Signal Denoising based on Dual Tree Complex Wavelet Transform and Goodness of Fit Test Khuram Naveed, Bisma Shaukat and Naveed ur Rehman Department of Electrical Engineering, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan. Emails:
[email protected],
[email protected],
[email protected] Abstract—We proposes a signal denoising framework algorithm which employs goodness of fit (GOF) test on complex wavelet coefficients obtained via dual tree complex wavelet transform (DT-CWT). Owing to its redundancy, DT-CWT is near translation invariant insuring better denoising performance over the classical discrete wavelet transform (DWT). The GOF test is used to identify the noisy DT-CWT coefficients whereby statistics based on empirical distribution function (EDF), namely Anderson Darling (AD) statistics, is employed to quantify the distance between the EDFs of local wavelet coefficients and reference white Gaussian noise (WGN) distribution. We pose the denoising as a hypothesis testing problem where null hypothesis corresponds to detection of noise while alternate hypothesis corresponds to the signal detection. Experimental results demonstrate that the proposed signal denoising method gives superior performance over the state of the art methods.
I. I NTRODUCTION Typically, real world signals are corrupted with noise during the process of acquisition, transmission etc. Hence, noise removal or denoising operation is performed as an essential preprocessing step in most signal processing applications. Let x denote the noisy signal obtained from a transmission medium or an acquisition system as x = s + η,
(1)
where s denotes the true signal and η denote the additive noise. The denoising operation seeks to estimate the true signal s from its noisy version x. To this end, researchers have developed multiple algorithms for denoising signals. Initially, linear filtering schemes were employed for denoising which proved to be suboptimal specifically in the case of commonly encountered nonlinear and nonstationary signals. This lead to the development of more evolved denoising strategies employing nonlinear filtering (i.e. thresholding) and sparse signal recovery methods (i.e. compressive sensing) in transform domain. Discrete wavelet transform (DWT) is a powerful tool for denoising nonstationary signals whereby its sparsity is exploited via nonlinear thresholding. The sparsity in DWT means that wavelet coefficients corresponding to signal have much larger amplitudes compared to the noisy coefficients. Consequently, a threshold value can be defined such that the wavelet coefficients less then threshold are identified as noise and coefficients greater then threshold are considered to be belonging to desired signal. Subsequently, noisy coefficients are discarded while signal coefficients are retained without
any change called the hard thresholding [1]. Mathematically, the DWT based denoising operation to obtain an estimate ˆs of the true signal is given as ˆs = W −1 (Υ(W(x))).
(2)
where W and W −1 denote the forward and inverse DWT operations respectively while Υ denotes thresholding. VisuShrink [2], the simplest DWT based denoising algorithm, employs a universal threshold on all the wavelet coefficients at each scale as a function of noise variance σ 2 , which is estimated a priori. Contrarily, adaptive thresholding was used in SureShrink [3], whereby separate thresholds are employed at each DWT scale based on the minimization of Steins unbiased risk estimator (SURE). In more evolved signal and image denoising strategies, statistical dependencies between DWT coefficients have exploited. To this end, various statistical tools such empirical Bayes (EB) [4], Bayesian false discovery rate (BLFDR) [5] and goodness of fit (GOF) testing have been employed to develop separate DWT based denoising algorithms for signals. Despite being an effective tool for signal denoising, DWT suffers from lack of translation invariance and directionality (for images). These shortcomings cause various artifacts in the DWT based denoised signal/image, deteriorating its quality. Dual tree complex wavelet transform (DT-CWT), a variant of DWT, enjoys near translation invariance and directional selectivity (for images) owing to its redundancy coupled with complex number representation. Hence, DT-CWT is better suited for signal or image denoising over the DWT. This fact is demonstrated in [6] whereby comparatively better performance is shown by the DT-CWT based denoising algorithms over DWT. Similarly, a tree structured approach employed in [7] for threshold selection based on local signal strength at multiple scales. Apart from the wavelet transform based methods, a wavelet like thresholding technique has also been employed on multispectral coefficients of empirical mode decomposition (EMD) [8]. In addition, signal denoising has also been performed by a minimizing cost function such that the total variation of the signal is reduced, an example of such a method is given in [9] whereby non-iterative total variational filtering based denoising (TVD) algorithm has been employed. In our previous work, goodness of fit (GOF) test was employed on DWT coefficients for signal as well as image
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redundancy. In the DT-CWT two separate trees of wavelet filters are employed to independently calculate the real and imaginary parts of the complex wavelet coefficients as
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Fig. 1: Block diagram of the proposed denoising method. denoising [10], [11]. In this paper, we extend the use of GOF test on DT-CWT coefficients for signal denoising. The AD statistics based on empirical distribution (EDF) is employed within the GOF test to calculate the similarity between the local DT-CWT coefficients and the reference noise distribution (Gaussian noise in our case). Here, denoising is defined as a hypothesis testing problem whereby a threshold value, which is estimated as a function of probability of false alarm (Pf a ), is compared against the AD measure to test the detection of the local wavelet coefficients as noise (null hypothesis) or desired signal (alternate hypothesis). Proposed methods demonstrate superior performance against the state of the art signal and image denoising methods. Note that, this extension is not trivial due to following reasons: (a) proposed framework calculates separate thresholds for real and imaginary parts of complex wavelet coefficients, (b) a parallel framework is developed for separately thresholding real and imaginary parts of the complex wavelet coefficients. In addition, the use of the GOF test has been introduced to complex-valued wavelet coefficients (of CWT, DT-CWT etc.) which is not attempted before in any application let alone signal or image denoising. Remainder of this paper is organized as follows: SectionII discusses the advantage of signal denoising with DT-CWT over the DWT and the GOF tests are introduced in Section-III. Section-IV presents the proposed denoising method, SectionV discusses experimental results while the paper is concluded in Section-VI. II. A DVANTAGE OF DENOISING VIA DT-CWT OVER DWT Maximal decimation in DWT introduces spectral aliasing in its coefficients which can only be avoided if the wavelet coefficients are not altered before reconstruction. This leads to the lack of translation invariance in DWT, meaning that slightest perturbation of wavelet coefficients will greatly perturb the reconstructed signal. Since, denoising operation alters the wavelet coefficients via thresholding, as a consequence various type of undesired artifacts arise in the denoised signal or image deteriorating its quality. The DT-CWT, which is an extension of the DWT, has been designed to be near translation invariant and directionally selective (for images) thanks to its
(3)
where W denote the DT-CWT transformation resulting in complex wavelet coefficients wk at scale k. Here, the redundancy is achieved by treating real and imaginary parts k k as independent wavelet coefficients denoted by w and w respectively. The redundant DT-CWT coefficients help diminish the effect of maximal decimation (in each tree) to an extent that the problematic spectral aliasing is compensated-for. Hence, most of the artifacts arising in the DWT based denoising are avoided when denoising with the DT-CWT. A. Goodness of Fit Tests The goodness of fit (GOF) test is used to test the hypothesis that whether given data observations originate from a particular (reference) distribution or not? Typically, the GOF tests are used in spectrum sensing to detect the case of noise only (null hypothesis H0 ) or the case of signal plus noise (alternate hypotheis H1 ) [12]. To this end, a measure τ is employed within the GOF test to estimate the distance between the data and the reference noise distribution. Subsequently, hypothesis testing is performed by comparing the measure τ against a threshold T which is estimated by minimizing the probability of falsely detecting noise as signal or probability of false alarm (Pf a ). If τ < T the difference between reference noise distribution and the data at hand is very little i.e. noise is detected (H0 ). Contrarily, if τ > T signal plus noise case is detected (H1 ). Multiple measures for GOF test are reported in literature having there own certain advantages in different cases, however, Anderson Darling (AD) statistical measure [13] is employed in this work due to its robustness and flexibility, whereby statistics based on empirical distribution function (EDF) are exploited. Assuming that the empirical cumulative distribution function (ECDF) of test data samples is denoted by F(x) and the cumulative distribution function (CDF) of reference distribution is denoted by the Fr (x), defined for the interval 0 ≤ x < 1, then the AD statistical measure is mathematically defined as ∞ τ= (Fr (x) − F(x))2 ψ(Fr (x))d(Fr (x)). (4) −∞
where ψ(Fr (x)) = Fr (x)(1 − Fr (x))−1 is the weighting function. The flexibility and robustness in the AD statistics, is enabled by the weighting function ψ(Fr (x)) which gives more weight to the tail of the distributions. Numerically, (5) is written in a convenient way as τ =L−
L (2n − 1) (ln(Fr (xn ) − ln(Fr (xL+1−n ))). (5) L n=1
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The threshold value is estimated as a function of the probability of false alarm (Pf a ) which is defined as the probability of erroneously detecting noise as desired signal i.e. Pf a = P rob{τ > T |H0 }. Hence, threshold is selected by minimizing the Pf a . III. P ROPOSED M ETHOD The GOF test was employed on the DWT coefficients of noisy signal for denoising purpose in our previous work, namely the DWT-GOF [10]. However, spectral aliasing in DWT results in the lack of translation invariance causing wide range of artifacts in the denoised signal/image. Contrarily, the DT-CWT (a variant of the DWT) exploits redundancy to compensate for the problematic spectral aliasing which in turn introduces the causi-translation invariance. In order to institute redundancy in the DT-CWT, the real and imaginary parts of the complex wavelet coefficients are considered as independent DT-CWT coefficients owing to their independent computation via separate trees of wavelet filters. Hence, denoising with DT-CWT will successfully avoid various artifacts otherwise present in the DWT based denoising results. These desirable characteristics of DT-CWT motivate our work whereby the use of the GOF test is introduced on complex wavelet coefficients. The linearity of the DT-CWT transformation prompts the use of GOF test at multiple DT-CWT scales because additive white Gaussian noise (AWGN) does not alter its distribution if operated upon by the linear mathematical operations. This implies that DT-CWT coefficients corresponding to the Gaussian noise obey Gaussian distribution. Hence, the GOF test can be used at multiple DT-CWT scales to check whether the local wavelet coefficients of a noisy image originate from Gaussian distribution and subsequently classify them as noise and desired signal. Since, the GOF test is a hypothesis testing tool by definition, we pose denoising as a hypothesis testing problem as H0 : τ ≤ T k H1 : τ > T k ,
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Fig. 3: Insight into the measure of AD statistics where (a) noise detection case as the CDF of WGN and the CDF of local wavelet coefficients is similar yielding smaller τ while (b) shows otherwise case of signal detection.
for signal denoising are discussed. We name our method GOFDTCWT for future reference. The proposed denoising framework employs parallel architecture for GOF based thresholding operation on the real and imaginary parts of the complex wavelet coefficients of the noisy image as shown in the Fig. 1. Firstly, a threshold based on the minimization of probability of false alarm (Pf a ) is empirically estimated separately for real and imaginary parts and then the GOF based thresholding is performed via the AD statistics in combination with the hard thresholding operation. The proposed GOF based signal denoising framework employs AD statistics based on EDF as a measure τ to quantify the distance between the local wavelet coefficients and the reference Gaussian noise distribution computed via 5. Subsequently, this measure τ is compared against a threshold value to make a decision regarding signal or noise detection as given in 6. Separate thresholds Tk and Tk are estimated in parallel for k k real and imaginary wavelet coefficients w and w . Here, thresholds based on minimization of Pf a are estimated empirically for Gaussian noise distribution. To this end, a large set of Gaussian noise was decomposed via DT-CWT at multiple scales and the corresponding trees of real and imaginary wavelet coefficients are divided into windows containing local coefficients. For a given threshold, the GOF test is applied on each of these windows and probability of falsely detecting noise as signal is recorded as Pf a for that particular threshold. This process continues up till a particular threshold yields lowest Pf a . The plot of threshold T k versus Pf a is shown in Fig. 2. Once, the thresholds T k verus Pf a table is estimated, it can be used for denoising all kinds of signals corrupted with Gaussian noise. Hence, threshold estimation in our method is one time process for a given noise distribution. In the next step; DT-CWT is used to decompose a noisy signal x into separate trees of real and imaginary wavelet k k coefficients w and w via (3). Afterwards, the DT-CWT coefficients in both trees are normalized by noise standard deviation σ as wk ˜k = w , (7) σ where σ = Median{w1 }/0.6745 and the normalized DT-
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Fig. 4: Input test signals employed to validate the performance of proposed method; (a) Bumps (b) Doppler (c) Heavy Sine (d) Blocks. k k ˜ ˜ ˜k = w +w . This normalization is done CWT coefficients w to achieve a unit variance noise at multiple scales, since the thresholds are computed for a unit variance Gaussian noise. In the parallel framework of the GOF based thresholding; k k ˜ ˜ firstly, windows around the coefficients w (i) and w (i) are selected simultaneously from both trees of the DT-CWT coefficients for locally operating the GOF test. In this regard, the AD statistics based on EDF is simultaneously employed on the local wavelet coefficients of both windows to quantify the distance between the CDFs (i.e. F (x) and F (x)) from the reference Gaussian distribution Fr (x) via (5). An insight into the AD statistical measure is presented in Fig. 3, whereby the CDF of local wavelet coefficients are plotted with the CDF of reference Gaussian noise. In Fig. 3(left), it can be seen that both CDFs are not so different suggesting smaller AD distance τ indicating the detection of noise. However, in Fig. 3(right) the otherwise case is depicted whereby two CDFs seem vastly different suggesting a larger τ which consequently indicates that desired signal is detected. Once the distance measure τ is computed, it is compared against the threshold value (empirically selected for minimum Pf a ) via (6) for testing the hypothesis of noise detection or signal detection. This means if τ < T k , noise is detected (i.e. H0 ) culminating in the removal centeral coefficient of the window via hard thresholding. However, if τ > T k the signal is detected (i.e. H1 ) whereby the central coefficient is retained. This process is repeated for all the DT-CWT coefficients resulting in noise suppressed multiscale coefficients k k ˆ ˆ ˆk = w +w . The last step would be to apply inverse DTw ˆ k in order to obtain the denoised signal CWT operation on w as ˆ k × σ), ˆs = W −1 (w (8)
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Fig. 5: Denoised test signals using the proposed method: (upper left) Bumps, (upper right) Doppler, (lower left) Heavy sine, (lower right) Blocks; for input SN R = 10 dB and input signal length N = 213 . where σ is multiplied to reverse the normalization process before reconstruction to obtain the denoised signal ˆs. IV. E XPERIMENTAL RESULTS In order to validate the performance of the proposed method, we employ standard signals like Bumps, Heavy Sine, Doppler and Blocks as input signals. The state of the art signal denoising methods employed for comparison against the proposed GOF-DTCWT are based on DTCWT [7], TVD [9], DWT-GOF [10] and TI-EMD [8]. The noisy versions of these signals having length l = 213 , corrupted with noise of SNR = 10 are shown in Fig. 4 where original signals are highlighetd in dark black while noise enveloping signals is represented in shaded black. Several experiments are conducted to analyse the performance of the proposed method whereby qualitative aspect of results is shown by plotting the denoised versions while the quantitative results are listed against the comparative state of the art methods. In this regard, signal to noise ratio (SNR) has been appointed as the quantifying measure of the quality of denoised signals. All the experiments are repeated J = 1000 times and all the reported output SNR values are the result of their ensemble average. The window length was chosen to be L = 28, for local operation of the GOF test whereby the Pf a was fixed to 0.005. The filters required to implement the dual tree complex wavelet transform are taken from [14], each of them having a length 10. The number of decomposition levels were chosen to be k = 5 for all methods to give a fair comparison. Fig. 5 displays the plots of the signals obtained by denoising the signals given in Fig. 4 via the proposed GOF-DTCWT. It is evident from the figure that the original signals have been
TABLE I: Performance evaluation in terms of output SN R/M SE of the proposed DT CW T -GOF method against various comparative methods for a range of input SN R by conducting this experiment for four standard signals each of length N = 213 . SNR
−2dB
T I-EM D DT CW T TV D DW T -GOF P roposed
9.14/0.397 8.18/0.494 8.21/0.490 11.55/0.228 15.69/0.087
T I-EM D DT CW T TV D DW T -GOF P roposed
8.92/0.011 8.11/0.013 8.43/0.012 10.00/0.008 14.78/0.002
T I-EM D DT CW T TV D DW T -GOF P roposed
9.17/1.172 8.08/1.485 4.71/3.201 10.47/0.857 16.34/0.222
T I-EM D DT CW T TV D DW T -GOF P roposed
8.96/1.125 8.02/1.393 5.00/2.788 9.82/0.920 14.43/0.318
2dB
6dB a) Bumps 13.07/0.161 16.69/0.070 12.16/0.197 16.18/0.078 13.29/0.151 16.35/0.074 15.24/0.097 19.00/0.041 18.84/0.042 21.18/0.024 b) Doppler 12.63/0.004 16.19/0.002 12.00/0.005 15.93/0.002 8.46/0.0099 8.55/0.010 13.74/0.003 17.20/0.001 17.14/0.001 18.54/ 0.001 c) Heavy Sine 12.67/0.523 16.22/0.230 12.10/0.589 16.16/0.231 11.91/0.605 18.16/0.155 14.12/0.370 18.18/0.145 20.27/0.090 23.85/0.039 d) Blocks 12.41/0.507 15.60/0.243 11.82/0.581 15.35/0.258 11.45/0.601 18.16/0.140 12.94/0.449 15.48/0.250 16.59/ 0.193 17.85/ 0.144
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14dB
20.38/0.030 20.11/0.032 17.67/0.055 22.52/0.018 22.56/0.018
24.04/0.013 24.06/0.013 18.29/0.045 25.69/0.008 23.35/0.015
19.93/8.8e-4 19.55/9.5e-4 8.49/0.0093 20.00/8e-4 19.26/ 0.001
23.59/3.8e-4 22.73/4.6e-4 8.51/0.0009 22.04/5e-4 19.59/ 9e-04
20.06/0.095 20.10/0.093 22.79/0.049 22.02/0.060 27.23/0.018
23.76/0.040 23.96/0.038 25.43/0.030 25.69/0.026 29.76/0.010
18.96/0.112 18.28/0.131 22.63/0.051 17.08/0.172 18.47/0.125
22.24/0.053 20.45/0.079 26.70/0.011 17.97/0.141 18.74/0.010
recovered mostly as the estimated signals hugely resemble the true ones. The corresponding higher output SNR values have also been displayed along with the denoised signals in order to support the argument of efficient recovery of the true signal via the proposed method. The recovered Doppler signal in Fig. 5(b) exhibits most resemblance with least number of artifacts. In the Bumps signal in Fig. 5(b) and Heavy Sine signals in Fig. 5(b), all the edges have been recovered perfectly giving impressive resemblance with their original signals, whereby most of the artifacts are present in the smoother regions and can be removed via low pass filtering. Least amount of resemblance (relatively) is shown by the Block signal with its original version, which may be due to the sharper discontinuities because of its piecewise constant nature. Table. I lists the output SNR values obtained by denoising the input test signals via various denoising algorithms. The input noisy signals were corrupted with noise of input SNR = −2 dB to 14 dB. In order to insure stability, each of the output SNR value, listed in Table. I, is computed by taking the ensemble average of the J = 1000 realization of the each experiment. The proposed GOF-DTCWT outperforms the state of the art in signal denoising methods at all noise levels for the input signal Heavy Sine yielding considerable margin of superior results. Similarly, proposed method shows superior results for Bumps signal for all input SNR = −2 → 10 dB except at SNR 14 dB where the DWT-GOF beats the proposed method but only marginally. The proposed method exhibits superior performance for lower input SNR values for the Doppler input signal. However, at higher input SNR values i.e. 10 & 14 dB, the DWTGOF beats the proposed method comprehensively. For Blocks
signal, proposed method performs well enough yielding better results at input SNR value −2 & 2 dB against all comparative methods. However, for input SNR 6 =→ 14 dB the TVD beats the rest of the methods. It is evident from Table. I that proposed method shows overall better performance compared to the rest of state of the art methods whereas the GOF-DWT remains second best. The TVD shows worst performance for Doppler and Bumps but Block it gives considerably better performance. V. C ONCLUSION The use of GOF test at mutiple scales of dual tree complex wavelet transform has been introduced, in this paper, for signal denoising. The proposed GOF based thresholding rule has been separately employed on real and imaginary part of the complex wavelet coefficients. Anderson Darling (AD) measure has been employed within the GOF test to compute the distance between the empirical distribution functions of the wavelet coefficients and Gaussian noise. A threshold value has been estimated based on probability of false alarm which is then compared against the AD measure to identify the case of signal or noise detection. The experimental results demonstrate comprehensively superior performance of the proposed method against the selected satate of the art methods. R EFERENCES [1] D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, vol. 81(3), pp. 425-455, 1994. [2] D. L. Donoho and I. M. Johnstone, “Wavelet shrinkage: Asymptotic?” Journal of Royal Statistical Society, vol. 57(2), pp. 301-369, 1995. [3] D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” Journal of the American statistical association, vol. 90(432), pp. 1200-1224, 1995. [4] H. A. Chipman, E. D. Kolaczyk, and R. E. McCulloch, “Adaptive Bayesian wavelet shrinkage”, Journal of American Statistical Association, vol. 92, pp. 1413-1421, 1997. [5] I. Lavrik, and Y. Y. Jung, and F. Ruggeri, and B. Vidakovic, “Bayesian False Discovery Rate Wavelet Shrinkage: Theory and Applications”, Communication in Statistics-Simulation and Computation, vol. 37, pp. 1086- 1100, 2008. [6] N. Kingsbury, “The dual-tree complex wavelet transform: a new efficient tool for image restoration and enhancement,” 9th European Conference In Signal Processing (EUSIPCO), pp. 1-4, 1998. [7] G. Chen, and W. -P. Zhu,“Signal denoising using neighbouring dual-tree complex wavelet coefficients”, IET Signal Processing, vol. 6, no. 2, pp. 143-147, 2012. [8] Y. Kopsinis, and S. Mclaughlin, “Development of EMD-Based Denoising Methods Inspired by Wavelet Thresholding,” IEEE Transactions in Signal Processing, vol. 57, pp. 1351-1362, 2009. [9] L. Condat, “Direct Algorithm for 1-D Total Variation Denoising”, IEEE Signal Processing Letters, vol. 20, no. 11, pp. 1054-1057, 2013. [10] N. ur Rehman, et al. “Translation invariant multi-scale signal denoising based on goodness-of-fit tests,” Signal Processing, vol. 131, pp. 220-234, 2016. [11] N. ur Rehman, K. Naveed, S. Ehsan, K. McDonald-Maier, “Multiscale image denoising based on goodness-of-fit tests,” European Signal Processing Conference (EUSIPCO), pp. 1548-1552, Budapest, Hungary, 2016. [12] S. Lei, and H. Wang, and L. Shen,“Spectrum sensing based on goodness of fit tests”,IEEE International Conference on Electronics, Communications and Control, pp. 485-489, Ningbo, 2011. [13] T.W. Anderson and D. A. Darling, “A Test of Goodness-of-Fit,” Journal of the American Statistic Association, vol. 49, pp. 765769, 1954. [14] N. Kingsbury, “Image processing with complex wavelets,” Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 357, no. 1760, pp. 2543-2560, 1999.
