meaning of image Fourier transform

The frequency of

        image is the index of the intensity of gray intensity in the image, and it is the gradient of the gray level in the plane space. such as: a large area of the desert in the image is a region of slow gray-scale changes, the corresponding frequency value is very low, and for the Surface attribute transformation of the sharp edge region in the image is a sharp change in the area of gray, the corresponding frequency value is higher. Fourier transform has a very obvious physical meaning in practice, and f is an analog signal with limited energy, then its Fourier transform represents the spectrum of F. From a purely mathematical point of view, Fourier transforms a function into a series of periodic functions to be processed. From the physical effect, the Fourier transform transforms the image from the spatial domain to the frequency domain, and its inverse transformation is to convert the image from the frequency domain to the spatial domain. In other words, the physical meaning of Fourier transform is to transform the image's gray distribution function 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 transform, an image (uncompressed bitmap) is a set of points that are sampled on a continuous space (real space), and we are accustomed to using a two-dimensional matrix to represent space points, the image can be represented by z=f (x, y). Since space is three-dimensional, the image is two-dimensional, so the relation of the object in the space is represented by the gradient, so that we can observe the image to know the correspondence of the object in three-dimensional space. Why do you want to raise gradients? Because in fact, the image of the two-dimensional Fourier transform to get the spectrum map, is the image gradient distribution map, of course, the spectrum map of the points and the image of the points there is no one by one corresponding relationship, even in the case of non-shift frequency is not. Fourier spectrum Map We see the bright and dark light, the actual image of a point and the difference between the neighboring points of strength, that is, the size of the gradient, that is, the frequency of the point (so to understand that the low-frequency part of the picture refers to the lower gradient point, high-frequency part of the opposite). Generally speaking, the gradient is large, the brightness of the point is strong, otherwise the point brightness is weak. In this way, by observing the Fourier transform Spectrum Map, also known as the Power map, we can first see that the image of the energy distribution, if the spectral map of the number of darker points, then the actual image is relatively soft (because the difference between the point and the neighborhood is not small, the gradient is relatively small), conversely, if the spectrum map, Then the actual image must be sharp, the boundary is clear and the boundary between the pixel difference is large. After the spectrum shift to the origin, it can be seen that the frequency distribution of the image is centered on the origin and symmetrically distributed. Shift the spectrum to the center of the circle in addition to clearly see the image frequency distribution, there is also a benefit, it can be separated by periodic rules of the interference signal, such as sinusoidal interference, a pair with sinusoidal interference, frequency shift to the origin of the spectrum map can be seen in addition to the center of a point as the center, symmetrical distribution of the highlight set, This set is caused by interfering noise, which can be intuitively eliminated by placing a band-stop filter at that position.

Note:

1, the image after two-dimensional Fourier transform, its transformation coefficient matrix shows:

If the shift matrix FN origin is located in the center, its spectral energy concentration is located near the center of the short matrix of the transformation coefficients (shaded area in the image). If the origin of the FN of the two-dimensional Fourier transform matrix is located in the upper-left 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 two-dimensional Fourier transform itself. At the same time, it is shown that the energy concentration in low frequency region.

2, after the transformation of the image at the origin of the shift before the four corners are low-frequency, the brightest, the middle part of the translation after the low-frequency, the brightest, the brightness of large low-frequency energy (large angle)

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The physical meaning of Fourier transform