The meaning and convolution of Fourier Transform

Very good article!

 

(1) physical significance of Fourier Transformation

Fourier transform is an important algorithm in digital signal processing. But what is the significance of this algorithm?

To understand the significance of the Fourier transform algorithm, we must first understand the significance of the Fourier principle. The Fourier principle indicates that any time series or signal of continuous measurement can be expressed as an infinite superposition of sine wave signals of different frequencies. The Fourier Transform algorithm based on this principle calculates the frequency, amplitude, and phase of different sine wave signals in this signal by accumulating the original signals directly measured.

The inverse Fourier transformation algorithm corresponds to the Fourier transformation algorithm. In essence, this inverse transformation is also a kind of accumulative processing, so that the separately changed sine wave signal can be converted into a signal.

Therefore, it can be said that Fourier transformation converts time-domain signals that were originally difficult to process into frequency-domain signals (signal spectrum) that are easy to analyze. Some tools can be used to process and process these frequency-domain signals. At last, we can use Fourier inverse transformation to convert these frequency-domain signals into time-domain signals.

From the perspective of modern mathematics, Fourier transformation is a special integral transformation. It can represent a function that meets certain conditions as a linear combination or integral of a sine base function. In different research fields, Fourier transformation has different variants, such as continuous Fourier transformation and discrete Fourier transformation.

Fourier transform is the content of harmonic analysis. The word "analysis" can be interpreted as an in-depth study. Literally, the word "analysis" is actually "segmentation. It provides a deep understanding and research on complex functions through "segmentation" of functions. From a philosophical point of view, "analytics" and "restorism" are aimed at enhancing the essential understanding of things through appropriate analysis within things. For example, modern atomic theory attempts to analyze the origins of all things in the world as atoms, while there are only several hundred atoms, which are infinitely richer than the material world, such analysis and classification undoubtedly provide a good way to understand the various properties of things.

This is also true in the field of mathematics. Although Fourier analysis was originally used as a tool for analyzing hot processes, its ideas and methods still have the characteristics of typical theories of reduction and analysis. Any function can be expressed as a linear combination of sine functions through certain decomposition, while sine functions are physically relatively simple and fully researched, this idea is similar to the atomic theory in chemistry! What's amazing is that modern mathematics finds that Fourier transform has a very good nature, making it so useful and useful that people have to lament the magic of creation:

1. Fourier transformation is a linear operator. If an appropriate norm is assigned, it is still a simple operator;

2. The inverse transformation of Fourier transformation is easy to find, and its form is very similar to that of positive transformation;

3. the sine base function is the intrinsic function of the differential operation, so that the solution of the linear differential equation can be converted to the solution of the constant coefficient algebra equation. in a linear physical system, the frequency is a constant property. Therefore, the system's response to complex excitation can be obtained by combining its response to different frequencies;

4. The well-known convolution Theorem points out that Fourier transformation can convert complex convolution operations into simple product operations, thus providing a simple means to calculate convolution;

5. the discrete form of Fourier transformation can be quickly calculated using a digital computer (the algorithm is called the fast Fourier transformation algorithm (FFT )).

Due to the good nature of the above, Fourier transform is widely used in physics, number theory, composite mathematics, signal processing, probability, statistics, cryptography, acoustics, optics, and other fields.

Application in image processing
Fourier transform is the most commonly used transformation in image processing. It is a powerful tool for image processing and analysis.

Mathematical definition of Fourier Transform
The Traditional Fourier transformation is a pure frequency domain analysis. It can represent the general function f (x) as the weighted sum of a cluster of standard functions, and the weight function is also the Fourier transformation of F. If F is a real value or complex value function on R, F is a simulated signal with limited energy. The specific definition is as follows:

2. Physical Meaning of image Fourier Transformation

The image frequency is an indicator that represents the intensity of gray scale changes in the image, and is a gradient of gray scale in the plane space. For example, a large desert area is a area with slow gray-scale changes in the image, and the corresponding frequency value is very low; edge areas with sharp changes in surface properties are areas with intense gray changes in the image, and the corresponding frequency value is high. Fourier transformation has obvious physical significance in practice. If f is a simulated signal with limited energy, its Fourier transformation represents the spectrum of f. In purely mathematical sense, Fourier transformation converts a function into a series of periodic functions. From the physical effect, Fourier transform converts an image from the spatial domain to the frequency domain. Its inverse transformation is to convert the image from the frequency domain to the spatial domain. In other words, the physical meaning of Fourier transformation is to transform the gray-scale distribution function of the image into the image's frequency distribution function, and the Fourier inverse transformation is to transform the image's frequency distribution function into a gray-scale distribution function.

(2) Significance of convolution
This is still well written in Cambridge courseware. By the way, I posted it ~

Convolution

Several important optical effects can be described in terms of convolutions.

Let us examine the concepts using 1D continuous functions.

The convolution of two functionsF(X) AndG(X), WrittenF(X)*G(X), Is defined by the integral

For example, let us take two top hat functions of the type described earlier. Let be the top hat function shown in Fig. & <60; 11,

And let be as shown in Fig. & <60; 13, defined

Fig.13 Another top hat:

 

• Is the reflection of this function in the vertical axis,
• Is the latter shifted to the right by a distanceX.
• Thus for a given valueX, Integrated over all is the area of overlap of these two top hats, as has unit height.
• An example is shownXIn the range in Fig. & <60; 14.

Fig.14 Convolving two top hats

If we now considerXMoving from to, we can see that

• For or, there is no overlap;
• AsXGoes from-1 to 0 the area of overlap steadily increases from 0 to 1/2;
• AsXIncreases from 0 to 1, the overlap area remains at 1/2;
• And finallyXIncreases from 1 to 2, the overwriting area steadily decreases again from 1/2 to 0.
• Thus the convolutionF(X) AndG(X),F(X)*G(X), In this case has the form shown in Fig. & <60; 15,

Fig.15 Convolution of two top hats