wavelet soft-thresholding of

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two nonlinear denoising techniques. The soft-thresholding al-gorithms we will consider can provide considerable noise r eduction without greatly impairing the timefrequency r e s -lution of the TFR. Wavelet soft-thresholding algorithms -multiresolution denoising techniques applicable to a widerange of signals and images - onstruct nonlinear esti-mates of signals or images embedded in additive white G au ssian noise using a simple three-step procedure [2]: (1) com-pute t he wavelet transform of the da ta; ( 2 ) translate (soft-threshold) the wavelet coefficients towards zero by a setthreshold value; (3) invert the modified wavelet coefficientsto obtain the final estimate.

When applied to the Wigner distribution (which canbe interpreted as a two-dimensional image), wavelet soft-thresholding corresponds to nonlinear, scalar processing ofthe coefficients of this distribution in a wavelet basis repre-sentation

2-d WT-r-,ps) - D , . (5 ), - B, thr*ld2-d WTHere B, represents the wavelet transform (WT) coefficientsof W,, I-, represents soft-thresholding with threshold 7, andD , represents the denoised TFR. In contrast, Cohens classTFRs result from linear scalar processing of the coefficientsof the Wigner distribution in the sinusoidal Fourier basisrepresentation (see (4)). Figure 6 illustrates a wavelet soft-thresholded TFR for the same noisy test signal utilized forFigures 3-5. Unlike the spect rogram and Choi-Williams dis-tribution, this TF R offers reduced noise levels without d egraded resolution.

After a brief review of wavelet soft-thresholding in Sec-tion 2, we discuss its application to T FRs in Section 3. Sincewavelet processing of the Wigner distribution sacrifices someof its desirable properties, in Section 4 , we introduce soft-thresholding of the ambiguity function representation from(4). We close in Section 5 with some preliminary conclu-sions. While tantalizing, we will find that since the Wignerdistribution of a noisy signal does not conform to the stan-dard additive white Gaussian noise model, the application(or misapplication!) of soft-thresholding techniques to t imefrequency analysis remains as ad hoc as previous nonlinearschemes such as Wigner distribution thresholding, medianfiltering, and so on.

2. WAVELET SOFT-THRESHOLDINGThe wavelet transform of a one-dimensional continuous-timesignal s s defined as

Bs(m, ) = 2-k2 s ( t ) $~ (2-~ t m) dt . (6 )When the dilates and translates of the wavelet function 11,form an orthonormal basis, we have the signal representation,or inverse wavelet transform

IS ( t ) = B,(m, k) 2-k24(2-5 -m). (7 )

m , k

Roughly speaking, th e wavelet transform of a sinooth signalis concentrated in a relatively small number of wavelet coef-ficients. On the other hand, the transform of a white noisesignal spreads out over al l coefficients.

The wavelet thresholding concept arose fror 1 combiningthese two observations with the conventional lvisdom thatsimple thresholding performs well as a data recovery tech-nique whenever the data lies above the noise fl( or. Waveletthresholding addresses the following data recov?ry problem(stated in one dimension for simplicity): Recover the smooth,discretetime signal s ( i ) , =1,. ,N , given the c xrupte d ob-servations s ( i )+n( i ) ,where n( i) s a white Gauss an sequenceof zero mean and variance U? The algorithm of Donoho andJohnstone [2] runs as follows:1. Compute the wavelet transform of s + n using a discrete-

time, finite-data analog to (6 ) (an interval adapted fil-terbank).

2. Translate al wavelet coefficients Bs+n (m , ) towardszero by the amount y =d w .

3. Invert the thresholded coefficients using the discrete-time, finite-data analog to (7 ) .

A multidimensional wavelet transform [3] extends this pr ecedure to image and other d ata in higher dimenPions.

3. WAVELET SOFT-THRESHOLDIPrG THEWIGNER DISTRIBUTIOB

In addition to being straightforward and intiutively rea-sonable, wavelet soft-thresholding possesses twc remarkableproperties [2], both potentially useful for TFI 1 denoising.First, with high probability, the data estimate i s at least assmooth as the desired noise-free data. Thus, gi v :n a smoothset of Wigner distribution signal components mbedded innoise, wavelet denoising should not introduce artifacts thatcould be interpreted as new components. Second, the es-timate achieves almost the minimax mean-squrue error overevery one of a wide variety of smoothness measuris, includingmany where linear estimators do not and cannot achieve theminimax value. Thus, nonlinear denoising of the Wigner dis-tribution should offer higher performance than hi ear smooth-ing. Simulations support this intuition; Figure 6 illustratesa wavelet soft-thresholded Wigner distribution for the samenoisy test signal utilized for Figures 3-5.

Unfortunately, it appears difficult to go beyond simulationsfor justifying wavelet soft-thresholding in this :ontext, be-cause our dat a recovery model does not match tha t for whichthe algorithm was developed. In particular, the Wigner dis-tribution of the signal s +n, given by

Ws+n = Ws +Wn + 2ReWs,n ( 8 )2The signal components in a Wigner distrimtion time-frequency image are always at least absolutely c01tinuous, dueto the integral in (2).

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where the last term involves the cross-Wigner distribution

corresponds to data W, plus interference Wn +2 Re Ptrs,n.This interference is anything but Gaussian and white: Ws,,, sGaussian, yet highly correlated and signal-dependent, whileW, is neither Gaussian nor uncorrelated. Further complicat-ing matters, note that since W,,, has variance proportionalto 1131 u2 and W,, has variance proportional to u4, ne termwill dominate depending on the particular value of SNR.

Nevertheless, as very little is known about the probabilitydensity of Wn,d hoc methods such as thresholding mustsuffice until a more complete theory for Wigner distributionestimation can be deri~ed.~ome progress has been madewith the stationary power spectrum [ti]; a similar approachmight prove useful here and result in an explicit formula forthe threshold y. At present, we take y either as a free pa-rameter or adjust it automatically to optimize some measureof TFR performance as in [6,7].4

The denoisiig provided by wavelet soft-thresholding comesat some expense in terms of the desirable mathematical properties of the Wigner distribution [I]. First, due to the nonlin-earity of the processing, the energy preservation and marginalproperties fai l to hold true. Second, the time-frequency shiftcovariance property is lost: Because the discrete wavelettransform is not covariant to shifts, a time-frequency shiftin the Wigner distribution will result in a different thresh-olding pattern and thus a slightly different denoised TFR.Third, with separable wavelet processing, the rotation co-variance property of the Wigner distribution abandons us aswell (although it should be noted that nonadaptive linearsmoothing cannot retain rotation covariance either).

4. SOFT-THRESHOLDING THE AMBIGUITYFUNCTION

The wavelet transform proves so useful as a soft-thresholdingbasis transformation, because wavelets form unconditionalbases for an incredible variety of signal spaces, including mostof those related to smoothness [2]. Sinusoids are more limitedin their utility for soft-thresholding, because they do not formunconditional bases for most of these spaces. Nevertheless, inhght of the loss of desirable TFR properties mentioned in theprevious section, it appears reasonable to consider also theFourier basis for soft-thresholding the Wigner distribution.

30bviously, denoising the signal and then computing theWigner distribution of the result avoids any problems with theadditive white Gaussian noise requirement. However, signal com-ponents modulated to lower frequencies are smoother than theircounterparts at higher frequencies, and hence they are preferen-tially treated by the wavelet transform before thresholding. Thisfavoritism is demonstrated in Figure 7, where we show the Wignerdistribution of the denoised test signal. For modulateds i g n a l s , theWilson bases [4]probably represent abetter alternative to waveletsfor soft-thresholding.4Even when the additive white Gaussian noise model is valid,determinationof the noise power 0 can be very difficult in practice,usually requiring some experimentation or optimization.

The resulting scheme fits in the framework of (4). but witha now signal-dependent kernel i P , that soft-thresholds theambiguity function of the signal

threshold 2-d FT-W, -AFT. -----+ r 7 ( A S ) - E , . (10)Note that E , belongs to Cohens class and is time-frquencyshift covariant; additional constraints can be imposed on thethresholding to ensure tha t it satisfies other prope :ties suchas energy preservation and marginah, if desired. Further-more, the RCnyi information measures [7] can be L tilized tooptimize the threshold value. Figure 8 illustrates a r FRaris-ing from soft-thresholding the ambiguity function of the noisytest signal; it closely resembles the wavelet-denoisel TF R ofFigure 6. Less ad hoc approaches to adaptive keritel designare detailed in [6,8].5

5. CONCLUSIONSWhile our results are preliminary and admittedly rsomewhatad hoc, nonlinear smoothing techniques have potential forproviding time-frequency analyses with Wigner-like resolu-tion down to low S N b . The hallmarks of th: waveletsoft-thresholding technique - implicity, use of in-ormationacross scales, smoothness preservation, and near c ptimalityfor additive white Gaussian noise- emain tantalming, butmore work is required in order to justify its applcation toT F b . I t is likely that a detailed analysis of the cxrela ted,nonGaussian interference will inspire modifications to the al-gorithm, with a corresponding performance increw e.

Finally, we note tha t soft-threshold4 represen ,ations oftime t and scale a (related to the continuous wavflet trans-form and the scalogram [3]) can be obtained simpJy by pro-cessing the reparameterked Wigner distribution W , ( t , o/a).

REFERENCES1. L. Cohen, Timefrequency distributions - review,2. D. Donoho, De-noising by soft-thresholding, xeprin t.3. 0. Rioul and M. Vetterli, Wavelets and signal process-

ing, IEEE Signal Proc. Mag., October, 1991.4. I. Daubechies, S. J a a r d , J. JournC, A simple Wilson

basis with exponential decay, SIAM J . Math. Anal.,vol. 22 , no. 2 , 1991.

5. P. Moulin, Wavelet thresholding techniques lor powerspectrum estimation, IEEE I zans . Signd Proc . ,preprint.

6. R. Baraniuk and D. Jones, A signal-dependent time-frequency representation: Optimal kernel des4 n, IEEETrans. Signal Proc., vol. 41, no. 4, 1993.

7. P. Flandrin, R. Baraniuk, 0. Michel, (Timefrequencycomplexity and information, Proc. ICASSP 94 .

8. A. Sayeed and D. Jones, Optimal kernels fo- Wigner-Ville spectral estimation, Proc. ICASSP 94 .

Proc. IEEE , vol. 77, no. 7, 1989.

51nfact, thealgorithm(10) s closelyrelatedtoarelatedversionof the 1/0 optimal kernel design from [SI.

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wavelet soft-thresholding of

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