"Get deep Once" wavelet transform popular interpretation-algorithm and the beauty of mathematics

Links: http://www.zhihu.com/question/22864189/answer/40772083

Article recommended by: Yang Xiaodong

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.
The following is the sequence of the Fourier-to-short Fourier transform--and the wavelet transform--to tell you why the wavelet is the thing, what is the idea of wavelets.
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.
Below we mainly will the Fourier transform insufficiency. That is, we know that the Fourier transform can analyze the spectrum of the signal, then why should we propose a wavelet transformation. The answer to "non-stationary process, Fourier transform has limitations." Look at a simple signal like this:

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 is a non-stationary signal with frequency changing over time.

As pictured above, 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 above image shows a normal event-related potential. 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. "The whole time-domain process is broken down into countless equal-length small processes, each small process is approximately stationary, and then Fourier transforms, to know at what point on the frequency of the occurrence." "This is the short-time Fourier transform.
Look at the picture:

When the time domain is divided into a section to do the FFT, do not know the frequency composition with the change of time.

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.
Using STFT There is a problem with how wide the window function should be.
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"

The

above shows the same signal (4 frequency components) with a different width of the window to do stft, the result is 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 might think, let the window size change, do a few more stft do not have it? Yes, the wavelet transform has such a way of thinking.
    But the fact that wavelets do not do this is that "wavelet transforms are based on algorithms that add unequal windows to each part of the Fourier transform", which is inaccurate. 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 do not get orthogonal, which is also a major flaw in it. The starting point and Stft of the

    Yu Sheppo transformation 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 can be positioned to the time ~

"Explanation"
    let's review the Fourier transform, why the Fourier transform can get the signal of the various frequency components of the students can also borrow my diagram to understand. The
    Fourier transform takes an infinitely long trigonometric function as a base function:

The base function will scale and translate (in fact, 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 that. With the wavelet, we never fear the unstable signal again. You can do time-frequency analysis from now on.
Fourier transform can only get a spectrum, do wavelet transform but get a time spectrum.

↑: Time Domain signal

↑: Fourier Transform results

-This image comes from "The WAVELET TUTORIAL"

↑: wavelet Transform Results

Wavelets also have some benefits:
1. We know that for a mutation signal, the Fourier transform has a Gibbs effect, and we use an infinitely long trigonometric function to fit a bad mutation signal:

However, the attenuation of the wavelet is not the same:

2. Wavelets can be orthogonal, short-time Fourier transform cannot.

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 on the link: INDEX to SERIES oftutorials 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 ~