For example, discrete cosine transform, gabor and wavelet also have important components in image processing.
In my impression, Fourier transform plays an important role in the following topics of image processing: 1. Most of the noise in image enhancement and image denoising is the high-frequency component of the image, and the high-frequency noise is filtered by the low-pass filter. Edge is also a high-frequency component of the image, and the edge of the original image can be enhanced by adding high-frequency components. 2. Edge detection of image segmentation, extracting high-frequency components of the image 3. Image feature extraction: shape feature: Fourier description of sub-texture feature: other features of texture feature are directly calculated by Fourier coefficient: the extracted feature value is Fourier transformed to make the feature invariant to translation, expansion and rotation. Image compression can directly compress data with Fourier coefficients; The commonly used discrete cosine transform is the real transform of Fourier transform;
Fourier transform Fourier transform is to decompose time domain signals into sine signals or cosine functions with different frequencies. In the continuous case, the original signal is required to meet the absolute integrable condition in one period. In the discrete case, Fourier transform must exist. Gonzalez edition; The explanation is vivid: an appropriate metaphor is to compare Fourier transform to a glass prism. Prism is a physical instrument that can decompose light into different colors, and the color of each component is determined by wavelength (or frequency). Fourier transform can be regarded as a mathematical prism, which decomposes a function into different components according to frequency. When we consider light, we discuss its spectrum or frequency spectrum. Similarly, Fourier transform enables us to analyze functions through frequency components. Fourier transform has many excellent properties. Such as linearity and symmetry (which can be used to calculate the Fourier transform of signals);
Time shift: the time shift of the function in time domain corresponds to its additional phase shift in frequency domain, while the amplitude spectrum remains unchanged;
Frequency shift: the function of multiplying e^jwt in time domain can shift the whole spectrum by w, which is also called modulation theorem, which is needed for signal frequency division multiplexing in communication (modulating different signals to different frequency bands and transmitting at the same time); Convolution theorem: time domain convolution is equal to frequency domain product; The time domain product is equal to the frequency domain convolution (plus a coefficient). (This is a key point in image processing)
The performance of signal in frequency domain is in frequency domain. The greater 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, and there is no change. Therefore, the magnitude of frequency reflects the speed of signal change. The high frequency component explains the abrupt part of the signal, while the low frequency component determines the overall image of the signal. In image processing, frequency domain reflects the intensity of image gray change in spatial domain, that is, the speed of image gray change, that is, the gradient of image. For an image, the edge of the image is abrupt and changes rapidly, so the response is a high frequency component in the frequency domain; The noise of the image is high frequency in most cases; The gently changing part of the image is a low frequency component. That is to say, Fourier transform provides another angle to observe the image, which can transform the image from gray distribution to frequency distribution to observe the characteristics of the image. In written form, Fourier transform provides a way to freely convert from spatial domain to frequency. For image processing, the following concepts are very important:
High-frequency components of image: image mutation; In some cases, it refers to image edge information, in other cases, it refers to noise, and more is a mixture of the two; Low-frequency components: the gently changing part of the image, that is, the Qualcomm filtering of the image contour information: let the image suppress the low-frequency components and let the high-frequency components pass through the low-pass filter; contrary to Qualcomm, let the image suppress the high-frequency components and let the low-frequency components pass through the band-pass filter; let the frequency information of a certain part of the image pass through and suppress all other components that are too low or too high. There is also a band-stop filter, which is the opposite of a band-pass filter.
Template Operation and Convolution Theorem Doing template operation in time domain is actually convolution of images. Template operation is a very important process in image processing. Many image processing processes, such as enhancement/denoising (both are unclear), are widely used in edge detection. According to convolution theorem, time domain convolution is equivalent to frequency domain product. Therefore, templating the image in time domain is equivalent to filtering the image in frequency domain. For example, the frequency response is the average template of the low-pass filter; Filtering the image in time domain is equivalent to low-pass filtering the image in frequency domain by using the frequency response of the average template.
Image denoising Image denoising is to suppress the noise in the image. Therefore, if the noise is high frequency, from the perspective of frequency domain, it is necessary to use low-pass filter to process the image. The low-pass filter can suppress the high-frequency components of the image. But in this case, it often leads to the suppression of edge information. Common denoising templates include mean template and Gaussian template. Both of these filters suppress the high-frequency components of the image in the local area, blur the edge of the image and suppress the noise at the same time. There is also a kind of nonlinear filtering-median filtering. Median filtering can remove impulse noise well. Because the pulse points are all mutation points, and the median value is output after sorting, then those maximum points and minimum points can be removed. Median filtering is not effective for Gaussian noise.
Salt and pepper noise: Salt and pepper can be well removed by median filtering. Using the average value can also achieve a certain effect, but it will cause blurred edges. Gaussian white noise: White noise is distributed in the whole frequency domain, which seems difficult.
Gonzalez version of image processing P 185: arithmetic average filter and geometric average filter (especially the latter) are more suitable for processing Gaussian or uniform random noise. Harmonic averaging filter is more suitable for dealing with impulse noise.
Image enhancement sometimes feels that image enhancement and image denoising are contradictory processes. Image enhancement often needs to enhance the edge of the image to obtain better display effect, which requires increasing the high-frequency components of the image. The purpose of image denoising is to eliminate image noise, that is, to suppress high-frequency components. Sometimes these two refer to similar things. For example, the image display effect has been obviously improved while eliminating noise, so this time it means the same thing. Common image enhancement methods include contrast stretching, histogram equalization and image sharpening. The first two are pixel-based transformations in the spatial domain, and the latter is processed in the frequency domain. My understanding of sharpening is to add Qualcomm filter directly to the image, that is, the edge effect of the image. Contrast stretching and histogram equalization are both to improve the contrast of the image, that is, to make the image look more different. In my opinion, after such processing, the image should also enhance the high-frequency components of the image and make the details of the image more different. At the same time, it also introduces some noise.