【正文】 ，……λJ]。=傳統(tǒng)的小波閾值方法被證明具有有效的和良好的視覺效果[ 1 ]。核磁共振(MR)課題類：94A12文獻(xiàn)標(biāo)識(shí)碼：A 文章編號(hào)：（2006）05047305denoise　　2000 MRSubject Classification : 94A12　　Document code : A　　　　Article ID : 0255—7797(2006)05—0473—051 　IntroductionSince the beginning of wavelet transforms in signal processing ,it has been noticed that wavelet thresholding is of considerable interest for removing noise from signals and images. Recently , Donoho and others [1 ,2] have presented the soft2thresholding(CST)and the hard—thresholding (CHT). Several years later ,promise function (CCT) and modulus squared function (CMST) [3 ,4] are also proposed.The conventional wavelet thresholding method is shown to be effective and have agood visual effect [1] . Following are the three steps in wavelet thresholding methods.1. Apply the discrete wavelet transform (DWT) to the vector Y and obtain theempirical wavelet coefficients at scale j ,where j =1,2, …J.2. Apply the thresholding function to the empirical wavelet coefficients at each scale j ,where j =1,2 …J. Then the estimate coefficients at each scale are obtained based on the selected threshold value λ= [λ1 ,λ2 , …,λJ ] . Note that λj is the threshold for wavelet coefficients at scale j.3. Use the inverse wavelet transform (IDWT) on thresh olded wavelet coefficients and obtain the estimation values of the signal.However ,the impulsive noise can’t be smoothed effectively in simulation this paper ,four modified thresholding function are proposed ,they can not only smooth the white Gaussian noise but also clean impulsive noise effectively ,which are shown by simulation experiments and proved theoretically. Moreover ,as far as SNR and MSE are concerned ,the results abtained by modified soft—thresholding function is obviously better than the results of three other modified thresholding functions.2 　Four modified thresholding functionsFormally,after Daubechies(1992) ,we define a wavelet to be any function ψ∈L2（R）,which satisfies the admissibility condition, and this condition implies ,in particular ,that∫Rψ(x) dx = 0. For the function f ∈L2 R its integral wavelet transform(IWT) .This consequence is useful in applications.We now turn to the question: How to improve the threshold function ? Assume the observed data vector Y = [y1 ,y2 , …, yN ] is given by yi = si + ni 　i = 1,2, …, N　　Where si is the value of original signal and ni is the value of Gaussian noise withindependent identical distribution N (0,σ) .Then it is time to apply the thresholding function to empirical wavelet coefficients at each scale j , where j = 1,2, …J. There are some thresholding functions such as hard—thresholding function ,softthresholding function ,promise function ,modulus squared function ,and the like. However ,they can not smooth the impulsive noise effectively though they can do the white Gaussian noise. So this paper proposes four improved thresholding functions ,which are available and effective in smoothing the white Gaussian noise and theimpulsive noise.Now ,taking the hard thresholding function as an example ,we will show the change of the hard thresholding function into the modified hard2thresholding hard thresholding is defined byωj,k = ωj,k，ωj, k≥λj , 0, ωj,k λj . （1）　　whereλj =σ 2logN/ log(j +1) 　j =1,2, …J.Since we always use one observation ωj,k to estimate the wavelet transform coefficients of the observed signal at a time , the noise elements whose wavelet transform coefficients are larger than the threshold are preserved. To improve it , we let where M is the length of the high—pass filter or the length of the support of the wavelet. For the Haar wavelet ,we choose M =1.Thus the modified hard2thresholding function (MHT) is given byωj,k = ωj,k , ωj,k ≥λand uj, k ≥λj 0, ωj,k λj or uj,k λj （2）The impulsive noise can be smoothed effectively by this modified hard2thresholding function ,For example , if the observed signal is corrupted by an impulsive noise at point k0[5] then the value ωj,k ≥λj 0, and Haar is selected , then according to the corollary1,uj,k = ωj, k m = ψ m = 0 λj So more noise can be removed if we use this modified hard2thresholding function. The sum uj,k0 =0 ≤λj ,so more noise can be removed if we use this modified hard2thresholding function.In the same way ,the other three modified thresholding functions are as follows:themodified soft2thresholding function (MST) isωj, k = sign(ωj,k) ( ωj,k λj ) , ωj,k ≥λj and uj,k ≥λj0, ωj,k λj or uj,k λj （3）and the modified modulus squared thresholding function (MMST) isωj,k = sign(ωj,k) (ωj,k) λj , ωj,k ≥λj and uj,k ≥λj0, ωj,k λj or uj,k λj （4）the modified promise thresholding function (MCT) isωj,k = sign(ωj,k) ( ωj,k αλj) , ωj,k ≥λj and uj,k ≥λj0, ωj,k λj or uj,k λj （5）Whereα∈(0,1) .According to the corollary 1 ,we are able to draw the conclusion that all of these three modified thresholding functions can smooth the implusive noise. Certainly they are all effective to denoise the white Gaussian noise.3. SimulationsIn order to examine the noise smoothing ability of the four modified thresholding functions ,the signals of Blocks ,Bumps and Heavy sine are selected as test signals ,which are corrupted with the white Gaussian noise and the impulsive noise respectively. In the process of experiment ,the Haar wavelet is used as wavelet basis and the largest DWT level J =4 and the author selects the standard deviation σ= median(abs(cd))/ 0. 6745 and the different threshold value λj =σ 2logN/ log(j +1) at each scale j , j =1,2, …,J. The SNR and MSE of different wavelet thresholding methods for estimation are respectively given inTable 1 and Table 2.From Table 1 and Table 2 ,we can make out that the eight thresholding functions havetheir own advantages ,which depend on the test signal. but