Fully understand Fourier transform and wavelet (1)--Master

Special Note: This series of articles has been combined with the mathematical principles of image Processing series of articles, now the Fourier transform part of the basic introduction is complete. You can refer to the mathematical principles in the image processing of the total catalogue (http://blog.csdn.net/baimafujinji/article/details/48467225)

Whether it is learning signal processing, or doing image, audio and video processing research, you can never avoid a content, is the Fourier transform and wavelet. But these two things are not easy to understand, or are actually very abstract and obscure.

To understand the Fourier transform and the wavelet, you need to know at least what kind of preliminary knowledge. Homepage June from now on will be a series of articles to tell you the ins and outs between them. This section is the first section of the whole series of articles-the master, and we will follow this idea 1.1 points to tell all the knowledge. It should be explained that this article is mainly for the computer professional or electronic information professional readers, we will try to take some very basic knowledge to help you understand. Therefore, the topic of "fully understand" is not from the point of view of physics or pure mathematics, because the Fourier transform was originally a French scientist Fourier in the study of physics (mainly thermodynamics) created a set of theories, if you want to say from this point of view, "thoroughly understand", must need a lot of physical knowledge, Complex transformation analysis, like the heavenly Book of Things. In fact, a lot of people who study computers or learn electronic information have no need at all. All we have to do is use what you already know to build the whole system.

Jin Yong in his martial arts novel "Tianlong Eight" in the shape of a Tibetan national division of the character image-"Mono wisdom." Mono Zhi Martial Arts Quick success, in order to in a short period more into several kung Fu, often the foundation does not fight, on the forced cultivation of superior military study. With the Taoist small non-phase power catalytic Shaolin 72 stunt, seemingly powerful, in fact, endless. In fact, the daily study is like this, if you come up to remember a few formulas, and then straightforward to eat Fourier transform or wavelet, in fact, can also make something, but because the foundation is empty, so the process of internal processes is almost broken, defect. Whether it is Fourier transform or wavelet, their most basic theory is all mathematics. It's just that most students have difficulty building up their connections. Let's start with math today.

The contents of the dashed box in the picture, should be in the higher mathematics must learn the content, this part has what does not understand, to ask the child headmaster should be most suitable. First of all, you should know the Fermat theorem (this is very simple, in fact, the function has the extremum of the condition), through the Fermat theorem, you can prove the Raul theorem, and then through the Raul theorem, you can prove the LaGrand value theorem, through the LaGrand value theorem, you can then prove Cauchy mean value theorem. The significance of the Cauchy median theorem is that it can be used to prove the Taylor formula. Taylor's formula, of course, was proposed by a man named Taylor, but the man who really proved Taylor's formula was Cauchy, because Cauchy knew Cauchy's mean theorem, and to prove that Taylor's formula needed to use Cauchy's mean value theorem. Taylor's formula has two uses for us, first it can be used to prove Euler's formula, and Euler's formula must be used in the Fourier transform. Secondly, the expansion of power series can be obtained by Taylor formula. This is a precursor, because whether it is a wavelet expansion, or Fourier expansion, if you understand the Taylor Show or power series expansion, then the corresponding is easy to explain, people design the Fourier Ye Zhan and wavelet display of the original intention and intention. There are two main types of series in the high number, except the power series, the other is the Fourier series. The high number of knowledge you need to get here is enough.

The yellow block diagram in the figure is what you should learn in the digital signal processing course. Before we studied Fourier transform, we must first learn a Fourier series in high numbers, but what is the relationship between Fourier series and Fourier transform? In plain words, their nature is the same, although the expressions of their respective formulas seem to differ greatly. By Fourier series formula, in fact you do some simplification and variable substitution (about this part of the content, if the reader is interested, the main page can give a detailed proof of the process), the Fourier series has become a Fourier transform, of course, is continuous. Then you will be able to change the Fourier transform, which is the DFT, according to the sampling theorem in the digital signal processing. But the DfT has a problem, if the formula calculation, efficiency is too low, the practical value is not high, later people invented a fast algorithm, FFT. This way down, if every step, you are very clear, then you are already in the Fourier transform understanding is very in place.

Then the wavelet, the wavelet can be compared with the Fourier transform to understand, and in fact, before the advent of the wavelet, people first created a short-time Fourier transform things, which can be considered as a bridge between the two. Of course, this part of the content, you do not know also does not matter, you can even use Taylor Formula and power series as a starting point to understand the wavelet. The Wavelet series expansion corresponds to the Fourier expansion, the continuous wavelet corresponds to the continuous Fourier transform, and the DWT corresponds to the DFT. These are very easy to understand. Similarly, people (in fact mainly Mallat) have also developed a fast algorithm for wavelets, FWT. FWT is like the position of FFT in Fourier transform. To understand FWT, you need to know two basic knowledge, one called MRA, that is, multi-resolution analysis, Learning MRA is also very meaningful for understanding wavelets. Because Mra is a method of constructing wavelets. Another thing you have to know is called "sub-band coding" or sub-band decomposition.

To understand the sub-band decomposition, you must know QMF, that is, orthogonal image filter, and QMF is the multi-sample rate signal processing inside the important content. So you have to have a knowledge base of multi-sample rate signal processing, and the basis of multi-sample rate signal processing is digital signal processing. In addition, there is a theoretical basis for the sub-band coding, or why the sub-band code is proven effective, it is necessary to know the theory of rate distortion of the relevant conclusions. The theory of rate distortion is also an important part of information theory. So, at the end of this line, your learning context should be from information theory (in fact, mainly mutual information and rate distortion of the content), then the signal processing, and then the multi-sample rate processing, and then the sub-band coding and QMF, which you understand, FWT is too easy. Of course, you directly learn FWT algorithm, and do not care how it comes, but also can, just like I said before, your foundation is basically empty, you learn also just learn a move, internal strength heart almost won't, so go up to walk may encounter many difficulties.

Any subject or knowledge development to now less say is a few decades, let us in just a few months to learn, in fact, it is not easy. So it seems that there are too many things to know, and the time for learning is so limited. But there is no way, the higher the building to cover, the foundation is bound to hit the deeper.

In addition: Add a little, follow up in the implementation of the specific code, we use the language C + +, in Visual C + + completed.

not finished, to be continued ....