Fourier transform in image processing (idle Version)

Fourier transform plays a very important role in image processing. Because not only does Fourier analysis involve many aspects of image processing, Fourier's improved algorithm,

For example, the discrete cosine transformation, Gabor and wavelet also have important components in image processing.

In the impression, Fourier transform plays an important role in the following topics of image processing:
1. Image Enhancement and Image Denoising
Most of the noise is the high-frequency component of the image. The low-pass filter is used to filter out high-frequency noise. The edge is also the high-frequency component of the image. You can add high-frequency components to enhance the edge of the original image;
2. Edge Detection of Image Segmentation
Extract high-frequency image Components
3. Image Feature Extraction:
Shape Feature: Fourier Descriptor
Texture features: texture features are calculated directly using Fourier coefficients.
Other features: Perform Fourier transformation on the extracted feature values to make the feature translate, scale, and rotate immutability.
4. Image Compression
Data can be compressed directly by Fourier coefficient. The common discrete cosine transformation is the real transformation of Fourier transformation;

Fourier transform
Fourier transform is the sum of Time-domain signals divided into sine signals or Cosine Functions of different frequencies. In continuous conditions, the original signal must meet the absolute product conditions within a period. In discrete cases, Fourier transformation must exist. The interpretation in the <image processing> version is very vivid: an appropriate metaphor is to compare Fourier transform to a glass prism. A prism is a physical instrument that splits light into different colors. The color of each component is determined by the wavelength (or frequency. Fourier transform can be regarded as a prism in mathematics. It splits functions into different components based on frequency. When we consider light, we will discuss its spectral or frequency spectrum. Likewise, Fourier transformation enables us to analyze a function through the frequency component.
Fourier transform has many excellent properties. For example, linearity and symmetry (which can be used in Fourier transformation of computing signals );

Time shifting: The time shifting of a function in the time domain corresponds to the phase shifting generated in the frequency domain, while the amplitude spectrum remains unchanged;

Frequency Shift: The function is multiplied by E ^ JWT in the time domain to shift the entire spectrum. This is also called the modulation theorem. This feature is required for the frequency division multiplexing of signals in communication (modulation of different signals to different frequencies for simultaneous transmission );
Convolution theorem: Time Domain convolution equals the product of the frequency domain; Time Domain product equals the convolution of the frequency domain (an additional coefficient ). (This is a key point in image processing)

 

Signal performance in Frequency Domain
The larger the frequency, the faster the original signal changes. The smaller the frequency, the smoother the original signal. When the frequency is 0, it indicates the DC signal, which is not changed. Therefore, the frequency reflects the signal variation speed. The high-frequency component interprets the abrupt changes of the signal, while the low-frequency component determines the overall image of the signal.
In image processing, the frequency domain reflects the intensity of gray-scale changes in the image in the airspace, that is, the speed of gray-scale changes, that is, the gradient size of the image. For an image, the edge part of the image is the abrupt part, which changes rapidly. Therefore, the response is a high-frequency component in the frequency domain. In most cases, the noise of the image is a high-frequency part; the gentle variation part of the image is the low frequency component. That is to say, Fourier transform provides another angle to observe the image, which can convert the image from the gray distribution to the frequency distribution to observe the image features. In writing, Fourier transform provides a way of free conversion from airspace to frequency. The following concepts are very important for image processing:

High-frequency components of an image: abrupt changes in the image. In some cases, it refers to the edge information of the image, in some cases it refers to noise, and more it refers to the mixture of the two;
Low-frequency components: the area where the image changes gently, that is, the image contour information.
High-pass filter: used to suppress low-frequency components and pass through high-frequency components.
Low-pass filter: opposite to high-pass filter, which allows the image to suppress high-frequency components and pass low-frequency components
Band-pass filter: enables the image to pass through a certain part of the frequency information, and other low or too high will suppress
There is also a band-blocking filter, which is the back of the band-pass.

Template operation and convolution Theorem
Performing template operations in the time domain is actually convolution of the image. Template operations are a very important processing process for image processing. Many image processing processes, such as enhancement/de-noise (these two points are unclear) are widely used in edge detection. According to the convolution theorem, time-domain convolution is equivalent to the product of the frequency-domain. Therefore, performing a template operation on the image in the time domain is equivalent to performing a filter on the image in the frequency domain.
For example, the frequency-domain response of an average template is a low-pass filter; performing mean filtering on images in the time domain is equivalent to performing a low-pass filtering on the image's frequency domain response using the mean template in the frequency domain.

Image Denoising
Image Denoising is the noise aspect of the image. Therefore, if the noise is of a high frequency, from the perspective of the frequency domain, it is necessary to use a low-pass filter to process the image. The low-pass filter can suppress the high-frequency components of the image. However, in this case, edge information is often restrained. Common denoising Templates include mean templates and Gaussian templates. Both filters suppress the high-frequency component of the image in the local area, while the blurred image edge also suppresses noise. There is also a non-linear filter-median filter. The median filter can be used to remove impulsive noise. Because the pulse points are abrupt points, and the output values are sorted, the maximum and minimum points can be removed. The median filter has poor effect on Gaussian noise.

Salt and pepper noise: It can be well removed by using median filter. The mean value can also be used to achieve certain results, but it may cause blurred edges.
Gaussian white noise: white noise is distributed throughout the frequency domain, which seems to be difficult.
Image processing P185: the arithmetic mean filter and Geometric Mean Filter (especially the latter) are more suitable for processing Gaussian or even random noise. The harmonic mean filter is more suitable for processing pulse noise.

Image Enhancement
Sometimes it is a conflict between image enhancement and image de-noise. Image Enhancement often requires image edge enhancement for better display. This requires increasing the image's high-frequency component. Image Noise reduction aims to eliminate image noise, that is, to suppress high-frequency components. Sometimes these two are similar things. For example, when noise is eliminated and the Image Display Effect is significantly improved, this means the same.
Common image enhancement methods include contrast stretching, histogram equalization, and image sharpening. The first two are pixel-based transformations in the airspace, and the latter is processed in the frequency domain. I understand that sharpening is to directly add the image to the image after high-pass filtering, that is, the image edge effect. Contrast stretching and histogram equalization both aim to improve the contrast of the image, that is, to make the image look more different. I think, after such processing, the image should also enhance the high-frequency component of the image, making the image details more different. Noise is also introduced.