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 Voicing 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 WAVELET 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 WAVELET 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 WAVELET 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 \
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) \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
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