外文翻译小波变换在图像处理中的仿真及应用

论文翻译

通信102 吴志昊

译文：

小波变换在图像处理中的仿真及应用

一、课题意义

在传统的傅立叶分析中, 信号完全是在频域展开的, 不包含任何时频的信息, 这对于某些应用来说是很恰当的, 因为信号的频率的信息对其是非常重要的。但其丢弃的时域信息可能对某些应用同样非常重要, 所以人们对傅立叶分析进行了推广, 提出了很多能表征时域和频域信息的信号分析方法, 如短时傅立叶变换, Gabor 变换, 时频分析, 小波变换等。而小波分析则克服了短时傅立叶变换在单分辨率上的缺陷, 具有多分辨率分析的特点, 使其在图像处理中得到了广泛应用。

传统的信号理论，是建立在Fourier分析基础上的，而Fourier变换作为一种全局性的变化，其有一定的局限性。在实际应用中人们开始对Fourier变换进行各种改进，小波分析由此产生了。小波分析是一种新兴的数学分支，它是泛函数、Fourier分析、调和分析、数值分析的最完美的结晶；在应用领域，特别是在信号处理、图像处理、语音处理以及众多非线性科学领域，它被认为是继Fourier分析之后的又一有效的时频分析方法。小波变换与Fourier变换相比，是一个时间和频域的局域变换因而能有效地从信号中提取信息，通过伸缩和平移等运算功能对函数或信号进行多尺度细化分析（Multiscale Analysis），解决了Fourier变换不能解决的许多困难问题。

小波变换是一种快速发展和比较流行的信号分析方法, 其在图像处理中有非常重要的应用, 包括图像压缩, 图像去噪, 图像融合, 图像分解, 图像增强等。小波分析是傅立叶分析思想方法的发展与延拓。除了连续小波(CWT)、离散小波(DWT), 还有小波包(Wavelet Packet)和多维小波。

小波分析在图像处理中有非常重要的应用, 包括图像压缩, 图像去噪, 图像融合, 图像分解, 图像增强等。小波变换是一种新的变换分析方法，它继承和发展了短时傅立叶变换局部化的思想，同时又克服了窗口大小不随频率变化等缺点，能够提供一个随频率改变的时间一频率窗口，是进行信号时频分析和处理的理想工具。它的主要特点是通过变换能够充分突出问题某些方面的特征，因此，小波变换在许多领域都得到了成功的应用，特别是小波变换的离散数字算法已被广泛用于许多问题的变换研究中。从此，小波变换越来越引进人们的重视，其应用领域来越来越广泛。

二、课题综述

（一）小波分析的应用与发展

小波分析的应用是与小波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。电子信息技术是六大高新技术中重要的一个领域，它的重要方面是图象和信号处理。现今，信号处理已经成为当代科学技术工作的重要部分，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、精确地重构(或恢复)。从数学地角度来看，信号与图象处理可以统一看作是信号处理(图象可以看作是二维信号)，在小波分析的许

多分析的许多应用中，都可以归结为信号处理问题。现在，对于其性质随时间是稳定不变的信号（平稳随机过程），处理的理想工具仍然是傅立叶分析。但是在实际应用中的绝大多数信号是非稳定的（非平稳随机过程），而特别适用于非稳定信号的工具就是小波分析。

事实上小波分析的应用领域十分广泛，它包括：数学领域的许多学科；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、去噪声、压缩、传递等。在图象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。

（1）小波分析用于信号与图象压缩是小波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象的特征不变，且在传递中可以抗干扰。基于小波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。

（2）小波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。

总之，由于小波具有低墒性、多分辨率、去相关性、选基灵活性等特点，小波理论在去噪领域受到了许多学者的重视，并获得了良好的效果。但如何采取一定的技术消除图像噪声的同时保留图像细节仍是图像预处理中的重要课题。目前，基于小波分析的图像去噪技术已成为图像去噪的一个重要方法。

（二）在图像处理的方面，小波变换存在以下几个优点：

（1）小波分解可以覆盖整个频域(提供了一个数学上完备的描述)

（2）小波变换通过选取合适的滤波器，可以极大的减小或去除所提取得不同特征之间的相关性

（3）小波变换具有“变焦”特性，在低频段可用高频率分辨率和低时间分辨率(宽分析窗口)，在高频段，可用低频率分辨率和高时间分辨率(窄分析窗口)

（4）小波变换实现上有快速算法(Mallat小波分解算法)

小波分析已经成为发展最快和最引人注目的学科之一，几乎涉及或者应用到信息领域的所有学科。

(三)方案论证

本文对基于小波变换的图像去噪方法进行了深入的研究分析,首先详细介绍了几种经典的小波变换去噪方法。对于小波变换模极大值去噪法,详细介绍了其去噪原理和算法,分析了去噪过程中参数的选取问题,并给出了一些选取依据;详细介绍了小波系数相关性去噪方法的原理和算法;对小波变换阈值去噪方法的原理和几个关键问题进行了详细讨论。最后对这些方法进行了分析比较,讨论了它们各自的优缺点和适用条件,并给出了仿真实验结果。

在众多基于小波变换的图像去噪方法中,运用最多的是小波阈值萎缩去噪法。传统的硬阈值函数和软阈值函数去噪方法在实际中得到了广泛的应用,而且取得了较好的效果。但是硬阈值函数的不连续性导致重构信号容易出现伪吉布斯现象;而软阈值函数虽然整体连续性好,但估计值与实际值之间总存在恒定的偏差,具有一定的局限性。鉴于此,本文提出了一种基于小波多分辨率分析和最小均方误差准则的自适应阈值去噪算法。该方法利用小波阈值去噪基本原理,在基于最小均方误差算法LMS 和Stein 无偏估计的前提下,引出了一个具有多阶连续导数的阈值函数,利用其对阈值进行迭代运算,得到最优阈值,从而得到更好的图像去噪效果。最后,通过仿真实验结果可以看到,该方法去噪效果显著,与硬阈值、软阈值方法相比,信噪比提高较多,同时去噪后仍能较好地保留图像细节,是一种有效的图像去噪方法。

小波基函数选择可从以下3个方面考虑。

（1）复值与实值小波的选择

复值小波作分析不仅可以得到幅度信息，也可以得到相位信息，所以复值小波适合于分析计算信号的正常特性。而实值小波最好用来做峰值或者不连续性的检测。

（2）连续小波的有效支撑区域的选择

连续小波基函数都在有效支撑区域之外快速衰减。有效支撑区域越长，频率分辨率越好；有效支撑区域越短，时间分辨率越好。

（3）小波形状的选择

如果进行时频分析，则要选择光滑的连续小波，因为时域越光滑的基函数，在频域的局部化特性越好。如果进行信号检测，则应尽量选择与信号波形相近似的小波。

小波变换与傅里叶变换的比较

小波分析是傅里叶分析思想方法的发展和延拓。自产生以来，就一直与傅里叶分析密切相关。它的存在性证明，小波基的构造以及结果分析都依赖于傅里叶分析，二者是相辅相成的。两者相比较主要有以下不同：

（1）傅里叶变换的实质是把能量有限信号()t f 分解到以{}

jwt e 为正交基的空间上去；而小波变换的实质是把能量有限的信号()t f 分解到由小波函数所构成的空间上去。两者的离散化形式都可以实现正交变换，都满足时频域的能量守恒定律。

（2）傅里叶变换用到的基本函数只有()wt sin ，()wt cos 或()iwt exp ，具有唯一性；小波分析用到的小波函数则不是唯一的，同一个工程问题用不同的小波函数进行分析时有时结果相差甚远。小波函数的选用是小波分析应用到实际中的一个难点问题也是小波分析研究的一个热点问题，目前往往是通过经验或不断的实验，将不同的分析结果进行对照分析来选择小波函数。一个重要的经验就是根据待分析信号和小波函数的相似性选取，而且此时要考虑小波的消失矩、正则性、支撑长度等参数。

（3）在频域中，傅里叶变换具有较好的局部化能力，特别是对于那些频率成分比较简单的确定性信号，傅里叶变换很容易把信号表示成各频率成分的叠加和的形式，但在时域中，傅里叶变换没有局部化能力，即无法从信号()t f 的傅里

叶变换()w F 中看出()t f 的在任一时间点附近的性态。因此，小波变换在对瞬态信号分析中拥有更大的优势。

（4）在小波分析中，尺度a 的值越大相当于傅里叶变换中w 的值越小。

（5）在短时傅里叶变换中，变换系数()τ,w G f 主要依赖于信号在时间窗内的情况，一旦时间窗函数确定，则分辨率也就确定了。而在小波变换中，变换系数()b a WT ,虽然也是依赖于信号在时间窗内的情况，但时间宽度是随尺度a 的变化而变化的，所以小波变换具有时间局部分析的能力。因此，小波变换也可以看成是信号局部奇异性分析的有效工具。

（6）若用信号通过滤波器来解释，小波变换与短时傅里叶变换不同之处在于：对短时傅里叶变换来说，带通滤波器的带宽w ?与中心频率w 无关。

（7）从框架角度来说傅里叶变换是一种非冗余的正交紧框架，而小波变换却可以实现冗余的非正交非紧框架。

总之小波变换是图像处理中图像特征分析的新方法，特别是在图像细节的处理及图像特征分析上具有良好的效果。它具有多分辨率分析的特点，且在时域和频域都具有表征信号局部特征的能力，是一种窗口大小固定不变但形状可变，时间窗和频率窗都可变的时域局部化分析方法。即在低频部分具有较高的频率分辨率和较低的时间分辨率，在高频部分具有较高的时间分辨率和较低的频率分辨率。对于大部分信息集中在低频的图像信号的分析而言，它具有明显的优势。因此，

到的,二维小波函数分解是把尺度j 的低频部分分解成四部分:尺度j + 1 的低频部分和三个方向(水平、垂直、斜线) 的高频部分。设输入图像为PA , Hx ( Z) , Gx ( Z) , Hy ( Z) , Gy ( Z)分别为行方向和列方向的高通滤波器和低通滤波器。

结论

图像的压缩有利于图像的传输和储存，本文对静止图像的融合方法进行了研究，分析了小波融合技校的发展和瑞代处理过程，算法采用了更简单的集合分割与排序策略，提高了编码速度，减少了内存的消耗，提高了图象复原的质量。并分析了二维图像的小波重构，针对性不清晰的图片进行合理化的融合。

原文

Wavelet transform in image processing in simulation and Application

1, task significance

In the traditional analysis of signal in frequency domain, is completely unfolded, does not contain any time frequency information, which for some applications it is appropriate, because the frequency of the signal to its information is very important. But its discarded time information may be possible for some applications also is very important, so the analysis of the promotion, put forward a lot of time domain and frequency domain information signal analysis methods, such as short Fourier transform, Gabor transform, time-frequency analysis, wavelet transform. Wavelet analysis overcomes the STFT in a single resolution on the defect, has the characteristics of multi-resolution analysis, which has been widely applied in image processing.

The traditional signal theory, is built on the basis of the analysis of Fourier, Fourier transform is a kind of global change, it has some limitations. In practical application, the people start to Fourier transform are improved, thus resulting in wavelet analysis. Wavelet analysis is a new branch of mathematics, it is a universal function, Fourier analysis, harmonic analysis, numerical analysis of the most perfect crystalline; in the fields of application, especially in signal processing, image processing, speech processing and nonlinear science domain, it is considered to be the Fourier analysis after another effective when frequency analysis method. Wavelet transform and

Fourier transform, is a time and frequency domain of the local transform which can effectively extracted from the signal information, through dilation and shift operation function to function or signal multiscale analysis ( Multiscale Analysis ), to solve the Fourier transform can not solve many difficult problems

Wavelet transform is a rapid development and more popular signal analysis method, the image processing is a very important application, including image compression, image denoising, image fusion, image decomposition, image enhancement. Wavelet analysis is the analysis method of thinking in the development and continuation. In addition to continuous wavelet, discrete wavelet transform ( CWT ) ( DWT ), and the wavelet packet ( Wavelet Packet ) and multidimensional wavelet

Wavelet analysis in image processing applications are very important, including image compression, image denoising, image fusion, image decomposition, image enhancement. Wavelet transform is a new transform analysis method, it has inherited and developed the STFT localization of thought, and also overcomes the window size does not vary with frequency and other shortcomings, to provide a frequency changing with time frequency window, is a time-frequency signal analysis and processing the ideal tool. It is mainly characterized by transform can highlight some aspects of characteristics, therefore, the wavelet transform in many areas have been successfully applied, especially wavelet transform discrete digital algorithm has been widely used in many of the problems of the transformation research. Since then, the wavelet transform is more and more the introduction of people's attention, its application fields more and more widely.

2, problem overview

( a ) the application of wavelet analysis and development

The application of wavelet analysis and wavelet analysis theory to work closely together. Now, it has been in the information technology industry has made the achievement attract people's attention. Electronic information technology is the six new and high technology an important field, which is an important aspect of image and signal processing. Nowadays, signal processing has become an important part of

the work of contemporary science and technology, the purpose of signal processing is: accurate analysis, diagnosis, coding and quantization, fast transmission or storage, accurately reconstruct ( or return ). From a mathematical perspective, signal and image processing can be unified as a letter

Course number processing ( image can be viewed as a two-dimensional signal ), the wavelet analysis of the many analysis for many applications, can be attributed to the signal processing problems. Now, for its properties with time stable signal ( stationary random process ), an ideal tool in processing is still a Fourier analysis. But in the practical application of the vast majority of signal is unstable ( non stationary random process ), and is especially suitable for the unstable signal wavelet analysis tool is.

In fact the wavelet analysis applied field is very extensive, it includes many disciplines: mathematics; signal analysis, image processing; quantum mechanics, theoretical physics; military electronic warfare and weapons computer intelligent; classification and recognition; music and language artificial synthesis; medical imaging and diagnosis; seismic data processing; mechanical the fault diagnosis and so on; for example, in mathematics, it has been used in numerical analysis, structure, fast numerical method of curve and surface structure, solving differential equations, control theory. In signal analysis, noise filtering, compression, transmission and so on. In the image processing of the image compression, classification, identification and diagnosis, such as the decontamination. In medical imaging, the reduction of B ultrasound, CT nuclear magnetic resonance imaging time, improve the resolution

(1) application of wavelet analysis in signal and image compression wavelet analysis is an important application of the. It is characterized by high compression ratio, compression speed, the compressed signal can be maintained and image feature invariant, and the transfer of anti interference. The compression method based on wavelet analysis, comparative success of wavelet packet best base method, wavelet texture model method, wavelet transform Zerotree compression, wavelet transform vector compression.

(2) the wavelet in the signal analysis are widely used. It can be used for boundary

processing and filtering, time-frequency analysis, signal-noise separation and extraction of weak signal, fractal index, signal recognition and diagnosis as well as the multi-scale edge detection.

In conclusion, because wavelet has low entropy, multi-resolution, decorrelation, selected medium characteristics such as flexibility, the theory of wavelet in denoising fields by many scholars, and obtained good results. But how to take certain technology to eliminate image noise while preserving image detail is an important topic in the image pretreatment. At present, based on wavelet analysis in image denoising image denoising technology has become an important method.

(b) in the image processing field, wavelet transform has the following advantages:

(1) wavelet decomposition can cover the whole frequency domain ( provides a mathematically complete description)

(2) wavelet transform by selecting appropriate filter, can greatly reduce or remove the correlation between different feature extraction

(3) wavelet transform has a" zoom" characteristics, in the low frequency band can be used with high frequency resolution and low resolution ( width analysis window ), in the high frequency band, the available low frequency resolution and high temporal resolution ( narrow analysis window )

(4) wavelet transform on a fast algorithm ( Mallat algorithm )Wavelet analysis has become one of the fastest and most attract sb.'s attention on one of the subjects, or applied to the field of information involved in almost all disciplines.

(c) demonstration program

This paper based on the wavelet transform image denoising methods carried out in-depth research and analysis, the paper introduces several classic wavelet transform denoising method. The wavelet transform modulus maximum denoising method, described in detail the denoising principle and algorithm, analyzes the denoising process parameter selection problem, and gives some basis; detailed correlation of wavelet coefficient denoising method principle and algorithm; wavelet transform threshold denoising method principle and several key the problem is discussed in detail. The last of these methods are analyzed and compared, and discusses their

respective advantages and disadvantages and applicable conditions, and gives the simulation results.

In many image denoising based on wavelet transform method, is the largest use wavelet shrinkage denoising method. The traditional hard threshold function and soft threshold function de-noising method in practice has been widely used, and achieved good results. But the hard threshold function is discontinuous resulting reconstructed signal prone to false phenomenon of Gibbs; and the soft threshold function although the overall good continuity, but the estimated value and the actual value of aggregate in the presence of constant deviation, with certain limitations. In view of this, this paper puts forward a method based on wavelet multiresolution analysis and minimum mean square error criterion for adaptive threshold denoising algorithm. The method uses wavelet threshold de-noising principle, based on minimum mean square error algorithm of LMS and Stein unbiased estimates of the premise, the derivation of a continuous derivative with multiple threshold function, the use of the threshold for iterative operation, get the optimal threshold, resulting in better image denoising effect. Finally, the simulation results can be seen, the denoising effect is remarkable, and hard threshold, soft threshold method is compared, the SNR improvement more, at the same time denoising can preserve image details, is an effective method for image denoising.

Wavelet basis function from the following3 aspects to consider.

(1)complex and real wavelet selectionComplex wavelet analysis can not only obtain the amplitude information, can also be obtained from the phase information, so the complex wavelet is suitable for the analysis and calculation of the normal signal characteristics. But the real wavelet is best used for peak or discontinuity detection.

(2) continuous wavelet effective support region selectionContinuous wavelet basis function in the effective support outside the area of rapid decay. Effective support area is longer, the frequency resolution is better; effective support region is shorter, the better time resolution.

(3) wavelet shape selectionIf the time-frequency analysis, to select a smooth continuous wavelet, because time more smooth basis functions, in the frequency

domain localization characteristic. If the signal is detected, then you should try to choose and signal waveform phase approximation of wavelet.

Wavelet transform and FFT comparison

Wavelet analysis, FFT analysis method of thinking in the development and extension of. Since its inception, has been closely related with fft. Its existence proof, wavelet construction as well as the results of analysis are dependent on Fourier analysis, the two complement. Comparing the two following different:

(1) transform is the essence of the finite energy signal decomposition to orthogonal base space up; and the wavelet transform is the essence of the limited energy of the signal is decomposed by wavelet function of space. Both the discrete form can realize orthogonal transformation, to meet the time frequency energy conservation law.

(2) transform used in the basic function only, or, with the only; wavelet analysis using wavelet function is not the only, the same engineering problems by using different wavelet function analysis the results are sometimes differ very far. The selection of wavelet function is the application of wavelet analysis to the practical problem of a difficulty in wavelet analysis is a hot issue for the study, it is often through experience or continuous experiments, different analysis results were analyzed and compared to select the wavelet function. An important experience is based on the analysis of signal and the wavelet function similarity selection, and to consider the wavelet vanishing moments, regularity, supporting length and other parameters.

(3) in the frequency domain, transform has good localization ability, especially for those frequency components compared to simple deterministic signal, FFT is easy to signal into various frequency components of the superposition and form, but in the time domain, Fourier-transform no localization ability, i.e. not from the signal FFT to see at any point in time the nearby state. Therefore, wavelet transform in the transient signal analysis with greater advantage.

(4) in wavelet analysis, scale larger values of equivalent transform of value is small.

(5) in the short-time Fourier transform, transform coefficient depends mainly on the signal in the time window inside the case, once the time window function is determined, the resolution can be defined. In wavelet transform, transform coefficients are dependent on the signal in the time window inside the case, but the time width with scale changes, so the wavelet transform has the time local analysis ability. Therefore, the wavelet transform can also be regarded as local signal singularity analysis tool.

(6) if the signal through the filter, wavelet transform and short time Fourier transform is different: the STFT, band-pass filter bandwidth and center frequency independent; on the contrary, wavelet band-pass filter bandwidth is proportional to the center frequency.

(7) from the frame angle for FFT is a non-redundant orthogonal compact frame, and the wavelet transform can realize the redundant non orthogonal non tight frame.In wavelet transform in image processing is a new method of image feature analysis, especially in the details of the image processing and image feature analysis has a good effect. It has the characteristics of multi-resolution analysis, and in both time domain and frequency domain ability of characterizing local features, is a window size is fixed but the shape of the variable, the time window and window are both variable frequency domain localization method. In the low frequency portion has a high frequency resolution and low time resolution in the high-frequency part, has a high time resolution and low frequency resolution. For most of the information is concentrated in the low frequency image signal analysis, it has obvious advantages. Therefore, one of the great advantages of wavelet analysis is to reflect the signal local character.

Three, task design

In 1, the basic knowledge of wavelet transform

n mathematics, the definition of the given function localization of wavelet function. Wavelets can be defined by a finite interval function in structure, become the mother wavelet and wavelet. A set of wavelet basis function { )(,x b a }, by zooming and

panning the basic wavelet generating, )(,x b a ψ=)(1a

b x a -ψ. Where a is a scaling parameter, responses to specifi

c basis function width; B for translational parameters, designate

d along th

e X axis o

f the translational position.Wavelet transform is a kind of signal time -- scale analysis method, it has the characteristics of multi-resolution analysis in time and frequency domain, and two have ability of characterizin

g local features, is a kind of window size fixed but its shape is variable, the time window and frequency windows are variable time-frequency analysis method. In the low frequency portion has a hig

h frequency resolution and time resolution, in the high frequency portion has a high time resolution and low frequency resolution, it is suitable for detection of transient entrainment of the normal signal anomaly and show their constituents, thus is called a microscope of signal analysis. Wavelet analysis of the signal is decomposed into low frequency and high frequency D1A1two part, in the decomposition, low frequency A1lost information by high frequency D1 capture. In the next layer decomposition, and the A1is decomposed into low frequency and high frequency parts of A2D2 two, A2lost in the low frequency information by high frequency D2 capture, and so on down, can deeper decomposition. Two dimensional wavelet function is obtained by one-dimensional wavelet function after tensor transformation to obtain the wavelet function, decomposition is the scale J low frequency component is decomposed into four parts: the scale j + 1low frequency component and the three directions (horizontal, vertical, diagonal) of the high frequency part. Set the input image into PA, Hx ( Z ), Gx ( Z ), Hy ( Z ), Gy ( Z ) respectively for the row and column directions of the high-pass filter and low-pass filter.

Conclusion

Image compression for image transmission and storage, this paper still image fusion method based on wavelet fusion, analyzes technical development and the Swiss generation process, algorithm uses a simpler set partitioning and sequencing strategy, improve the coding speed, reduce the memory consumption, improve the quality of image restoration. And the analysis of 2D image wavelet reconstruction, targeted not

clear picture of rationalization of the fusion.

机械设计设计外文文献翻译、中英文翻译、外文翻译

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会计内部控制中英文对照外文翻译文献

会计内部控制中英文对照外文翻译文献(文档含英文原文和中文翻译)

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外文翻译 (2)

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外文翻译中文

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外文翻译--如何监测内部控制

附录A

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queuing modeling排队论的matlab仿真(包括仿真代码)

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