小波分析中英文对照外文翻译文献

小波分析中英文对照外文翻译文献
(文档含英文原文和中文翻译)
译文：
一小波研究的意义与背景
在实际应用中，针对不同性质的信号和干扰，寻找最佳的处理方法降低噪声，一直是信号处理领域广泛讨论的重要问题。

目前有很多方法可用于信号降噪，如中值滤波，低通滤波，傅立叶变换等，但它们都滤掉了信号细节中的有用部分。

传统的信号去噪方法以信号的平稳性为前提，仅从时域或频域分别给出统计平均结果。

根据有效信号的时域或频域特性去除噪声，而不能同时兼顾信号在时域和频域的局部和全貌。

更多的实践证明，经典的方法基于傅里叶变换的滤波，并不能对非平
稳信号进行有效的分析和处理，去噪效果已不能很好地满足工程应用发展的要求。

常用的硬阈值法则和软阈值法则采用设置高频小波系数为零的方法从信号中滤除噪声。

实践证明，这些小波阈值去噪方法具有近似优化特性，在非平稳信号领域中具有良好表现。

小波理论是在傅立叶变换和短时傅立叶变换的基础上发展起来的，它具有多分辨分析的特点，在时域和频域上都具有表征信号局部特征的能力，是信号时频分析的优良工具。

小波变换具有多分辨性、时频局部化特性及计算的快速性等属性，这使得小波变换在地球物理领域有着广泛的应用。

随着技术的发展，小波包分析(Wavelet Packet Analysis)方法产生并发展起来，小波包分析是小波分析的拓展，具有十分广泛的应用价值。

它能够为信号提供一种更加精细的分析方法，它将频带进行多层次划分，对离散小波变换没有细分的高频部分进一步分析，并能够根据被分析信号的特征，自适应选择相应的频带，使之与信号匹配，从而提高了时频分辨率。

小波包分析(wavelet packet analysis)能够为信号提供一种更加精细的分析方法，它将频带进行多层次划分，对小波分析没有细分的高频部分进一步分解，并能够根据被分析信号的特征，自适应地选择相应频带,使之与信号频谱相匹配，因而小波包具有更广泛的应用价值。

利用小波包分析进行信号降噪，一种直观而有效的小波包去噪方法就是直接对小波包分解系数取阈值，选择相关的滤波因子，利用保留下来的系数进行信号的重构，最终达到降噪的目的。

运用小波包分析进行信号消噪、特征提取和识别是小波包分析在数字信号处理中的重要应用。

二小波分析的发展与应用
小波包分析的应用是与小波包分析的理论研究紧密地结合在一起的。

近年来，小波包的应用范围也是越来远广。

小波包分析能够把任何信号映射到一个由基本小波伸缩、平移而成的一组小波函数上去。

实现信号在不同时刻、不同频带的合理分离而不丢失任何原始信息。

这些功能为动态信号的非平稳描述、机械零件故障特征频率的分析、微弱信号的提取以实现早期故障诊断提供了高效、有力的工具。

（1）小波包分析在图像处理中的应用
在图像处理中，小波包分析的应用是很成功的，而这一方面的著作和学术论文也特别多。

二进小波变换用于图像拼接和镶嵌中，可以消除拼接缝。

利用正交变换和小波包进行图像数据压缩。

可望克服由于数据压缩而产生的方块效应，获得较好的压缩效果。

利用小波包变换方法可进行边缘检
测、图像匹配、图像目标识别及图像细化等。

（2）小波包分析在故障诊断中的应用
小波包分析在故障诊断中的应用已取得了极大的成功。

小波包分析不仅可以在低信噪比的信号中检测到故障信号，而且可以滤去噪声恢复原信号，具有很高的应用价值。

小波包变换适用于电力系统故障分析，尤其适用于电动机转子鼠笼断条以及发电机转子故障分析。

用二进小波Mallat算法对往复压缩机盖振动信号进行分解和重构，可诊断出进、排气阀泄漏故障。

利用小波包对变速箱故障声压信号进行分解，诊断出了变速箱齿根裂纹故障等。

（3）小波包分析在语音信号处理中的应用
语音信号处理的目的是得到一些语音参数以便高效地传输或存储。

利用小波包分析可以提取语音信号的一些参数，并对语音信号进行处理。

小波包理论应用在语音处理方面的主要内容包括：清浊音分割、基音检测、去躁、重建与数据压缩等几个方面。

小波包应用于语音信号提取、语音台成语音增加波形编码已取得了很好的效果。

三基础知识介绍
近年来,小波理论得到了非常迅速的发展,而且由于其具备良好的时频特性,实际应用也非常广泛。

这里希望利用小波的自身特性,在降低噪声影响的同时,尽量保持图像本身的有用细节和边缘信息,从而保证图像的最佳效果。

小波合成
连续小波变换是一种可逆的变换，只要满足方程2。

幸运的是，这是一个非限制性规定。

如果方程2得到满足，连续小波变换是可逆的，即使基函数一般都是不正交的。

重建可能是使用下面的重建公式：
公式1小波逆变换公式
其中C_psi是一个常量，取决于所使用的小波。

该重建的成功取决于这个叫做受理的常数，受理满足以下条件：
公式2受理条件方程
这里 psi^hat(xi) 是 FT 的psi(t)，方程2意味着psi^hat(0) = 0，这是:
公式3
如上所述，公式3并不是一个非常严格的要求，因为许多小波函数可以找到它的积分是零。

要满足方程3，小波必须振荡。

连续小波变换
连续小波变换作为一种替代快速傅里叶变换办法来发展，克服分析的问题。

小波分析和STFT 的分析方法类似，在这个意义上说，就是信号和一个函数相乘，{它的小波}，类似的STFT的窗口功能，并转换为不同分段的时域信号。

但是，STFT和连续小波变换二者之间的主要区别是：
1、Fourier转换的信号不采取窗口，因此，单峰将被视为对应一个正弦波，即负频率是没有计算。

2、窗口的宽度是相对于光谱的每一个组件变化而变化的，这是小波变换计算最重要的特征。

连续小波变换的定义如下：
公式4
从上面的方程可以看出，改变信号功能的有两个变量，τ和s，分别是转换参数和尺度参数。

psi(t)为转化功能。

小波包分析的基本原理
目前大多数数字图像系统中，输入图像都是采用先冻结再扫描方式将多维图像变成一维电信号，再对其进行处理、存储、传输等加工变换。

最后往往还要在组成多维图像信号，而图像噪声也将同样受到这样的分解和合成。

噪声对图像信号幅度、相位的影响非常复杂，有些噪声和图像信号是相互独立不相关的，而有些则是相关的，并且噪声本身之间也可能相关。

因此要有效降低图像中的噪声，必须针对不同的具体情况采用不同方法，否则就很难获得满意的去噪效果。

一般图像去噪中常见的噪声有以下几种：
1）加性噪声：加性噪声和图像信号强度是不相关的，如图像在传输过程中引进的“信道噪声”电视摄像机扫描图像的噪声等。

这类带有噪声的图像可看成是理想的没有被噪声“污染”的图像与噪声。

2）乘性噪声：图像的乘性噪声和图像的加性噪声是不一样的，加性噪声和图像信号强度是不相关的，而乘性噪声和图像信号是相关的，往往随着图像信号的变化而发生变化，如飞点扫描图像中的噪声、电视扫描光栅、胶片颗粒噪声等。

3）量化噪声：量化噪声是数字图像的主要噪声源，它的大小能够表示出数字图像和原始图像的差异程度，有效减少这种噪声的最好办法就是采用按灰度级概率密度函数选择量化级的最优量化措施。

4）“椒盐”噪声：此种噪声很多，例如在图像切割过程中引起的黑图像上的白点、白图像上的黑点噪声等，还有在变换域引入的误差，在图像反变换时引入的变换噪声等。

实际生活中还有多种多样的图像噪声，如皮革上的疤痕噪声、气象云图上的条纹噪声等。

这些噪声一般都是简单的加性噪声，不会随着图像信号的改变而改变。

这为实际的去噪工作提供了依据。

2.图像去噪效果的评价
在图像去噪的处理中，常常需要评价去噪后图像的质量。

这是因为一个图像经过去噪处理后所还原图像的质量好坏，对于人们判断去噪方法的优劣有很重要的意义。

目前对图像的去噪质量评价主要有两类常用的方法：一类是人的主观评价，它由人眼直接观察图像效果，这种方法受人为主观因素的影响比较大。

目前由于对人的视觉系统性质还没有充分的理解，对人的心理因素还没有找到定量分析方法。

因此主观评价标准还只是一个定性的描述方法，不能作定量描述，但它能反映人眼的视觉特性。

另一类是图像质量的客观评价。

调试环境-MATLAB开发平台
MATLAB是Math Works公司开发的一种跨平台的，用于矩阵数值计算的简单高效的数学语言，与其它计算机高级语言如C, C++, Fortran, Basic, Pascal等相比，MATLAB语言编程要简洁得多，编程语句更加接近数学描述，可读性好，其强大的圆形功能和可视化数据处理能力也是其他高级语言望尘莫及的。

四综述
众所周知，由于图像在采集、数字化和传输过程中常受到各种噪声的干扰，从而使数字图像中包含了大量的噪声。

能否从受扰信号中获得去噪的信息，不仅与干扰的性质和信号形式有关，也与信号的处理方式有关。

在实际应用中，针对不同性质的信号和干扰，寻找最佳的处理方法降低噪声，一直是信号处理领域广泛讨论的重要问题。

小波包分析的应用是与小波包分析的理论研究紧密地结合在一起的。

现在，它已经在科技信息产业领域取得了令人瞩目的成就。

如今，信号处理已经成为当代科学技术工作的重要组成部分，信号处理的目的就是：准确的分析、诊断、编码、压缩和量化、快速传递或存储、精确的恢复（或重构）。

从数学的角度来看，信号与图像处理可以统一看作是信号处理，在小波包分析的许多分析的许多应用中，都可以归结为信号处理问题。

小波包分析的应用领域十分广泛，它包括：信号分析、图象处理、量子力学、理论物理、军事电子对抗与武器的智能化、计算机分类与识别、音乐与语言的人工合成、医学成像与诊断、地震勘探数据处理、大型机械的故障诊断等方面。

例如，在数学方面，它已用于数值分析、构造快速数值方法、曲线曲面构造、微分方程求解、控制论等。

在信号分析方面的滤波、去噪、压缩、传递等。

在图像处理方面的图象压缩、分类、识别与诊断，去污等。

在医学成像方面的减少B超、CT、核磁共振成像的时间，提高分辨率等。

小波包分析用于信号与图像压缩是小波包分析应用的一个重要方面。

它的特点是压缩比高，压缩速度快，压缩后能保持信号与图像的特征不变，且在传递中可以抗干扰。

基于小波包分析的压缩方法很多，比较成功的有小波包最好基方法，小波域纹理模型方法，小波变换零树压缩，小波变换向量压缩等。

小波包在信号分析中的应用也十分广泛。

它可以用于边界的处理与滤波、时频分析、信噪分离与提取弱信号、求分形指数、信号的识别与诊断以及多尺度边缘检测等。

A ·The wavelet study the meaning and background
In practical applications, the different nature of the signal and interference, to find the best processing method to reduce noise, the important issue is widely discussed in the field of signal processing. Currently, there are many methods can be used to signal noise reduction, such as median filtering, low pass filtering, Fourier transform, etc., but they are filtered out by the useful part of the signal details. The traditional signal de-noising method smooth signal only from the time domain or frequency domain are given the results of the statistical average. Time domain or frequency domain characteristics of the effective signal to noise removal, but not taking into account the local and the whole picture of the signal in the time domain and frequency domain. More Practice has proved that the classical approach based on the Fourier transform of the filter, and can not be non-stationary signal analysis and processing, denoising effect can not meet the requirements of engineering application development. In recent years, many papers non-stationary signal wavelet threshold de-noising method. Donoho and Johnstone contaminated with Gaussian noise signal
de-noising by thresholding wavelet coefficients. Commonly used hard threshold rule and soft threshold rule set to filter out the noise from the signal high-frequency wavelet coefficients to zero. Practice has proved that these wavelet thresholding method with approximate optimization features, has a good performance in the field of non-stationary signals. The threshold rule mainly depends on the choice of parameters. For example, the hard threshold and soft threshold depends on the choice of a single parameter - global threshold lambda lambda adjustment is critical However, due to the non-linearity of the wavelet transform. Threshold is too small or too large, will be directly related to the pros and cons of the signal de-noising effect. When the threshold value is dependent on a number of parameters, the problem will become more complex. In fact, the effective threshold denoising method is often determined based on wavelet decomposition at different levels depending on the threshold parameter, and then determine the appropriate threshold rule. Compared with the wavelet analysis, wavelet packet analysis (Wavelet Packet Analysis) to provide a more detailed analysis for the signal, it will band division of multi-level, multi-resolution analysis
is no breakdown of the high-frequency part of the further decomposition, and according to the characteristic of the signal being analyzed, adaptive selection of the corresponding frequency band, to match with the signal spectrum, thereby increasing the time - frequency resolution. The wavelet packet transform is the promotion of the wavelet transform in signal with more flexibility than the wavelet transform. Using wavelet packet transform to the signal decomposition, the low-frequency part and
high-frequency components are further decomposed. Wavelet packet signal de-noising threshold method combined with good application value.
At present, both in engineering applications and theoretical study, removal of signal interference noise is a hot topic. Extract valid signal is band a wide interference or white noise pollution signal mixed with noise signal, has been an important part of signal processing. The traditional digital signal analysis and processing is to establish the basis of Fourier transform, Fourier transform stationary signals in the time domain and frequency domain algorithm to convert each other, but can not accurately represent the signal time-frequency localization properties. For non-stationary signals people use short-time Fourier transform, but it uses a fixed short-time window function is a single-resolution signal analysis method, there are some irreparable defect. Wavelet theory is developed on the basis of Fourier transform and short-time Fourier transform, and it has the characteristics of multi-resolution analysis, have the ability to characterize the local signal characteristics in the time domain and frequency domain, is an excellent tool for signal analysis . Wavelet transform (Wavelet transform) emerged in the mid 1980s when the frequency domain signal analysis tools, since 1989 S.Mallat the first time since the introduction of wavelet transform image processing, wavelet transform its excellent time-frequency local capacity and good to go related capacity in the field of image compression coding has been widely used, and achieved good results. Multi-resolution wavelet transform, time-frequency localization characteristics and calculation speed and other attributes, which makes the wavelet transform has been widely applied in the field of geophysics. Such as: using wavelet transform gravity and magnetic parameters of the extraction, the magnitude of the error of the reconstructed signal with the original signal after the wavelet analysis as a standard to select the wavelet basis
Seismic data denoising. As technology advances, the wavelet packet analysis (Wavelet Packet Analysis) method developed wavelet packet analysis is the expansion of the wavelet analysis, with a very wide range of application. It is able to signal to provide a more detailed analysis of the method, it is the band
multi-level framing is not broken down at high frequency portion of the discrete wavelet transform is
further analyzed, and according to the characteristics of the signal to be analyzed, adaptively selecting the frequency band corresponding to , with the signal matching, thereby increasing the time-frequency resolution. The wavelet packet analysis (wavelet packet analysis) signal to be able to provide a more detailed analysis of the method, it is divided band multi-level wavelet analysis no breakdown of the high frequency portion is further decomposed, and according to the characteristic of the signal being analyzed, adaptively select the appropriate frequency band, the signal spectrum to match, thus wavelet packet has a wider range of applications. Fractal theory of wavelet packet by U.S. scientists BBMandelbrot in the
mid-1970s the creation of "self-similarity" and "self-affine fractal object, dimension to quantitatively describe the complexity of the signal, it is mainly research, widely used in many fields of science, including the recent wavelet analysis and fractal theory, is used to determine the overlap complex chemical signals in the group scores and the peak position and fractal characteristics of the DNA sequence. Using wavelet packet analysis for signal noise reduction, an intuitive and effective wavelet packet de-noising method is the direct thresholding wavelet packet decomposition coefficients, select the filter factor coefficient signal reconstruction preserved, and ultimately to drop The purpose of the noise. Signal de-noising using wavelet packet analysis, feature extraction and recognition is an important application of wavelet packet analysis in digital signal processing.
B·The development and application of wavelet analysis
Wavelet packet analysis of the application of theoretical research and wavelet packet analysis closely together. Now, it has been made in the field of science and technology information industry made remarkable achievements. Electronic information technology is an area of six high-tech focus, image and signal processing. Today, the signal processing has become an important part of the contemporary scientific and technical work, the purpose of signal processing: an accurate analysis, diagnosis, compression coding and quantization, rapid transfer or storage, accurately restore (or reconstructed). From the point of view of mathematically, signal and image processing can be unified as a signal processing, wavelet packet analysis many many applications of the analysis, can be attributed to the signal processing problem. Now, for its nature with practice is stable and unchanging signal processing ideal tool still Fourier analysis. However, in practical applications, the vast majority of the signal is stable, while the tool is especially suitable for
non-stationary signal is wavelet packet analysis.
In recent years, the combined fund research projects and corporate research projects. China in the
application of wavelet packet analysis carried out some exploration.
First, wavelet packet signal analysis, the the boundary singularity processing method and wavelet packet processing in the frequency domain positioning is perfect from the application point of view. Harmonic wavelet packet analysis method, and the harmonic wavelet packet and fractal combined to solve practical problems in engineering.
Secondly, in the operation of the rotor vibration signal detection of the fault feature analysis simulation and practical research. Motor noise analysis method using wavelet packet analysis theory to identify the impact threshold to noise singular signal of the acceleration of the vehicle, using the method of wavelet packet analysis and come to a satisfactory conclusion, while the harmonic wavelet packet combined with the fractal theory. Automobile gearbox nonlinear crack fault feature, the first application of the method of combining wavelet analysis and fractal theory and the technical design of the vehicle driveline. Middle and low agricultural transport light goods vehicle driveline job stability is not good, the problem of short working life, in the practical application of engineering to explore a new way.
Next, using theoretical analysis, experiments and software implementation phase junction station, namely the use of wavelet packet analysis and computer programs to achieve the digital signal processing. In the analysis of non-stationary signals, respectively, using existing technology and wavelet packet analysis method, the fractal method is used, expect improvements in digital signal processing. To reflect the complex characteristics of the information to improve the accuracy of the signal analysis and detection, reached the advanced level. On the basis of cooperation with others to complete a set of signal processing methods and techniques of high-speed data processing system.
In recent years, the range of applications of the wavelet packet is increasingly far and wide. Wavelet packet analysis any signal can be mapped to a basic wavelet telescopic pan from the wavelet function up. Signal to achieve a reasonable separation of the different frequency bands at different times, without losing any of the original information. These features for non-stationary dynamic signal description, analysis of the mechanical parts fault characteristic frequency, weak signal extraction provides an efficient and powerful tool to achieve early fault diagnosis. In recent years, through the continuous efforts of the scientific and technical personnel in China have achieved encouraging progress, successfully developed a wavelet transform signal analyzer, to fill the gap with the international advanced level. In theoretical and applied research on the basis of the generally applicable to non-stationary detection and diagnosis of mechanical equipment online and offline technologies and devices to obtain economic benefits. The National Science
and Technology Progress Award.
(1) wavelet packet analysis applications in image processing
In image processing, the application of wavelet packet analysis is very successful, and this aspect of books and academic papers are particularly high. Dyadic wavelet transform for image mosaic and mosaic, can eliminate the seam. Orthogonal transform and wavelet packet image data compression. Is expected to overcome the the blocking effects arising due to compression of data, to obtain better compression results. Wavelet packet transform method for edge detection, image matching, image target recognition and image thinning.
(2) The wavelet packet analysis application in fault diagnosis
Wavelet packet analysis in fault diagnosis has been made a great success. Wavelet packet analysis can not only be detected in the low signal-to-noise ratio of the signal to the fault signal, and can filter out the noise to restore the original signal has a high application value. Wavelet packet transform is applied to power system fault analysis, particularly suitable for motor rotor cage broken bars and generator rotor failure analysis. With the dyadic wavelet Mallat algorithm reciprocating compressor cover vibration signal decomposition and reconstruction can be diagnosed into the exhaust valve leakage fault. Gearbox failure sound pressure signal using wavelet packet decomposition, diagnose gearbox root crack fault.
Wavelet packet analysis in speech signal processing. The purpose of the speech signal processing is to get some of the speech parameters for efficient transmission or storage. Wavelet packet analysis can extract some of the parameters of the speech signal, speech signal processing. The main contents include: the theory of wavelet packet used in voice processing V oicing segmentation, pitch detection, to impatient to rebuild data compression and other aspects. Wavelet Packet used in speech signal extraction, the voice station into increased voice waveform coding has achieved very good results.
Wavelet packet analysis in mathematics and physics. In the field of mathematics, wavelet packet analysis is a powerful tool for numerical analysis, a simple and effective way to solve partial differential equations and integral equations. Also good for solving linear and nonlinear problems. The resulting wavelet finite element method and wavelet boundary element method, greatly enriched the contents of the numerical analysis method.
In the field of physics, wavelet packet represents a new condensed matter in quantum mechanics. In the adaptive optics. There are currently study wavelet packet transform wavefront reconstruction. In addition, the suitability of wavelet packet transform to portray irregularities, provides a new tool for turbulence
research.
Wavelet analysis in medical applications. Micronucleus identification has important applications in medicine. Environmental testing, pharmaceutical and other sets of objects can be used for toxin detection. In the micronucleus computer automatic identification, continuous wavelet can accurately extract the edge of the nucleus. Currently, it is being studied by using wavelet packet transform brain signal analysis and processing, This will effectively eliminate the transient interference and EEG short-term, low-energy transient pulse is detected.
Wavelet packet analysis neural network. Wavelet packet theory provides a prequel network analysis and theoretical framework that the wavelet form in the network structure is used to make specific spectral information contained in the training data. Wavelet packet transform designed to handle network training can greatly simplified. Unlike traditional ago
The case of a neural network structure, where the function is convex. Global grant urinate only the wavelet packet analysis and neural network node sets up the equipment intelligent diagnosis. The use of wavelet packet analysis can be given the initial alignment of the linear and nonlinear models of the inertial navigation system.
Wavelet packet analysis in engineering calculations. The matrix operations frequently encountered problems in the project, such as dense matrix acting on the vector (discrete) or integral operator acting on the calculation of the function (continuous). Sometimes computation great, fast wavelet transform, so that the operator is greatly reduced. In addition, CAD / C AM, large-scale engineering finite element analysis, mechanical engineering optimization design, automatic test system design aspects of wavelet packet analysis should be examples.
Wavelet packet analysis equipment protection and status detection system can also be used, such as
high-voltage line protection and generator stator inter-turn short circuit protection. In addition, the wavelet packet analysis is also used in astronomical research, weather analysis, identification and signal sending.
C·BASIC THEORY
In recent years,wavelet theory has been very rapid development,but also because of its good
time-frequency character istics of awide range of practical applications. Here wish to take advantage of the self-wavelet features,in the reduction of noise at the same time,to keep the details of the image itself and the edge of useful information,thus ensuring the best image.one of image wavelet thresholding denoising method can be said that many image denoising methods are the best.
THE W A VELET THEORY: A MATHEMATICAL APPROACH
This section describes the main idea of wavelet analysis theory, which can also be considered to be the underlying concept of most of the signal analysis techniques. The FT defined by Fourier use basis functions to analyze and reconstruct a function. Every vector in a vector space can be written as a linear combination of the basis vectors in that vector space , i.e., by multiplying the vectors by some constant numbers, and then by taking the summation of the products. The analysis of the signal involves the estimation of these constant numbers (transform coefficients, or Fourier coefficients, wavelet coefficients, etc). The synthesis, or the reconstruction, corresponds to computing the linear combination equation.
All the definitions and theorems related to this subject can be found in Keiser's book, A Friendly Guide to Wavelets but an introductory level knowledge of how basis functions work is necessary to understand the underlying principles of the wavelet theory. Therefore, this information will be presented in this section.
THE WA VELET SYNTHESIS
The continuous wavelet transform is a reversible transform, provided that Equation 2 is satisfied. Fortunately, this is a very non-restrictive requirement. The continuous wavelet transform is reversible if Equation 2 is satisfied, even though the basis functions are in general may not be orthonormal. The reconstruction is possible by using the following reconstruction formula:
Equation 1 Inverse Wavelet Transform
where C_psi is a constant that depends on the wavelet used. The success of the reconstruction depends on this constant called, the admissibility constant , to satisfy the following admissibility condition :
Equation 2 Admissibility Condition
where psi^hat(xi) is the FT of psi(t). Equation 2 implies that psi^hat(0) = 0, which is:
Equation 3
As stated above, Equation 3 is not a very restrictive requirement since many wavelet functions can be found whose integral is zero. For Equation 3 to be satisfied, the wavelet must be oscillatory.
THE CONTINUOUS W AVELET TRANSFORM
The continuous wavelet transform was developed as an alternative approach to the short time Fourier transform to overcome the resolution problem. The wavelet analysis is done in a similar way to the STFT analysis, in the sense that the signal is multiplied with a function, {it the wavelet}, similar to the window
function in the STFT, and the transform is computed separately for different segments of the time-domain signal. However, there are two main differences between the STFT and the CWT:
1. The Fourier transforms of the windowed signals are not taken, and therefore single peak will be seen corresponding to a sinusoid, i.e., negative frequencies are not computed.
2. The width of the window is changed as the transform is computed for every single spectral component, which is probably the most significant characteristic of the wavelet transform.
The continuous wavelet transform is defined as follows
Equation4
As seen in the above equation , the transformed signal is a function of two variables,τ and s ,the translation and scale parameters, respectively. psi(t) is the transforming function, and it is called the mother wavelet . The term mother wavelet gets its name due to two important properties of the wavelet analysis as explained below:
The term wavelet means a small wave . The smallness refers to the condition that this (window) function is of finite length (compactly supported). The wave refers to the condition that this function is oscillatory . The term mother implies that the functions with different region of support that are used in the transformation process are derived from one main function, or the mother wavelet. In other words, the mother wavelet is a prototype for generating the other window functions.
The term translation is used in the same sense as it was used in the STFT; it is related to the location of the window, as the window is shifted through the signal. This term, obviously, corresponds to time information in the transform domain. However, we do not have a frequency parameter, as we had before for the STFT. Instead, we have scale parameter which is defined as $1/frequency$. The term frequency is reserved for the STFT. Scale is described in more detail in the next section.
MULTIRESOLUTION ANALYSIS
Although the time and frequency resolution problems are results of a physical phenomenon (the Heisenberg uncertainty principle) and exist regardless of the transform used, it is possible to analyze any signal by using an alternative approach called the multiresolution analysis (MRA) . MRA, as implied by its name, analyzes the signal at different frequencies with different resolutions. Every spectral component is not resolved equally as was the case in the STFT.
MRA is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies. This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long durations. Fortunately, the signals that are encountered in practical applications are often of this type. For example, the following shows a signal of this type. It has a relatively low frequency component throughout the entire signal and relatively high frequency components for a short duration somewhere around the middle.
he basic principle of wavelet packet analysis
image noise classification
Most digital imaging systems, the input image are based on the first freeze and then scan the multi-dimensional image into a one-dimensional electrical signal, its processing, storage, transmission and processing transform. Finally, they often have in the composition of multi-dimensional image signal, image noise will be equally subject to such decomposition and synthesis. The impact of noise on the image signal amplitude and phase is very complicated, some noise and image signals are independent of each other Irrelevant, while others are related to, and the noise itself may also be relevant. Therefore, to effectively reduce the noise in the image, using different methods must be specific for the type, otherwise it is difficult to obtain a satisfactory denoising effect. Common in the general image denoising noise are the following: 1) is not relevant to additive noise: the additive noise and the image signal intensity, such as the image introduced during transmission channel noise of the scanned image of the television camera noise. Such with noise of the image can be seen as the ideal no noise pollution "image noise.
2) multiplicative noise: image multiplicative noise and image additive noise is not the same, the additive noise and image signal strength is not related to the multiplicative noise and image signals are related, often with the image signal change change, flying point in a scanned image noise, the TV raster scanned film grain noise.
3) quantization noise: the quantization noise is the main noise source of a digital image, its size can show the degree of difference of the digital image and the original image, effectively reducing this noise the best way is to select grayscale probability density function quantified level optimal quantitative measures.
4) "salt and pepper" noise: Many of such noise, such as white spots on the black image in the the image cutting process caused the white image on the black point noise, the error introduced in the transform domain, the inverse transform of the image introducing the transformed noise.
Real life there are a variety of image noise, such as leather scar noise, weather maps stripe noise. These noises are generally simple additive noise will not change with the change of the image signal. This provides a basis for actual denoising.
2. Evaluation of the effectiveness of image denoising
In the image denoising processing is often necessary to evaluate the quality of the image denoising. This is because an image after denoising restore the image quality is good or bad, has a very important significance for the people to judge the merits of de-noising method. Current image denoising quality evaluation mainly there are two commonly used methods: one is the subjective evaluation, it is directly observed by the human eye image effects, which, due to the relatively large human subjective factors. Due to the nature of the human visual system is not fully understood, the psychological factors have yet to find a quantitative analysis method. Subjective evaluation criteria is only a qualitative description can not be quantitative description, but it reflects the human visual characteristics. The other is an objective evaluation of the image quality. It is a mathematical statistics on the processing method, its disadvantage is that it does not always reflect the human eye's real feeling. A compromise approach in assessing the pros and cons of image denoising algorithm, the subjective and objective two standards considered together.
debugging environment-MATLAB development platform
MATLAB Math Works, Inc. to develop a cross-platform, used for the the matrix numerical calculation of the simple and efficient mathematical language, compared with other high-level computer language such as。