Wavelet Denoising and Speech Enhancement

Transcription

1 1 Wavelet Denoising and Speeh Enhanement V. Balakrishnan, Nash Borges, Luke Parhment Abstrat It has reently been shown that by taking advantage of the sparseness of natural signals in the wavelet domain, wavelet-based tehniques show promise in oping with the diffiult problem of denoising. We review three tehniques for image denoising via wavelet oeffiient thresholding. To enhane speeh orrupted by additive white Gaussian noise we take the results of the reviewed tehniques and employ BayesShrink in a timeadapted thresholding framework, demonstrating an improvement over a state of the art algorithm. I. INTRODUCTION During transmission and reeption signals are often orrupted by noise whih an ause severe problems for downstream proessing and user pereption. Therefore an automated means of removing the noise would be an invaluable first stage for many signal proessing tasks. Denoising has long been a fous of researh and yet there always remains room for improvement. Simple methods originally employed the use of time-domain filtering of the orrupted signal, however, this is only suessful when removing high frequeny noise from low frequeny signals and does not provide satisfatory results under real world onditions. To improve performane, modern algorithms filter signals in some transform domain suh as z or Fourier. Over the past two deades, a flurry of ativity has involved the use of the wavelet transform after the ommunity reognized the possibility that this ould be used as an superior alternative to Fourier analysis [1]. Numerous signal and image proessing tehniques have sine been developed to leverage the power of wavelets. These tehniques inlude the disrete wavelet transform, wavelet paket analysis, and most reently, the lifting sheme. II. WAVELET THRESHOLDING Assuming additive white Gaussian noise (AWGN), any signal y(t) an be represented by the summation of the original x(t) and the noise n(t) y(t) = x(t) + n(t). (1) After performing the wavelet transform, we end up with Y j,k = X j,k + N j,k (2) Y j,k is the k th wavelet oeffiient in the sale j. There are two ommon ways to threshold the resulting wavelet oeffiients. The first is referred to as hard thresholding whih sets the oeffiients to zero whose absolute value is below the threshold Ŷ hard j,k = { Yj,k Y j,k > λ otherwise The authors are with the Department of Eletrial and Computer Engineering, The Johns Hopkins University, Baltimore, MD balaji.iitm1@gmail.om, nashborges@jhu.edu, lparhm1@jhu.edu.. Soft thresholding goes one step further and dereases the magnitude of the remaining oeffiients by the threshold value Ŷ soft j,k = sign(y j,k )max( Y j,k λ, ). Hard thresholding maintains the sale of the signal but introdues ringing and artifats after reonstrution due to a disontinuity in the wavelet oeffiients. Soft thresholding eliminates this disontinuity resulting in smoother signals but slightly dereases the magnitude of the reonstruted signal. III. NOISE ESTIMATION When dealing with images and other smooth signals it is safe to assume that the highest frequeny band does not ontain muh signal ontent, aside from some information about sharp edges. However, white noise gets evenly spread throughout the wavelet oeffiients inluding the highest band. Therefore, we an estimate the overall noise level from these oeffiients. The simplest way to do this is to ompute the variane of the samples in the high-pass band σ 2 = 1 ( ˆN k µ) 2. The problem with this method is that is it suseptible to outliers. A more robust estimate an be derived by assuming the noise is Gaussian pdf( ˆN) = 1 σ 2π e ˆN 2 2σ 2 whose probability density funtion we then take the absolute value of pdf( ˆN ) = u( ˆN) 2 σ ˆN 2 2π e 2σ 2. We know that the median of any random variable x is simply the point x suh that x so, median( ˆN ) pdf(x)dx = σ ˆN 2 2π e 2σ 2 d ˆN = 1 2. This median an then be expressed as median( ˆN ) = σ ( ) 1 2 erf σ 2 erf(x) = 2 x e x2 dx. π

2 2 Therefore, we an estimate the standard deviation of the noise using the equation σ = median( Y HH 1 ).6745 sine we assume Y HH1 ˆN HH1 IV. VISUSHRINK One of the first wavelet-based methods of denoising was developed by Donoho and Johnstone [2]. Given a signal of length d, it seeks to minimize the maximum error over all possible samples. This method assumes (1) with n(t) having some known standard deviation σ. The proposed universal threshold λ univ = σ 2 log(d) (4) is shown to be asymptotially optimal in the minimax sense when employed as a hard threshold. V. SURESHRINK Donoho and Johnstone [3] also proposed a more advaned strategy based on Stein s unbiased risk estimate (SURE) [4]. Here, soft thresholding is used beause it is more mathematially tratable (i.e. ontinuous) and the lean signal is estimated as ˆX j = Y j + H λ (Y j ) (5) H λ (Y j ) = Y j I( Y j < λ) λi( Y j λ)sign(y j ) (6) using the indiator funtion I. Then, (3) d dy j H λ (Y j ) = I( Y j < λ). (7) Risk is defined as the expeted value of the square of the differene between the estimate ˆX j and X j [ R(λ, Y j ) = E ( ˆX j X j ) 2] (8) whih we seek to minimize. This an be rewritten as [ E (N j + H λ (Y j )) 2] (9) = E[N 2 j ] + E[H 2 λ(y j )] + 2E[N j H λ (Y j )]. (1) Evaluating these addends one at a time, we assume E[N 2 j ] = σ 2 (11) and sine we observe Y j, we an determine H λ (Y j ). Then, given samples of the random variable Y j, the best estimate of the expeted value of its square an be written as E[Hλ(Y 2 j )] = 1 H 2 λ(y j,k ). (12) Finally, aording to Stein, if H λ (Y j ) is a weakly differentiable funtion, as is true in this ase due to the use of soft thresholding, we an estimate the expeted value of the funtion weighted by any random variable in terms of its derivative. That is E[N j H λ (Y j )] = E[Nj 2 d ] H λ (Y j ). (13) dy j Combining the terms (11), (12), and (13) in (1), we get SURE(λ, Y j ) = E[N 2 j ] + E[H 2 λ(y j )] + 2 E[N j H λ (Y j )] = σ H 2 λ(y j,k ) + 2 E[N 2 d j ] H λ (Y j,k ) dy j = σ [ Y 2 j,k I( Y j,k < λ) + λ 2 I( Y j,k λ) ] 2σ2 = σ σ2 I( Y j,k < λ) [min( Y j,k, λ)] 2 I( Y j,k < λ). Minimizing the above expression with respet to λ, we finally arrive at the optimal threshold λ SURE = argmin λ 2 log(d) SURE(λ, Y j ). (14) VI. BAYESSHRINK BayesShrink is an adaptive wavelet thresholding method proposed by Chang, Yu, and Vetterli [5] using a Bayesian estimate of the risk. Threshold alulations are based on the assumption that wavelet oeffiients an be desribed by a generalized Gaussian (GG) distribution with shaping parameter β. These distribution funtions fit the oeffiients of most natural images very well. Given and GG σxj,β(x) = C(σ Xj, β)e [α(σ X j,β) x ] β α(σ Xj, β) = σ 1 Γ(3/β) X j Γ(1/β) C(σ Xj, β) = β α(σ X j, β) ( ), 2Γ the aim is again finding the threshold that minimizes the risk 8. There is no losed form solution to this problem, but Chang et al. estimated this threshold to be 1 β λ Bayes = σ2 σ Xj, (15) whih ompares very well to the optimal threshold found numerially. This estimate is also independent of the shaping parameter β, eliminating the need to estimate any other parameters. As β grows larger than 1, the estimated threshold

3 3 does depart from the optimal, but the error was found to be within 5% of optimal up to β = 4. This provides a very good approximation for images sine most images have a β in the range [.5, 1]. A. Implementation The BayesShrink threshold is applied independently to eah subband of an image s wavelet oeffiients using soft thresholding, exluding the lowest resolution LL subband. This yields a frequeny adaptive threshold that is more aggressive in the absene of signal and less aggressive when the signal is present within a subband. Using this threshold, Chang et al. optimize performane by removing the most noise possible while still preserving the original signal. Assuming that the HH 1 subband is almost entirely devoid of signal data, the noise level is estimated using the robust median estimator (3). For every other subband the noise is assumed to be AWGN, and by (2) we have σy 2 j = σx 2 j + σ 2. The variane σy 2 j is alulated diretly from the subband oeffiients, and an estimate of σx 2 j is then obtained by subtrating the estimated noise variane. If the estimated noise variane is greater than the observed subband variane, the threshold is set to infinity, i.e. all oeffiients are set to. This yields a losed form, omputationally simple expression for the threshold ˆσ Xj = λ Bayes = ˆσ2 ˆσ Xj max(ˆσ 2 Y j ˆσ 2, ). VII. IMAGE DENOISING RESULTS The algorithms desribed above were applied to four test images using the 9/7 biorthogonal wavelets of JPEG 2 [6], and to four artifiial signals [3] using the Daubehies 8 wavelets [7], all orrupted with varying levels of AWGN. Both hard and soft thresholding were used in all three tehniques even though eah algorithm was developed to optimize only one type of thresholding. The resulting average signal to noise ratios (SNRs) are shown in Table I with some examples images in Fig. 1. Overall, SureShrink seems to perform the best, but it omes with a signifiant omputational ost, while BayesShrink is muh more effiient and performs almost equally as well. VIII. SPEECH ENHANCEMENT Sine band-limited ( - 4 khz) speeh oupies the entire frequeny spetrum, we an no longer assume that the noise level an be estimated from the high-pass wavelet oeffiients. This is espeially true during periods of unvoied speeh (Fig. 2). One approah [8] is to divide up the frequeny range evenly into many bins using the wavelet paket transform (WPT) and estimate the noise level in eah one. Unlike the disrete wavelet transform, whih only iterates on the low-pass subband, the WPT iterates both branhes at eah stage. Sine speeh is not always present in the signal, we would like to adapt the thresholding over time so that it is larger TABLE I DENOISING SNR RESULTS FOR VARIOUS TEST IMAGES AND SIGNALS VisuShrink SureShrink BayesShrink Input Hard Soft Hard Soft Hard Soft lena σ 2 = σ 2 = σ 2 = boat σ 2 = σ 2 = σ 2 = tank σ 2 = σ 2 = σ 2 = aerial σ 2 = σ 2 = σ 2 = bloks σ 2 = σ 2 = σ 2 = bumps σ 2 = σ 2 = σ 2 = heavisine σ 2 = σ 2 = σ 2 = doppler σ 2 = σ 2 = σ 2 = during periods without speeh and smaller for those with speeh. This will eliminate as muh of the noise as possible while still maintaining speeh intelligibility. To do this we need some estimate of the signal energy over time. A popular method in the speeh proessing literature [8], [9] uses the Teager energy operator (TEO) [1] given by TEO i,k = Y 2 i,k Y i,k 1 Y i,k+1 (16) i is a partiular subband of the WPT. These oeffiients give an estimate of the energy required to generate the signal. Sine periods of speeh do not our in short disontinuous bursts, we need to smooth the TEOs to redue their sensitivity to noise. Originally this was done in this framework (Fig. 3), using an IIR filter [8] but Chen and Wang [9] have reently proposed onvolving the TEOs with a Hamming window M i,k = TEO i,k H p of length 256/2 p p is the depth of the wavelet paket transform. During periods of noise or silene the full magnitude of the threshold is desired, while during periods of speeh it should ideally approah. To do this we first normalize the smoothed

4 4 Fig. 2. Spetrogram of a speeh signal illustrating that unvoied phonemes suh as the /f/ sound in the word dolphin and the /s/ sound in the word some look similar to noise. Original Image Noisy, SNR=14.3dB Fig. 3. Time-adapted thresholding using smoothed and normalized Teager energy operators. VisuShrink(Hard) 2.9dB VisuShrink(Soft) 18.dB is essentially the universal threshold (4) adapted to a noise estimate σi of eah subband using the robust median estimator (3). We then used this TAT (Fig. 4) to soft threshold the oeffiients using the Daubehies 4 wavelets. IX. S PEECH E NHANCEMENT R ESULTS SureShrink(Hard) 2.7dB Empirial evidene (Fig. 5, 6, 7) suggests that this threshold (18) is overly aggressive ausing signifiant distortion of the speeh. This is both bothersome to listen to and an ause a derease in SNR under low noise onditions (Fig. 8). We propose using the TAT framework, but instead of the level-dependent threshold (18) we employ the threshold (15) whih minimizes the Bayesian risk (8). Some of the noise remains but the result is muh less distortion of the speeh signal, evident both when listening to the reonstruted signals and when examining the denoised SNR. SureShrink(Soft) 23.7dB X. C ONCLUSION BayesShrink(Hard) 21.4dB BayesShrink(Soft) 23.6dB Fig. 1. Example images illustrating the results of three wavelet-based denoising algorithms. TEO oeffiients Mi,k Mi,k =. max(mi,k ) The normalized oeffiients are now lose to 1 for periods of speeh and small for periods of noise. In order to use a higher threshold for non-speeh and lower threshold for speeh, Bahoura and Rouat ompute a time-adapted threshold (TAT), λi,k = λi (1 Mi,k ) (17) λi = σi p 2 log(d) (18) Of the three wavelet shrinkage denoising tehniques presented, SureShrink and BayesShrink yielded the best results when denoising a range of test images and signals. We propose a novel method of enhaning speeh in a timeadapted thresholding framework using BayesShrink due to its omputationally effiient implementation. When this threshold is used, we improve on the previous results of Chen and Wang [9], garnering better performane over a wide range of noise levels without any prior knowledge of the SNR. R EFERENCES [1] B. Jawerth and W. Sweldens, An overview of wavelet based multiresolution analysis, SIAM Review, vol. 36, no. 3, pp , [2] D. L. Donoho, I. M. Johnstone, G. Kerkyaharian, and D. Piard, Wavelet shrinkage: Asymptopia?, Journal of the Royal Statistis Soiety, vol. 57, pp , [3] D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, Journal of the Amerian Statistial Assoiation, vol. 9, no. 432, pp , 1995.

5 5 Fig. 6. Denoised speeh signal using the level-dependent threshold. Fig. 4. Plots showing the smoothed and normalized TEOs and the resulting time-adapted thresholds. Fig. 7. Denoised speeh signal using the BayesShrink threshold. Fig. 5. Spetrograms of denoised speeh using time-adapted wavelet thresholding to highlight the overly aggressive results of the level-dependent threshold (top) versus our methodology whih employs the BayesShrink threshold (bottom). Fig. 8. Comparison of the average SNR performane of universal TAT and Bayesian TAT.

6 6 [4] Charles M. Stein, Estimation of the mean of a multivariate normal distribution, The Annals of Statistis, vol. 9, no. 6, pp , [5] G. Chang, B. Yu, and M. Vetterli, Adaptive wavelet thresholding for image denoising and ompression, IEEE Trans. Image Proessing, vol. 9, no. 9, pp , 2. [6] Diego Santa Cruz and Touradj Ebrahimi, The JPEG 2 image oding standard, Dr. Dobb s Journal, vol. 26, no. 4, pp , Apr. 21. [7] I. Daubehies, Ten letures on wavelets, SIAM, Philadelphia, PA, [8] M. Bahoura and J. Rouat, Wavelet speeh enhanement based on the teager energy operator, IEEE Signal Proessing Letters, vol. 8, no. 1, pp. 1 12, 21. [9] S. Chen and J. Wang, Speeh enhanement using pereptual wavelet paket deomposition and teager energy operator, Journal of VLSI Signal Proessing, vol. 32, pp , 24. [1] James F. Kaiser, On a simple algorithm to alulate the energy of a signal, in International Conferene on Aoustis, Speeh, and Signal Proessing, 199, vol. 1, pp