(2) Fourier transform: the aperiodic continuous signal in time domain is converted into aperiodic continuous signal in frequency domain.
(3) Relationship among frequency domain, time domain and phase:
(4) Euler formula:
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Disadvantages of (1) Fourier transform
That is, we know that Fourier transform can analyze the frequency spectrum of signals, so why put forward wavelet transform? The answer is what Fang Qinyuan said, "Fourier transform has limitations for non-stationary processes."
As shown in the figure below:
After FFT (Fast Fourier Transform), four clear lines can be seen in the spectrum, and the signal contains four frequency components.
Everything's fine. But what if it is a non-stationary signal whose frequency changes with time?
As shown above, the top is a stationary signal with constant frequency. The following two are non-stationary signals whose frequency varies with time, and they also contain four components with the same frequency as the top signal.
After FFT, we found that the frequency spectrum (amplitude spectrum) of these three signals with great differences in time domain is very consistent. Especially the following two non-stationary signals, we can't distinguish them from the spectrum, because the components of the signals with four frequencies they contain are really the same, but the order of appearance is different.
It can be seen that Fourier transform has inherent defects in processing non-stationary signals. It can only obtain the frequency components of a signal as a whole, without knowing the time when each component appears. Therefore, two signals with very different time domains may have the same frequency spectrum.
However, most stationary signals are artificially generated, and a large number of signals in nature are almost non-stationary, so naive method is basically not seen in papers in biomedical signal analysis and other fields, such as simple Fourier transform.
(2) Short Time Fourier Transform (STFT).
A simple and feasible method is to add windows. I want to paraphrase Fang Qinyuan's description, "Decompose the whole time domain process into countless small processes of equal length, each of which is approximately stable, and then Fourier transform will know which time point and what frequency appears." This is a short-time Fourier transform.
Do FFT in time domain, and you will know the change of frequency components with time!
In this way, the time-frequency diagram of the signal can be obtained:
We can not only see the four frequency domain components of 10Hz, 25Hz, 50Hz and 100Hz, but also see the time when they appear. The two rows of peaks are symmetrical, so you only need to look at one row.
Isn't it great? Time-frequency analysis results are available. But STFT is still flawed.
There is a problem with using STFT. How wide a window function should we use?
The window is too wide and too narrow;
If the window is too narrow and the signal in the window is too short, it will lead to inaccurate frequency analysis and poor frequency resolution. The window is too wide, the time domain is not fine enough and the time resolution is low.
By the way, this truth can be explained by Heisenberg's uncertainty principle. Just as we can't get the momentum and position of a particle at the same time, we can't get the absolutely accurate time and frequency of the signal at the same time. This is also a pair of contradictions that cannot have both. We don't know which frequency component exists at a certain moment. We only know that the components of a certain frequency band exist in a time period. So the absolute instantaneous frequency does not exist. )
Therefore, the narrow window has high time resolution and low frequency resolution, and the wide window has low time resolution and high frequency resolution. For time-varying non-stationary signals, high frequency is suitable for small windows and low frequency is suitable for large windows. However, the window of STFT is fixed, and the width will not change within one STFT, so STFT still can't meet the requirements of the frequency of unsteady signal change.
(3) Wavelet transform
Then you might think, let the window size change and do STFT more times, don't you? ! Yes, wavelet transform has such an idea.
But in fact, wavelet does not do this (in this regard, Fang Qinyuan said that "wavelet transform is to add unequal windows according to the algorithm and carry out Fourier transform on each small part" is inaccurate. Wavelet transform does not adopt the idea of window, let alone Fourier transform. )
As for why STFT with variable window is not used, I think it is because it will be too redundant and STFT cannot be orthogonalized, which is also a big defect.
So the starting point of wavelet transform is different from STFT. STFT windowed the signal and performed FFT in segments, while wavelet directly changed the basis of Fourier transform-finite attenuation wavelet basis instead of infinite trigonometric function basis. This can not only get the frequency, but also locate the time ~
This is why it is called "wavelet", because it is a very small wave ~
As can be seen from the formula, unlike Fourier transform, the variable is only frequency ω, while wavelet transform has two variables: scale a and τ (translation τ. Scale a controls the expansion and contraction of wavelet function, and translation τ controls the translation of wavelet function. Scale corresponds to frequency (inverse ratio), and translation τ corresponds to time.
When stretching and translating to such a coincidence, it will also multiply to get a large value. At this time, unlike Fourier transform, we can not only know that the signal has such a frequency component, but also know its specific position in the time domain.
When we translate and multiply the sum signal on each scale, we know what frequency components the signal contains at each position.
Did you get a look at him? With wavelet, we are no longer afraid of signal instability! Time-frequency analysis can be done from now on!
Fourier transform can only get a spectrum, while wavelet transform can get a time spectrum!
Wavelet has some advantages. For example, we know that there is Gibbs effect in the Fourier transform of abrupt signals, and we can't fit abrupt signals with infinite trigonometric functions.
Link:/question/22864189/answer/40772083
(1) PSNR (peak signal-to-noise ratio)
PSNR: Peak Signal-to-Noise Ratio (PSNR), which refers to the image quality evaluation index comprehensively. ?
Where MSE represents the mean square error of the current image X and the reference image Y, and H and W are the height and width of the image respectively; N is the number of bits per pixel, which is generally 8, that is, the unit of pixel gray level is 256. PSNR, the larger the value, the smaller the distortion.
PSNR is the most common and widely used objective evaluation index of images, but it is based on the error between corresponding pixels, that is, the image quality evaluation based on error sensitivity. Because the visual characteristics of human eyes are not considered (human eyes are more sensitive to the contrast difference of low spatial frequency, human eyes are more sensitive to the contrast difference of brightness, and the perception of an area will be affected by its adjacent areas, etc.). ), the evaluation results are often inconsistent with people's subjective feelings.
(2) SSIM (similar in structure)
SSIM: Structural similarity index is an index to measure the similarity between two images. It measures the similarity of images from brightness, contrast and structure respectively.
The range of structural similarity is-1 to 1. When the two images are the same, the value of SSIM is equal to 1.
Other indicators:/smallstones/article/details/42198049.