Principles and physical significance of Fourier transform for images

Original article address:[Conversion] principles and physical significance of Fourier transformation of images Author:Uleen

The Fourier transformation of the image. The original image consists of N rows and n columns, and N must be base 2. This N * n points containing the image are called real parts, N * n points are also required to be called virtual parts, because the FFT is based on the plural, as shown in:

      (// Real-number DFT converts N points in the time domain into N/2 + 1 points in the two groups in the frequency domain (corresponding to the real part and the virtual part respectively ))

The process of calculating the image Fourier transformation is very simple: first, one-dimensional FFT is performed for each row, and then one-dimensional FFT is performed for each column. Specifically, first perform FFT (the real part has a value, and the virtual part is 0) on the N points of The 0th rows, and put the real part of the FFT output back to the real part of the original 0th rows, the imaginary part of the FFT output is put back to the imaginary part of The 0th rows. After the full part is calculated, the real and imaginary parts of the image contain intermediate data, then, use the same method to perform the FFT transformation on the column direction, so that the N * n image passes through the FFT to obtain a n * N spectrum.

The following shows the two-dimensional FFT transformation of an image:

Negative values can be contained in the frequency field. Gray indicates 0 in the image, black indicates negative values, and white indicates positive values. It can be seen that the black on the four corners is darker and the white is more white, indicating a larger margin. In fact, the coefficients on the four corners represent the low-frequency components of the image, the center is the high-frequency component of the image. In addition, the FFT coefficient is disorganized and hardly visible.

The Cartesian coordinates are converted to polar coordinates, which is easier to understand. The white areas on the four corners of the amplitude indicate a large margin, in phase, there is basically no difference between high frequency and low frequency.

The above shows the image spectrum in a different way, which translates the low frequency part to the center of the spectrum (// implement the function fftshift in MATLAB ). This is actually quite understandable, because the signals obtained through 2d-fft are discrete images, and the output of 2d-fft is a periodic signal, that is, the previous figure is periodically tiled, the image is centered on low frequencies. There are many benefits to placing the origin point in the center, such as being more intuitive and more cyclical, but in this section we will explain it with the diagram before the translation.

Rows n/2 and columns n/2 divide the frequency domain into four blocks. For the real part and amplitude, the upper-right and lower-left sides form an image relationship, and the upper-left and lower-right sides also form an image relationship. For the virtual part and phase, it is similar, the only difference is that the symbols must be reversed ?), This symmetry is similar to the 1-dimensional Fourier transformation. You can look forward.

For the sake of simplicity, we should first consider 4*4 pixels, and the right side is its gray value, which is a two-dimensional FFT transformation of these gray values.

The range of H and K is-n/2 to n/2-1.

Generally, I (n, m) is a real number, F () is always a real number (// dc component), and F (H, k) is parity.

If it is written in the plural form, that is:

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Physical Meaning of image Fourier Transformation

If only the amplitude close to the center is retained, the image details are lost, but different regions still have different gray scales.

If the range is left away from the center, the image details can be seen, but the gray scales of different regions are the same.

Consider the Fourier transformation of a black rectangle. The background of this black rectangle is white.

If the high-frequency components in the vertical direction in the frequency domain are truncated, the black and white components in the image are not so clear, and the result is an oscillation. (//???)

We can draw a conclusion:

The closeness of the Fourier transform coefficient to the Center describes the characteristics of slow changes in the image, or the characteristics of relatively slow gray-scale transformation (slow frequency );

The far-from-center Fourier transform coefficient describes the fast-changing feature of the image, or the intense feature of the gray-scale transform (the faster part ).

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Information contained in the Fourier transform phase

There are two images. If we use the first image's Fourier transformation amplitude and the second image's Fourier transformation phase for inverse transformation, what will the image look like?

If, in turn, the first image's phase and the second image's amplitude are reversed, what is the image like?

Here we will use the 1-dimensional Fourier transformation to explain:

In the 1D Fourier transformation, we can see that the phase contains the information about when the edge appears! In the same way in the Fourier transformation of the image, the phase determines the edge of the image, so it determines the way you see the object in the image!

You can understand the information contained in the phase as follows:

The edge is formed when many rising edges of the sine wave occur at the same time, that is, the phase of these sine waves is the same time. Therefore, the information contained in the phase determines the position of the edge, the edge determines the image.

This is a difference between an image signal and a sound signal. Most of the information of a sound signal is included in the amplitude of its Fourier transformation, that is, the magnitude of different frequencies, that is to say, what you hear depends on the frequency of the signal you hear, and it is not very important for these signals when and when they happen.

[Note: // Add comments for yourself later]

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