Disclaimer: perfect wavelet transform popular interpretation, it is the first "wavelet transform and motion signal processing" series.
The original source for the windstorm website http://www.kunli.info/, not the original site, but the master explain in simple terms the wavelet transform, are you in the process of learning wavelets, a must-see for!
"Wavelet transform and motion signal processing" contains three series:
First: universal basis (wavelet transform perfectly popular interpretation)
Second: in-depth wavelet
Part III: Application of wavelet
Due to space limitations relationship, here we only introduce the first part. The following is the text:
I remember when I was a senior, hesitated for a moment and go abroad in the application security research, the bones of the conservative conclusion, let me choose the first security research. Of course, later dropped out, but it is something. At that time security research to find a boss, lab, their own bad luck, very fast hardware into a school laboratory work in the road. Our laboratory is mainly engaged in the image, the strength of the country is very strong, go in and talk to senior sister apprentice brothers, we are engaged in what wavelet transform , H264 and the like. Then I do not mind in this regard, to make out what operating system migration, ARM + FPGA these things. On wavelet transform understanding will remain in the "king of the images of the video compression algorithm" mysterious above.
Later I found, in a wide range of other areas, the wavelet gradually became popular. Frequency domain signal processing, such as saying a long time ago, we are basically in contact Fourier and Laplace world. But over the years, and popularity began to flourish in the wavelet signal analysis. This forced people to feel curious, what characteristics make it in image compression, signal processing these critical applications get more trust it? To be honest, I was home, I began to wonder this issue, so the dogs found , poisoning search, searched the Chinese speaking wavelet transform science articles, found that few speak clearly, when curiosity is not so heavy, nor engage in this study, too lazy to find voluminous papers in English, and then gave up. Later came here, some projects use signal processing, forced to come into contact with something wavelet transform, began to bite the bullet and see. Read some material, I listened to some classes, only to find, or that cliché argument: foreign and domestic information technology TNND really is not a grade. The same thing, others say very clearly that I do not even smart people can read; domestic material is going around speaks mess, except for a few gifted few people can grasp in a short time.
The complaints do not continue to play. In this series, I want to briefly explain the wavelet transform, Fourier transform and compare it, and do it in a mobile platform motion applications of detection. Unless otherwise specified, are discrete wavelet as an example. Considering I used to see the extent of the suffering of Chinese data, I will try to use simple, intuitive way to introduce. Some of the necessary formula is not small, but I minimize the use of formulas, multi-map. In addition, I am not a good translator, so for some it is not clear translation of the term, I will be directly in English. I do not claim I will make it clear throughout the wavelet transform, which is impossible, I can only try to revolve around key points, such as the relative benefits of the wavelet transform Fourier transform, why are these benefits, several wavelet transform What is the fundamental nature, what is behind the derivation yes. I hope to achieve the purpose of the wavelet transform is a beginner in after reading this series, you can do basic analysis of signals with wavelet transform matlab or other analysis tools and know that this is probably how it was.
Final note, I am not a professional study of signal processing, so there will be omissions or errors in the text, as found also please let us know.
Talk about wavelet transform, we must understand the Fourier transform. To understand the Fourier transform, we need to figure out what is "transformation." A lot of treatment, whether it is compressed or, filtering or, graphics or, essentially transform. Transformation is what is it? Is the base, which is the basis. If you temporarily forget some definitions basis, then simply say, linear algebra, basis is the space where a series of linearly independent vectors, but this space of any other vector can be a linear combination of these vectors representation. That basis in the transformation inside what use is it? For instance, right, Fourier expansion of the essence, is to signal a space represented by a linear combination of a basis of the space, he expressed reason for this is because the nature of the Fourier transform ,Yes. Wavelet transform is no exception and the related basis. Another example is you use Photoshop to process images inside the image stretch, reverse, and so a series of operations, and to change the basis of all relevant.
Since these transformations are engaged in the base, then we naturally easy to think that the basis of selected important because determine the characteristics of the specific basis of calculation. A space may have many forms of basis, what the basis is better, depends largely on the basis of what service to use. For example, if we want to choose in favor of compression, then just I hope that this basis can be very few of the vector to maximize a signal indicative of, so that even if the other vector to cut, the signal will not lose a lot. If graphics processing common linear transformation, the most perfect basis province Eigenvector computation is a basis, because the transformation matrix T thereof and is equivalent to a diagonal matrix (Tv_n = av_n, a is an eigenvalue). Overall, aside specific applications aside, all the basis, we hope that they have a common characteristic, that is, easy to calculate, showing up signal characteristics with the most simple way.
Well, now we have a basic understanding of the transformation, in fact, they know gay. Of course, beauty gay is in the form of points, different transforms, gay vary. Then take a look at the Fourier transform is doing.
Joseph Fourier Fourier series was first put forward this man, he found that the basis not only exist with the vector space, is also present in the function space. This function space still essentially a linear vector space, can be limited, may be infinite, but in this space, vector is to function, while the corresponding scalar is real or complex. In the vector space where you have a vector v can be written as a linear combination of the vector basis, in that in the function space, function f (x) may be written as a linear combination of the corresponding function basis, there norm. You may be orthogonal vector basis, my function basis may also be orthogonal (such as sin (t) and sin (2t)). The only difference is that my function basis are endless, because my function space dimension is infinite. Well, specifically, it is now that we have a function, f (x). We hope it will be written in some forms and some of the functions cos sin function like this
again, this is an infinite loop function. Wherein the 1, cosx, sinx, cos2x ... .. Such is the Fourier series. Fourier series so widely used one main reason is that they are a bunch of sub-function basis are orthogonal, this is an interesting place. Why function basis orthogonal so important? We say two orthogonal vector, that is, They inner product is zero. That function basis for it? Seek inner product function basis how it?
Now to review the definition of vector orthogonal. We say that two vector v, w if orthogonal, they should comply with:
What is the function of orthogonal it? Suppose we have two functions f (x) and g (x), what is that? We follow the thinking vector to think about, seeking two vector inner product, it is to correspond to the same position on them the product of an accumulation of points to make. That move over, that is, for each point x, f and g do the corresponding product, and then accumulated. But the problem is, f and g are infinite function A, x is a continuous value. How to do it? Vector is discrete, the cumulative function is continuous, that is ....... Points!
We know that this calculation is within the function of the product, naturally easy to prove, according to this form to write the Fourier expansion, these series are indeed twenty-two orthogonal. The proof here is not started. Well, the next question is, why they are orthogonal basis so important? This involves solving a coefficient. We have studied the function f, study series, a bunch of trigonometric functions and constants 1, that factor does? A0, a1, a2 these coefficients which determine how it? Well, here I am ready to seek such a a1. What I know now signal f (x) is known, the Fourier series is known, how do we find a1 is very simple, all part of the equation are summed across the inner product cosx, that is??:
Then we found that because of the nature of orthogonal, non-a1 all the right items all disappeared, because their inner product and cosx are zero! All is reduced to
In this way, a1 on solving out. Here, you will see the wonderful nature orthogonal to it :)
Well, now we know that the Fourier transform is to use a series of triangular wave signal to a developed equation, this signal can be continuous, may be discrete. Fourier function basis is used specially selected, are orthogonal, it is conducive to calculate the coefficients. But do not misunderstand basis for the stretch transform used are orthogonal, it all depends on the specific needs, such as Taylor expansion on the basis simply of non-orthogonal polynomials.
With the Fourier transform of the basis, then, we will see what is the wavelet transform. First of all what is said wavelet. The so-called waves, that is, the time domain or spatial domain oscillation equations, such as a sine wave, is a wave. What is the wave analysis? Analysis pull (embarrassed) for the wave. Wavelet analysis is not to say it belongs to the wave analysis, Fourier analysis is wave analysis, because the sine wave is also a thing. So what is a wavelet? The "small" is aimed at in terms of the Fourier wave. What Fourier sine waves are used, this stuff has to infinite energy, the same amplitude oscillations throughout the infinite range inside, like this?:
What is it that wavelet? It is a very concentrated energy in the time domain wave. Its energy is limited and concentrated in a point nearby. For example, the following:
This wavelet what good is it? It becomes very useful for the analysis of signal instantaneous. It effectively extract the information from the signal, multi-scale detailed analysis of the function or signal scaling and translation of other operations, solve difficult problems, the Fourier transform can not be solved. Well, that's usually on domestic websites you can search the wavelet transform article to tell you. But why? This is what I want to make it clear in this series of articles. However, in this article, I will go beyond that, the important features and advantages of the wavelet transform of the cover, the next article more specific derivation of these characteristics.
The nature of Fourier transform and wavelet transform the like, a signal is represented by equation carefully selected basis. Each wavelet transform, there will be a mother wavelet, we call mother wavelet, as well as a scaling function, scaling function is Chinese, also known as the father of wavelets. Any basis function of wavelet transform, in fact, is a collection of the father and the mother wavelet wavelet after zooming and panning. This is below a certain wavelet schematic drawings:
Here it is seen from the number of stages is 2 where the zoom magnification, the size and extent of the translation of the current scaled concerned. The advantage is that both high frequency wavelet basis functions have low frequency, while also covering the time domain. For this, we will elaborate later.
Wavelet expanded form usually do (note that this is only approximate expression, rigorous The second expanded form please refer to):
one of them
就是小波级数，这些级数的组合就形成了小波变换中的基basis。和傅立叶级数有一点不同的是，小波级数通常是orthonormal basis，也就是说，它们不仅两两正交，还归一化了。小波级数通常有很多种，但是都符合下面这些特性：
1. 小波变换对不管是一维还是高维的大部分信号都能cover很好。这个和傅立叶级数有很大区别。后者最擅长的是把一维的，类三角波连续变量函数信号映射到一维系数序列上，但对于突变信号或任何高维的非三角波信号则几乎无能为力。
2. 围绕小波级数的展开能够在时域和频域上同时定位信号，也就是说，信号的大部分能量都能由非常少的展开系数，比如a_{j,k}，决定。这个特性是得益于小波变换是二维变换。我们从两者展开的表达式就可以看出来，傅立叶级数是
，而小波级数是
。
3. 从信号算出展开系数a需要很方便。普遍情况下，小波变换的复杂度是O(Nlog(N))，和FFT相当。有不少很快的变换甚至可以达到O(N)，也就是说，计算复杂度和信号长度是线性的关系。小波变换的等式定义，可以没有积分，没有微分，仅仅是乘法和加法即可以做到，和现代计算机的计算指令完全match。
可能看到这里，你会有点晕了。这些特性是怎么来的?为什么需要有这些特性?具体到实践中，它们到底是怎么给小波变换带来比别人更强的好处的?计算简单这个可能好理解，因为前面我们已经讲过正交特性了。那么二维变换呢?频域和时域定位是如何进行的呢?恩，我完全理解你的感受，因为当初我看别的文章，也是有这些问题，就是看不到答案。要说想完全理解小波变换的这些本质，需要详细的讲解，所以我就把它放到下一篇了。
接下来，上几张图，我们以一些基本的信号处理来呈现小波变换比傅立叶变换好的地方，我保证，你看了这个比较之后，大概能隐约感受到小波变换的强大，并对背后的原理充满期待:)
假设我们现在有这么一个信号：
看到了吧，这个信号就是一个直流信号。我们用傅立叶将其展开，会发现形式非常简单：只有一个级数系数不是0，其他所有级数系数都是0。好，我们再看接下来这个信号：
简单说，就是在前一个直流信号上，增加了一个突变。其实这个突变，在时域中看来很简单，前面还是很平滑的直流，后面也是很平滑的直流，就是中间有一个阶跃嘛。但是，如果我们再次让其傅立叶展开呢?所有的傅立叶级数都为非0了!为什么?因为傅立叶必须用三角波来展开信号，对于这种变换突然而剧烈的信号来讲，即使只有一小段变换，傅立叶也不得不用大量的三角波去拟合，就像这样：
看看上面这个图。学过基本的信号知识的朋友估计都能想到，这不就是Gibbs现象么?Exactly。用比较八股的说法来解释，Gibbs现象是由于展开式在间断点邻域不能均匀收敛所引起的，即使在N趋于无穷大时，这一现象也依然存在。其实通俗一点解释，就是当变化太sharp的时候，三角波fit不过来了，就凑合出Gibbs了:)
接下来我们来看看，如果用刚才举例中的那种小波，展开之后是这样的：
看见了么?只要小波basis不和这个信号变化重叠，它所对应的级数系数都为0!也就是说，假如我们就用这个三级小波对此信号展开，那么只有3个级数系数不为0 。你可以使用更复杂的小波，不管什么小波，大部分级数系数都会是0。原因?由于小波basis的特殊性，任何小波和常量函数的内积都趋近于0。换句话说，选小波的时候，就需要保证母小波在一个周期的积分趋近于0。正是这个有趣的性质，让小波变换的计算以及对信号的诠释比傅立叶变换更胜一筹!原因在于，小波变换允许更加精确的局部描述以及信号特征的分离。一个傅立叶系数通常表示某个贯穿整个时间域的信号分量，因此，即使是临时的信号，其特征也被强扯到了整个时间周期去描述。而小波展开的系数则代表了对应分量它当下的自己，因此非常容易诠释。
小波变换的优势不仅仅在这里。事实上，对于傅立叶变换以及大部分的信号变换系统，他们的函数基都是固定的，那么变换后的结果只能按部就班被分析推导出来，没有任何灵活性，比如你如果决定使用傅立叶变换了，那basis function就是正弦波，你不管怎么scale，它都是正弦波，即使你举出余弦波，它还是移相后的正弦波。总之你就只能用正弦波，没有任何商量的余地。而对于小波变换来讲，基是变的，是可以根据信号来推导或者构建出来的，只要符合小波变换的性质和特点即可。也就是说，如果你有着比较特殊的信号需要处理，你甚至可以构建一个专门针对这种特殊信号的小波basis function集合对其进行分析。这种灵活性是任何别的变换都无法比拟的。总结来说，傅立叶变换适合周期性的，统计特性不随时间变化的信号; 而小波变换则适用于大部分信号，尤其是瞬时信号。它针对绝大部分信号的压缩，去噪，检测效果都特别好。
看到这里，你应该大概了解了小波变换针对傅立叶变换的优点了。你也许对背后的原因还存在一些疑问，并希望深入了解一些小波的构建等知识，请移步本系列第二篇：傅立叶变换，小波变换和motion信号处理：第二篇