【正文】 any 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 bination of the basis vectors in that vector space , ., 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 puting the linear bination equation. All the definitions and theorems related to this subject can be found in Keiser39。 it is related to the location of the window, as the window is shifted through the signal. This term, obviously, corresponds to time 外文文獻(xiàn)原文 3 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 ponent 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 ponents for short durations and low frequency ponents 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 ponent throughout the entire signal and relatively high frequency ponents for a short duration somewhere around the middle. WAVELE PACKETS Performance of any transform in a particular application is highly dependent on the basis functions chosen. The choice of quadrature mirror ?lters (QMF) used in a wavelet packet analysis should be taken into consideration in a pression scheme. The choice of appropriate QMF is not only signal dependent, but scale dependent within a given generalization, known as Hybrid Wavelet Packet analysis, which includes the choice of optimal QMF at each level of deposition is explored. Computational results show that optimized hybrid wavelet packet bases provide better pression for a class of signals and indicate methods to develop search strategies to ?nd these optimal bases. The discrete wavelet transform (DWT) can be characterized as a recursive application of the highpass and lowpass ?lters that form a QMF pair. The calculation of the DWT begins by ?ltering a signal by the highpass and lowpass ?lters and then downsampling the output. The putation proceeds by applying the QMF pair to the output of the lowpass recursion, then, is simply just a repeated application of the QMF pair to the lowpass ?ltered output of the previous level. Wavelet packets are generated by only slightly changing this operation. To calculate a wavelet packet deposition, the procedure begins as before, with the application of the QMF to the data followed by downsampling. However, now the putation proceeds by applying the QMF to not only the owpass output but to the highpass output as well. The recursion is simply to ?lter and downsample all output of the previous level. The calculation of wavelet packets is often schematically 外文文獻(xiàn)原文 4 characterized by the formation of a binary tree with each branch representing the highpass and lowpass ?ltered output of a root node. De?nition A tableau is a wavelet packet tree. It is a structure for anizing the output of the recursive applications of a single QMF pair in a wavelet packet expansion. HYBRIDWAVELET PACKETS Accepting that the choice of QMF inherently affects the performance of a pression scheme, a simple solution is to perform a best basis analysis for m different QMFs and then choose the “best of the best.” This offers the possibility of improved pression, but this simple approach is merely a super?cial use of multiple QMFs. The central concept of this research is that the choice of appropriate QMF is not only signal dependent but scale dependent within a given signal as well. That is, given pression as the ultimate goal, the choice of QMF which yields the best performance may change at different levels within a wavelet packet analysis. (Ⅱ ) COMPLEX RIDGELETS FOR IMAGE DENOISING INTRODUCTION Wavelet transforms have been successfully used in many scientific fields such as image pression, image denoising, signal processing, puter graphics,and pattern recognition, to name only a and his coworkers pioneered a wavelet denoising scheme by using soft thresholding and hard thresholding. This approach appears to be a good choice for a number of applications. This is because a wavelet transform can pact the energy of the image to only a small number of large coefficients and the majority of the wavelet coeficients are very small so that they can be set to zero. The thresholding of the wavelet coeficients can be done at only the detail wavelet deposition subbands. We keep a few low frequency wavelet subbands untouched so that they are not thresholded. It is well known that Dono