Ne Gonzalez's explanation of image processing is vivid: an appropriate metaphor is the glass prism of Fourier transform. Prism can be a physical device that decomposes light into different colors, and the color of each component is determined by wavelength (or frequency).

Fourier transform can be regarded as a function on the mathematical prism, which is based on the frequency decomposition of different components. When we consider light, we discuss its spectrum or frequency spectrum. Similarly, through the frequency component analysis of Fourier transform function.

Physical Meaning of Fourier Transform of Image

The frequency intensity of an image is the gray change of the image, and it is a characterization index of the gray gradient in plane space. For example:? The desert is a large area in the image, and the gray level changes slowly, corresponding to low frequency values; For the region where the gray level of the edge region with strong surface characteristic transformation changes in the image, there is a corresponding value with high frequency. Fourier transform has obvious physical significance in practice. Let f be the frequency spectrum of finite energy analog signal and f be Fourier transform. In a purely mathematical sense, the Fourier transform of a function is regarded as a series of periodic functions. Physically speaking, Fourier transform is the inverse transformation of an image from spatial domain to frequency domain, and the image is transformed from frequency domain to spatial domain. In other words, the image gray distribution function of Fourier transform in physical sense is to transform the image of frequency distribution function of inverse Fourier transform into gray distribution function.

/& gt； Before Fourier transform, a series of images (uncompressed bitmaps) of continuous space (real world space) on the acquisition point are obtained from the samples. We are used to using two-dimensional matrix to represent points, and the image can be Z = F (x, y) in space. Because space is a three-dimensional and two-dimensional image, and gradient represents the relationship of objects in another dimensional space, we can learn the corresponding relationship of objects in three-dimensional space through observed images. Why mention the gradient? Because in fact, the spectrogram obtained by Fourier transform of a two-dimensional image, that is, the distribution map of image gradient, of course, does not correspond to every point in the spectrogram of every point on the image, even if the frequency is fixed. Fourier spectrum diagram, we can see a bright spot with different light and dark, which is actually an image near a small point. This is the difference of intensity, the size of gradient, and the point of frequency, that is, the size (in order to understand the low gradient of the low-frequency part of the image, the high-frequency part is the opposite). Generally speaking, points with large gradient and brightness or points with weak brightness. Therefore, by observing the spectrogram after Fourier transform, that is, the so-called power diagram, we can see that, first of all, the energy distribution of the image, if many points in the spectrogram are dark, then the actual image is soft (because there is little difference between each point and the neighborhood, a relatively small gradient), on the contrary, if the brightest point in the spectrogram, then the actual image must be clear, with sharp boundaries and large boundaries on both sides. When the frequency shifts to the origin of the spectrum, we can see that its frequency distribution is symmetrical based on the origin of the image center. At the center of spectral frequency shift, we can clearly see the frequency distribution of the external image, which has the advantage that the separated interference signal, such as sinusoidal interference, can be moved to the origin. In addition, there is an acquisition point outside the center that is symmetrical with the periodic law of sinusoidal interference. As can be seen from the spectrum of bright spots, this set is a place where noise generated by interference can be eliminated by placing band-stop filters.

In addition, I would like to make the following points:

For the image after 2D Fourier transform, the transformation coefficient matrix is as follows:

The Fn origin of the transformation matrix is located in the city center, and its spectral energy is concentrated near the center of the discontinuous transformation coefficient (shaded area in the figure). If the origin of the two-dimensional Fourier transform matrix Fn is located in the upper left corner, the image signal energy is concentrated in the four corners of the coefficient matrix. This is determined by the nature of the two-dimensional Fourier transform itself. It also shows images of low frequency areas with concentrated energy.

2. The four corners of the converted image before the origin translation are the brightest in low frequency, and the middle part after panning is the brightest and brightest low frequency energy (large amplitude angle).