i--wavelet Transform of the image-understandable explanation algorithm

Https://zhuanlan.zhihu.com/p/22450818?refer=dong5

The first answer: Can you explain the relationship between Fourier analysis and wavelet analysis in a popular mode? -Dong-dong knows the answer to the thump.
Current income column.


From the Fourier transform to the wavelet transform, is not a completely abstract thing, can speak very image. The wavelet transform has definite physical meaning, if we look at the problems that we face when we put forward it, we can sort out a very clear idea.

So I'm going to follow the sequence of the Fourier-to-short Fourier transform--and the wavelet transform--and tell me why this thing is going to happen, what's the idea of wavelets. (Anyway, the main requirement is the popular image, not to mention short, hope not too long not to see.) ）

One, Fourier transform
About the basic concepts of Fourier transform here I will not repeat, the default is now in the understanding of the Fourier but still do not understand the path of the wavelet. (in the third section of the wavelet transform, I'll say a little bit more about the Fourier transform.)

Below we mainly will the Fourier transform insufficiency. That is, we know that Fourier changes can analyze the spectrum of the signal, so why do we propose a wavelet transform? The answer is what Fang Qin said, "There are limitations to the non-stationary process, the Fourier transform." Here is a simple signal: After the FFT (Fast Fourier transform), you can see a clear four lines on the spectrum, the signal contains four frequency components.

Everything is fine. But what if it's a non-stationary signal with frequency changing over time?

For example, the top is a stationary signal with constant frequency. The bottom two is a non-stationary signal that changes frequency over time, and they also contain four components at the same frequency as the highest signal.
When we do the FFT, we find that there are very different signals in these three time domains, but the spectrum (amplitude spectrum) is very consistent. In particular, the two non-stationary signals below, we cannot distinguish them from the spectrum, because they contain four frequencies of the components of the signal is indeed the same, but the order of occurrence is different.

It can be seen that the Fourier transform processes non-stationary signals with innate defects. It can only get the components of the frequency that a signal is generally contained, but it does not know the moment when the components appear. Therefore, the time domain has a large difference of two signals, probably the same spectrum map.

However, the stationary signal is mostly artificially produced, and the vast number of signals in nature are almost non-stationary, so in the field of biomedical signal analysis, the basic method of naive is not seen in the paper.
The event-related potentials of a normal person are shown. For such non-stationary signals, it is not enough to know what frequency components are included, and we also want to know the time when each component appears. Knowing how the signal frequency changes over time, the instantaneous frequency and amplitude of each moment-this is the time-frequency analysis.


Second, short-time Fourier transform (short-time Fourier Transform, STFT)
A simple and feasible way is to add windows. I want to apply the description of the Fang Qin Yuan classmate, "The whole time domain process into countless equal length of small process, each small process approximately smooth, and then Fourier transform, you know at which point on the frequency of what happened." "This is the short-time Fourier transform.
Look at the picture:
The time domain is divided into a section to do the FFT, do not know the frequency composition over time change situation!
In this way, you can get a signal time-frequency graph:
-This image comes from "The WAVELET TUTORIAL"
You can see the four frequency domain components of 10Hz, Hz, Hz and Hz, as well as the time of occurrence. The two rows of peaks are symmetrical, so everyone just looks at a row.

Isn't that awesome? The results of the frequency analysis. But Stft still has flaws.

There is a problem with STFT, how wide should we use the window function?
The window is too wide and too narrow to have a problem:
The window is too narrow, the signal in the window is too short, will cause the frequency analysis is not accurate, the frequency resolution is poor. The window is too wide, time domain is not fine enough, time resolution is low.
(In this case, this can be explained by the Heisenberg Uncertainty principle.) Similarly, we cannot acquire the momentum and position of a particle at the same time, and we cannot simultaneously obtain the absolute exact moment and frequency of the signal. This is also a pair of contradictions can not be combined. We do not know which frequency component exists in an instant, and we know only that the component of a certain frequency band exists in a time period. So the instantaneous frequency of absolute meaning does not exist. ）

Let's look at the example effect:
-This image comes from "The WAVELET TUTORIAL"
For the same signal (4 frequency components) using a different width of the window to do Stft, the results are as shown on the right. With narrow windows, the Shimantu in the time axis resolution is very high, a few peaks basically into a rectangle, and with a wide window has become a long stretch of hill. But on the frequency axis, the narrow window is obviously not as accurate as the two wide windows below.

So the narrow window time resolution is high, the frequency resolution is low, the wide window time resolution is low, the frequency resolution is high. For the time-varying unsteady signal, the high frequency is suitable for small window, the low frequency is suitable for large window. However, the Stft window is fixed and the width does not change in a single stft, so Stft is still unable to meet the demand for the frequency of unsteady signal changes.


Three, wavelet transform

Then you may think, let the window size to change, do more several times Stft not be OK?! Yes, the wavelet transform has such a way of thinking.
But the fact that the wavelet is not doing so (on this point, Fang Qin Garden classmate of the expression "wavelet transform is based on the algorithm, plus unequal window, for each part of the Fourier transform" is not accurate.) The wavelet transform does not adopt the idea of the window, and does not do Fourier transform. ）
As for why not use variable window STFT, I think it is because the redundancy will be too serious, stft can not be orthogonal, which is also a major flaw in it.

So the starting point and the stft of the wavelet transform are still different. The STFT is to add a window to the signal, to do the FFT of the segment, and the wavelet transforms the Fourier transform's base directly-the infinite trigonometric function base is replaced by a finite long attenuation wavelet. This will not only be able to get the frequency, but also to locate the time ~

Explain
Let's go back to the Fourier transform, and don't know why the Fourier transform can get the signal each frequency component of the students can also borrow my diagram to understand.
The Fourier transform takes an infinitely long trigonometric function as a base function:

This base function will scale and translate (in fact it is not translational, but the decomposition of two orthogonal bases). Narrow, correspond to high frequency, extend wide, correspond to low frequency. The base function is then multiplied with the signal. A certain scale (width) of the turned out out of the results, it can be understood that the signal contains the current scale corresponding to the frequency component of how much. As a result, the base function is multiplied by the signal at some scales to get a large value, because there is a coincident relationship between the two. Then we know how much the signal contains the component of that frequency.

A closer look reveals that this step is actually calculating the correlation between the signal and the trigonometric function.

See, these two scales can multiply a large value (high correlation), so the signal contains more of these two frequency components, in the spectrum of these two frequencies will appear two peaks.


Above, is the principle of Fourier transform in the superficial sense.




As mentioned above, the change of the wavelet is to replace the infinitely long trigonometric function base with a finite long attenuation wavelet.
That's why it's called "wavelet," because it's a very small wave.


As can be seen from the formula, unlike the Fourier transform, the variable has only the frequency Ω, and the wavelet transform has two variables: scale a (scales) and translation amount tau (translation). Scale a controls the scaling of the wavelet function, and the translation of tau controls the translation of the wavelet function. The scale corresponds to the frequency (inverse), and the translational amount of tau corresponds to the time.



When scaling and panning to such a coincident condition, it is multiplied to get a large value. This time, unlike the Fourier transform, it is possible not only to know that the signal has such a frequency component, but also to know where it exists in the temporal domain.

And when we are translating and multiplying the signals at each scale, we know what frequency components the signal contains at each location.

Did you see it? With the wavelet, we never fear the unsteady signal again! You can do time-frequency analysis from now on!

Do Fourier transform can only get a spectrum, do wavelet transform but can get a time spectrum!
↑: Time Domain signal
↑: Fourier Transform results

-This image comes from "The WAVELET TUTORIAL"
↑: wavelet Transform Results

Wavelet also has some advantages, for example, we know that for the mutation signal, the Fourier transform has the Gibbs effect, we use the infinite trigonometric function How to also fit the bad mutation signal:
However, the attenuation of the wavelet is not the same:




Above, is the meaning of wavelet.

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The above is only with the image to show you a little bit of the idea of wavelet, I hope we can bring some help to the introduction. After all, if you know nothing about wavelets, go straight to see those formula, copy the language textbooks, it will be painful.
Here are recommended a few introductory reading, are based on the perceptual introduction, easy to understand but not in-depth, the initial understanding of the wavelet will be very helpful. Some of the ideas and diagrams in this article are also selected from:
1. The WAVELET TUTORIAL (strongly recommended, click Link: INDEX to SERIES of tutorials to WAVELET TRANSFORM by ROBI Polikar)
2. Wavelets:seeing the FOREST and the TREES
3. A really friendly guide to wavelets
4. Conceptual wavelets

But really understand the wavelet transform, these are still very far. For example, at least you have to know that there is a "scale function", which is the key to the construction of the "wavelet function", and it is together with the small wave function to form a multi-resolution analysis of wavelets, it is possible to use the wavelet to do some digital signal processing; you also understand discrete wavelet transform, orthogonal wavelet transform, wavelet transform, Wavelet packet ... The contents of the domestic teaching materials are also very bad, we have 1.1 bite it ~











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Answers to some questions:
1. About Heisenberg Uncertainty principle
The principle of uncertainty, or the principle of non-determinism, was first derived from quantum mechanics, meaning that in microcosm, the position and momentum of a particle cannot be determined simultaneously. But this principle is not confined to quantum mechanics, there are many physical quantities have such characteristics, such as energy and time, angular momentum and angle. The signal field is the time domain and the frequency domain. However, a more accurate statement should be: A signal can not be in the space-time domain and frequency domain at the same time too concentrated; the more "narrow" a function is, the more "wide" it is after the frequency domain of the Fourier transform.
If you are interested in delving into it, this principle is actually very intriguing. Some of the new theories in signal processing are fundamentally connected to it, such as compression perception. If you peel off its complex mathematical description, you will eventually find that it is intrinsically relevant to the principle of uncertainty. And don't you think that something so contradictory is a philosophical sense?


2. About Orthogonality
What is orthogonality? Why is it advantageous to achieve orthogonality of wavelets?
Simply stated, if the orthogonal basis, the transformation domain coefficients will not have redundant information, the signal energy is equal before and after the transformation, is equal to the least data to express the largest amount of content, conducive to numerical compression and other fields. JPEG2000 compression is the use of orthogonal wavelet transform.
For example, the typical orthogonal base: two-dimensional Cartesian coordinate system (1,0), (0,1), using them to express a signal is obviously very efficient, simple calculation. If the three vectors are expressed in 120°, there will be information redundancy and repeated expression.
But it does not mean that orthogonality is better than not orthogonal. For example, if the image is enhanced, sometimes it is desirable to have some redundant information, which is more conducive to the suppression of noise and the enhancement of some features.

3. About instantaneous frequency
The original problem: the moment point in the graph corresponds to a frequency value, a moment point only a signal value, how can we get his frequency?
That's a good question. As the text says, the instantaneous frequency of absolute meaning actually does not exist. Look at a signal value of a moment point, of course not get its frequency. We are only using a short signal frequency as the frequency of the moment, so we are only a limited time resolution of the approximate analysis results. This idea is evident in the STFT. The wavelet uses the attenuation basis function to measure the instantaneous frequency of the signal, and the thought is similar. (But to the Hilbert transformation, the idea is not the same, later have the opportunity to speak)

4. About the deficiency of wavelet transform
It depends on who is compared with.
A. As an image processing method, and multi-scale geometric analysis method (super-wavelet) ratio:
For the two-dimensional image of the signal, the two-dimensional wavelet transform can only be carried out in 2 directions, the information of the midpoint of the image can be expressed, but the line is relatively poor. And the most important information in the image is those edge lines, this time Ridgelet (ridge), Curvelet (Qu Bo) and other multi-scale geometric analysis method is more advantageous.
B. As the time-frequency analysis method, and the Hilbert-Huang transform (HHT) ratio:
Compared to HHT and other time-frequency analysis methods, the wavelet still does not deviate from the Heisenberg uncertainty principle of the binding, a certain scale, can not be at the same time and frequency of high precision, and the wavelet is non-adaptive, the base function is selected will not be changed.

5. On the rigor of the presentation in the text
Many friends in the comments mentioned that some of my statements were not accurate enough. That's for sure, and I know it too. For example, the Fourier transformation of the understanding of the part, I said the kind of "take a large value" expression is certainly not rigorous. I also made an explanation in the reply to the comment. I would like to say is easy to understand and precise rigorous is difficult to get, if the pursuit of rigorous, the best is the textbook mathematical expression, they are impeccable, but for beginners, I am afraid there is a threshold. If we want to explain in a popular sense, we must focus on only one key point, and there is a loophole. I think this is why textbooks never write these popular explanations--the authors are not ignorant, but afraid to write wrong. So want to deeply understand the Fourier transform and the wavelet transform friend also please earnestly study the textbook, if this article can give some beginners a little help, I am contented.

i--wavelet Transform of the image-understandable explanation algorithm