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1、Wavelet De-noising First, the wavelet threshold de-noising the signal estimate Signal processing signal de-noising is one of the classic. De-noising methods include traditional linear filtering method and nonlinear filtering methods, such as median filter and wiener filtering. De-noising method is n

2、ot traditional is the entropy of the signal increased after transformation, can not describe the characteristics of non-stationary signals and can not get the signal correlation. To overcome these shortcomings, people began to signal de-noising using the wavelet transform to solve the problem. Wavel

3、et transform has the following favorable characteristics: (1) Low Entropy of: the sparse distribution of wavelet coefficients, so that reduces the entropy of the transformed signal; (2) Multi-resolution features: Yu to characterize the signal can be very non-stationary features such as edges, spikes

4、, breakpoints, etc.;(3) To relevance: the relevance of the signal can be removed, and the noise in wavelet transform has whitening trend, the more beneficial than the time-domain de-noising;(4) Selected based flexibility: the flexibility to choose the wavelet basis function can therefore be required

5、 according to the signal characteristics and select the appropriate wavelet de-noisingIn the field of wavelet de-noising has been more widely used. Thresholding method is a simple, better methods of wavelet de-noising. Thresholding method is the idea of layers of wavelet decomposition coefficients o

6、f the model is larger than and smaller than a certain threshold value of the coefficient of treatment, and then re-processed the wavelet coefficients of an anti-transformation, through the reconstructed de-noised Signal. The following functions from the threshold and threshold estimation of both thr

7、esholding methods are introduced. 1.Threshold function Commonly used threshold function is mainly hard and soft threshold function threshold function.(1) Hard threshold function. Expression is(w)=wI(wT).(2) Soft threshold function. Expression is(w)=(w-sgn(w)T)I(wT)In general, the hard thresholding m

8、ethod can preserve the signal edge of the other local features, soft threshold is relatively smooth, but will cause the edge of the blurring distortion. To overcome these shortcomings, recently proposed a semi-soft threshold function. It can take into account the soft threshold and hard threshold me

9、thod has the advantage, and its expression is (w)=sgn(w) The basis of the soft threshold, you can improve them with their more advanced. It can be seen in the noise (wavelet coefficients) and the useful signal (wavelet coefficients) there is a smooth transition between the areas, more in line with t

10、he natural signal / image of continuous features. Its expression is (w)=2. Threshold estimation Donoho proposed in 1994 VisuShrink method (or uniform thresholding method). It is for the multi-dimensional joint distribution of independent normal variables, when the dimension tends to infinity the con

11、clusions of the maximum estimate of the minimum constraints derived optimal threshold. The choice of thresholds meets: T=Donoho prove that given estimates of the signal is Besov set, obtained in a number of risks similar to the ideal function of the risk of noise reduction. A unified method of Donoh

12、o threshold effect in the practical application unsatisfactory, resulting in the phenomenon of over kill, put forward in 1997 Janse unbiased estimate based on the threshold calculation. Risk function is defined as:Orthogonality of wavelet transform, the risk function can be written in the same form

13、in the wavelet domain Set So Finally, the expression of risk function can be obtained: Where is the indicator function, taking the number of two small. Thus, the best threshold selection can be obtained by minimizing the risk function, i.e. MATLAB to achieve the threshold of signal de-noising, inclu

14、ding the threshold and the thresholding for the two parties . The following description of them. Second, the wavelet de-noising function in MATLAB 1) Thresholds Implemented in MATLAB function of signal threshold for a ddencmp, thselect, wbmpen and wdcbm, following the use of their simple instruction

15、s. Ddencmp call the format of the following three (1)THR，SORH，KEEPAPP，CRIT=ddencmp(IN1，IN2, X) (2)THR，SORH，KEEPAPP，CRIT=ddencmp(IN1，wp，X) (3)THR，SORH，KEEPAPP=ddencmp(IN1，wv，X)Function ddencmp used to obtain in the process of de-noising or compression the default threshold. Input parameter X is one o

16、r two dimensional signals; IN1 value for the den or crop, den, said the de-noising, crop that is compressed; IN2 value for the wv or wp, wv, said selection of wavelet , wp said the choice of wavelet packets. Return value is the return threshold THR; SORH is soft or hard threshold threshold selection

17、 parameters; KEEPAPP that kept low frequency signal; CRIT is the entropy of name (only used in the choice of wavelet packet).Function thselect call the following format:THR=thselect(X，TPTR)THR=thselect(X，TPTR) according to the definition of the string TPTR threshold selection rules to select the sig

18、nal X of the adaptive threshold.Adaptive threshold selection rules include the following four. TPTR = rigrsure, adaptive threshold choose to use Steins unbiased risk estimate principle. TPTR = heursure, using the heuristic threshold selection. TPTR = sqtwolog, the threshold value is equal to sqrt (2

19、 * log (1ength(X). TPTR = minimaxi, with the minimax principle of selection threshold.Threshold selection rule based on the model, A is the Gaussian noise N (O, 1).Function wbmpen call the following format: THR = wbmpen (C, L, SIGMA, ALPHA) THR = wbmpen (C, L, SIGMA, ALPHA) returns the global de-noi

20、sing threshold THR. THR by a given selection rules calculated wavelet coefficients, wavelet coefficients selection rule using the Birge-Massart penalty algorithm. C, L is the de-noising of the signal or the wavelet decomposition structure; SIGMA is a zero mean Gaussian white noise of standard deviat

21、ion; ALPHA adjust the parameters used for punishment, it must be a real number greater than 1, a Shares take ALPHA = 2. Let t * is the crit (t) =- sum (c (k) 2, k = t) +2 * SIGMA 2 * t * (ALPHA + log (n / t) minimum, where c ( k) are ordered from largest to smallest absolute value of wavelet packet

22、coefficients, n is the number of coefficients, the THR = c (t *). wbmpen (C, L, SIGMA, ALPHA, ARG) calculated the threshold and draw the three curves. 2 * SIGMA 2 * t * (ALPHA +10 g (n / t) Sum (c (k) 2, k = t) crit (t) Function wdcbm call the following two formats: (1) THR, NKEEP = wdcbm (C, L, ALP

23、HA) (2) THR, NKEEP = wdcbm (C, L, ALPHA, M) Function wdcbm using Birge-Massart method for one-dimensional wavelet transform to obtain the threshold. Return value THR is the threshold and scale independent, NKEEP is the number of coefficients. C, L is to carry out signal de-noising or compression in

24、the j = length (L) -2 layer breakdown structure; ALPHA and M must be a real number greater than 1; THR is about j of the vector, THR (i) is the i-layer threshold; NKEEP is a vector on the j, NKEEP (i) is the coefficient of i layer number. 1.5 for the general compression ALPHA, ALPHA de-noising take

25、3. 2) Signal threshold de-noising MATLAB, the threshold for signal de-noising function has wden, wdencmp, wthresh, wthcoef, wpthcoef and wpdencmp. Following the usage of their brief. Function wden call the following two formats: (1) XD, CXD, LXD = wden (X, TPTR, SORH, SCAL, N, wname) (2) XD, CXD, LX

26、D = wden (C, L, TPTR, SORH, SCAL, N, wname) Function wden for the automatic one-dimensional signal de-noising. X is the original signal, C, L for the signal decomposition, N is the number of layers of wavelet decomposition. TPTR the threshold selection rules, TPTR the following four values: TPTR = r

27、igrsure, by Stein unbiased likelihood estimation. TPTR = heursure, using heuristic threshold selection. TPTR = sqtwolog, take universal threshold TPTR = minimaxi, using the maximum threshold for the minimum value selection. SORH is soft or hard threshold threshold selection (corresponding to s and h

28、). SCAL refers to the threshold used by the need to re-adjust, including the bottom three: SCAL = one, do not adjust. SCAL = sln, according to the first layer of the estimated coefficients to adjust the noise floor threshold. SCAL = mln, according to different estimates to adjust the noise level thr

29、eshold. XD for the noised signal, CXD, LXD for the signal after de-noising wavelet decomposition structure. Format (1) returns the signal X through N layers decomposed wavelet coefficients after thresholding and signal de-noising signal XD XD the wavelet decomposition structure CXD, LXD. Format (2)

30、return parameters and format (1), but its structure by direct decomposition of the signal structure of C, L obtained by threshold processing. Function wdencmp call the following three formats: (1)XC, CXC, LXC, PERF0, PERFL2 = wdenemp(gbl, X, wname, N,THR, SORH, KEEPAPP) (2) XC, CXC, LXC, PERF0, PERF

31、L2 = wdencmp (1 vd , X, wname , N, THR, SORH) (3) XC, CXC, LXC, PERF0, PERFL2 = wdencmp (1 vd , C, L, wname , N, THR, SORH) Function wdencmp for one or two dimensional signal de-noising or compression. wname wavelet function is used, gbl (global abbreviation) that each have adopted a threshold for t

32、he same treatment, lvd that each use different thresholds for treatment, N said that the number of layers of wavelet decomposition, THR is the threshold vector For Format (2) and (3) requires each department has a threshold value, so the threshold vector length THR N, SORH that choice of soft or har

33、d threshold threshold (value, respectively, for the s and h) , the parameter KEEPAPP value to 1, the frequency factor is not quantified by threshold, on the contrary, the low-frequency coefficients of the threshold to be quantified. XC is the elimination of noise or the compressed signal, CXC, LXC i

34、s the XC of the wavelet decomposition structure, PERF0 and PERFL2 is to restore and compress the percentage of the norm. If C, L is the wavelet decomposition structure of X, then ; If X is a one-dimensional signal, wavelet wname is a wavelet, then the Function wthresh call the following format: Y =

35、wthresh (X, SORH, T) Y = wthresh (X, SORH, T) returns the input vector or matrix of X by the soft threshold (if SORH = s) or Hard threshold (if SORH = h) after the signal. T is the threshold. Y = wthresh (X, s, T) returns , namely, the absolute value of the signal compared with the threshold value,

36、less than or equal to the threshold point to 0, the point becomes greater than the threshold value The point value and the threshold of the difference. Y = _wthresh (X, h, T) returns , namely, the absolute value of the signal compared with the threshold value, less than or equal to the threshold poi

37、nt to 0, greater than the threshold value of the point remains the same .An, the use of hard threshold signal after treatment than the soft threshold signal is more rough.Function wpthcoef call the following format: T = wpthcoef (T, KEEPAPP, SORH, THR)NT = wpthcoef (T, KEEPAPP, SORH, THR) by the coe

38、fficients of wavelet packet tree T after the threshold value returns a new wavelet packet tree NT. If KEEPAPP = 1, then the details of the signal factor is not the threshold processing; Otherwise, it is necessary for threshold processing. If SORH = s, using the soft threshold, if SORH = h, then use

39、the hard threshold. THR is the threshold. Call function wthcoef following four formats: (1) NC = wthcoef (d, C, L, N, P) (2) NC = wthcoef (d, C, L, N) (3) NC = wthcoef (a, C, L) (4) NC = wthcoef (t, C, L, N, T, SORH)Function wthcoef for one dimensional signal thresholding wavelet coefficients. Forma

40、t (1) returns the wavelet decomposition structure C, L defined by the vector of N and P after the compression rate of decomposition of the new vector NC, NC, L that constitutes a new wavelet decomposition structure. N contains the details to be compressed vector, P is set to 0, the smaller the perce

41、ntage of coefficient vectors of information. N and P must be the same length, the vector N must satisfy 1 N (i) length (L) -2. Format (2) returns wavelet decomposition structure C, L after the vector N is specified in detail coefficients set to 0 after the wavelet decomposition vector NC.Format (3)

42、returns wavelet decomposition structure C, L after approximate coefficients set to 0 after the wavelet decomposition vector NC.Format (4) returns wavelet decomposition structure C, L N as the vector after treatment, the wavelet threshold vector NC. If SORH = s, was soft threshold; if SORH = h, was a

43、 hard threshold. N contains the details of the scale vector, T is the N vector of the corresponding threshold. N and T must be equal in length.Function wpdencmp call the following two formats:(1) XD, TREED, PERF0, PERFL2 = wpdencmp (X, SORH, N, wname, CRIT, PAR, KEEPAPP)(2) XD, TREED, PERF0, PERFL2

44、= wpdencmp (TREE, SORH, CRIT, PAR, KEEPAPP) Function wpdencmp for the signal using wavelet packet compression or de-noising. Forma (1) returns the input signal X (one and two dimensional) of the signal after de-noising or compression XD. XD TREED output parameters are the best wavelet packet decompo

45、sition tree; PERFL2 and PERF0 is the energy recovery and the percentage of L2 compression. If X is a one-dimensional signal, wname is an orthogonal wavelet, the . SORH values for the s or h, that is soft or hard threshold threshold. Input parameter N is the number of layers wavelet packet decomposit

46、ion, wname string that contains the wavelet name. Function uses the definition of entropy by the string CRIT criteria and threshold parameters for optimal decomposition of PAR. If KEEPAPP = 1, then the approximation of wavelet coefficients are not quantified by threshold; Otherwise, proceed to quant

47、ify.Format (2) format (1) of the output parameter, the input options are the same, but it from the signal using wavelet packet decomposition tree TREE directly de-noising or compression.Third, the wavelet threshold de-noising examples of signal An to say, signal de-noising include the following three-step basic steps:(1) signal decomposition;(2) high-frequency coefficients of wavelet thresholding;(3) Signal wavelet reconstruction. Use of low frequency coefficients of wavelet decomposition and thresholding the high frequency coefficients after wavelet reconstruction.13