Gravitational-Wave Burst Signals Denoising Based on the Adaptive Modification of the Intersection of Confidence Intervals Rule

Gravitational-wave data (discovered first in 2015 by the Advanced LIGO interferometers and awarded by the Nobel Prize in 2017) are characterized by non-Gaussian and non-stationary noise. The ever-increasing amount of acquired data requires the development of efficient denoising algorithms that will enable the detection of gravitational-wave events embedded in low signal-to-noise-ratio (SNR) environments. In this paper, an algorithm based on the local polynomial approximation (LPA) combined with the relative intersection of confidence intervals (RICI) rule for the filter support selection is proposed to denoise the gravitational-wave burst signals from core collapse supernovae. The LPA-RICI denoising method’s performance is tested on three different burst signals, numerically generated and injected into the real-life noise data collected by the Advanced LIGO detector. The analysis of the experimental results obtained by several case studies (conducted at different signal source distances corresponding to the different SNR values) indicates that the LPA-RICI method efficiently removes the noise and simultaneously preserves the morphology of the gravitational-wave burst signals. The technique offers reliable denoising performance even at the very low SNR values. Moreover, the analysis shows that the LPA-RICI method outperforms the approach combining LPA and the original intersection of confidence intervals (ICI) rule, total-variation (TV) based method, the method based on the neighboring thresholding in the short-time Fourier transform (STFT) domain, and three wavelet-based denoising techniques by increasing the improvement in the SNR by up to 118.94% and the peak SNR by up to 138.52%, as well as by reducing the root mean squared error by up to 64.59%, the mean absolute error by up to 55.60%, and the maximum absolute error by up to 84.79%.


Introduction
The first detection of a gravitational-wave signal from a compact binary coalescence (CBC) system [1,2] was made in 2015 by the Advanced LIGO (Laser Interferometer Gravitational-Wave Observatory) detectors [3]. This detection marked a turning point in gravitational-wave astronomy and initiated intensive scientific research in the field of gravitational-wave data analysis leading to the Nobel Prize in Physics in 2017. Besides the Advanced LIGO detectors, the Advanced Virgo corrupted by either synthetic or real noise [44][45][46][47]. The study in [48] presented the deep transfer learning method with deep CNNs for glitch classification and automatic clustering of new classes of anomalies occurring in data from the gravitational-wave detectors. Transient noise glitch classification necessary for interferometer characterization was also studied in [49] where deep CNNs were used to classify simulated glitches based on their time-frequency representations in the form of images. Detection and classification of noise transients were further studied in [50][51][52][53][54].
However, the aforementioned machine learning-based algorithms require extensive databases of different gravitational-wave signal templates. In the case of burst signals produced by core collapse, the generation of such databases that would be large enough is unfeasible due to the computational efforts needed to calculate such waveforms.
Therefore, the noise removal methods that do not require any a priori information about the underlying gravitational-wave signals, such as their signal morphology and astrophysical source, also have interesting potential applications. In [55], total-variation (TV)-based algorithms were applied for denoising of two types of gravitational-wave signals: signals from BBH mergers and burst signals produced by the CCSN. TV denoising methods, mainly applied in the field of image processing, are based on the L 1 -norm minimization and Rudin-Osher-Fatemi (ROF) variational model [56][57][58][59][60][61][62][63]. The application of the TV denoising method in [55] resulted in successful noise removal. However, the study was limited to signals corrupted by the idealized additive Gaussian noise. This work was extended in [64] where the TV-based denoising method was successfully applied for denoising of gravitational-wave signals embedded in real noise data acquired from Advanced LIGO detectors, providing a detailed analysis of the model regularization parameter selection.
In this paper, we propose an algorithm based on the relative intersection of confidence intervals (RICI) rule combined with the local polynomial approximation (LPA) for denoising of gravitational-wave burst signals. The LPA is used as a filter design method in which a polynomial is fitted locally to the noisy measurement data within a data-driven, varying sliding window. The adaptive window size is selected by employing the asymmetrical RICI rule, which represents an improvement of the intersection of confidence intervals (ICI) rule. The LPA-RICI algorithm provides nearly optimal filter supports in terms of minimizing the estimation mean squared error (MSE). It does not require any information on the input signal, the noise, or the estimation of the bias, but only the noise variance estimation. This easy-to-implement algorithm is locally adaptive to the unknown and varying smoothness of the signal, with the estimation accuracy close to the one obtained when the original signal's smoothness is known in advance.
We apply the LPA-RICI algorithm to the denoising of gravitational-wave bursts from CCSNs. The data are obtained by injecting numerically generated signals into the real-life non-Gaussian and non-stationary noise data obtained by the Advanced LIGO Livingston detector. The denoising procedure is performed for three different CCSN burst signals at three different distances (5,10, and 20 kpc) corresponding to different (low) SNR levels. The numerical analysis done on the experimental results indicates that the proposed denoising method provides an accurate estimation of the original gravitational-wave signal corrupted by real-life noise data, by efficiently removing the noise and simultaneously preserving the characteristic features of CCSN bursts. Moreover, the LPA-RICI method outperforms several competing and conventionally applied denoising techniques, suggesting that it may be successfully applied in the preprocessing of gravitational-wave data characterized by intensive noise. The rest of the paper is organized as follows. Section 2 provides the theoretical background and mathematical framework of the LPA method used for the filter design and the RICI algorithm used for filter support selection. In Section 3, the experimental results obtained by several case studies are presented and discussed. Finally, the paper's conclusion is given in Section 4.

Materials and Methods
Let us consider the noise-corrupted signal x(k), composed of the noise-free signal s(k) and the additive white Gaussian noise, η(k) ∼ N (0, σ 2 η ), The goal of the signal denoising is to estimateŝ(k) from the noisy measurements x(k), such that the estimateŝ(k) is as close to s(k) as possible, i.e., the estimation error is minimized. This results in the instantaneous slope changes and other features in s(k) being well preserved. In order to achieve this goal, we applied the LPA method [65][66][67][68][69] as the filter design technique and proposed the adaptive RICI algorithm for the filter support selection.

The LPA Filter Design Method
The LPA method provides the estimateŝ(k) from the noisy measurements x(k), defined in (1), by fitting a polynomial to measurement data within a sliding window defined in the vicinity of the considered measurement. The polynomial, obtained as a linear combination of basis vectors, is fitted locally for each considered measurement so that it minimizes the following loss function using the weighted least squares (WLS) criterion [65][66][67][68]: where k 0 is the point of interest (center of the LPA), k is a signal sample, N k is the signal length, ψ w (k) is the scaled window function, w is the window size (filter support size), C is a vector of the polynomial coefficients, Φ w (k) is a polynomial basis vector, and n is the order of the LPA.
The window function ψ w (k) defines the location of the polynomial fitting with respect to the central point k 0 and, in normalized form, satisfies the conventional kernel properties [65,66]: The minimization of criterion J LPA (k 0 , w, C), defined in (2), with respect to C, leads to the coefficientĈ(k 0 , w) [65,67]:Ĉ which represents the estimate of s(k 0 ) with respect to a window function of size w. The estimates may be represented in the following form [67]: where q(k, k 0 , w) is the estimator kernel defined as [67]: with matrix Φ w defined as: The window function ψ w (k) is a scaled version of the normalized function defined in (3), where the scaling parameter w > 0 represents the window length (filter support size) [65]: The type of window function affects the weights associated with the signal samples, taken into consideration by local polynomial fitting inside the sliding window. Namely, the rectangular window function provides the same weights for all signal samples. In contrast, non-rectangular functions use higher weights for samples closer to the considered point k 0 and smaller weights for these farther away from the point of interest [68].
The filter support size w controls denoising quality and the smoothness of the signal estimatê s(k, w): larger values of w signify including more samples in the LPA procedure, which leads to the increase of the estimation bias (and at the same time, the decrease in estimation variance), while smaller values of w cause the estimation variance to increase (as well as a smaller bias) [65,66]. Therefore, the selection of filter support size w is a crucial part of the denoising procedure. The main goal is to find such w that provides the optimal trade-off between estimation bias and variance, which is here done by applying the RICI algorithm.

The RICI Algorithm
The absolute value of the estimation error ε m (k, w) obtained by LPA estimators is defined as: whereŝ m (k, w) denotes the estimate of the signal sample value calculated using m samples in the vicinity of the considered sample and w(k) is the adaptive filter support size. The point-wise mean squared estimation error, ρ(k, w), is given as [66,67]: where b m (k, w) is the estimation bias and σ m (k, w) is the standard deviation of the estimateŝ m (k, w). According to (10), the performance of the LPA estimators strongly depends on the adaptive filter support size w(k). Therefore, the main task of the denoising procedure consists of finding such a filter support size, w o (k), that provides the optimal bias-variance trade-off, hence minimizing ρ(k, w) [66,67]: where the proportion parameter κ is defined as: In the case of LPA estimators, the estimation error is given in the following form [65,67,70]: where ζ m (k, w) ∼ N (0, σ 2 m (k, w)) is the random error.
The following inequality holds true with the probability p = 1 − β [65,67]: where χ 1−β/2 is the (1 − β/2)th quantile of the standard Gaussian distribution, N (0, 1). Combining (14) and (15) suggests that the following inequality holds true with the same probability p [65,66]: As shown in [66,67], based on the estimation bias and standard deviation properties, Using (16) and (17), the following inequalities for the estimation error are obtained [66,67,70]: where parameter Γ is defined as the ICI threshold value: With the same probability p, (19) can be expressed as: which introduces the confidence intervals, ∆ m (k, w), that contain the noise-free signal values s(k) with the confidence p, defined as [65,67,70]: where parameter Γ represents the critical value of the confidence interval ∆ m (k, w), while ∆ l,m (k, w) and ∆ u,m (k, w) are the lower and the upper confidence limits, respectively. The estimated signal valueŝ m (k, w) is calculated using the LPA-based filters. For instance, in the case of zero-order LPA, the estimate is obtained by averaging sample values in the neighborhood of the considered sample, where the number of samples that are taken into calculation is determined by the RICI algorithm.
The first stage of the RICI algorithm is the ICI rule [71], which, for each signal sample k, calculates a sequence of M growing filter support values [65,67,70]: and the accompanying confidence intervals ∆ m (k, w), defined in (23).
The ICI rule provides the filter support candidates w m (k), such that w m (k) ≤ w − (k). The support w − (k) denotes the largest support for which the intersection of the confidence intervals is still non-empty [65,67,70]: This condition is met if the following inequality still holds true [65,67,70]: where ∆ u,m (k, w) is the smallest upper and ∆ l,m (k, w) the largest lower confidence interval limit, defined as: Several approaches on the filter support selection using the RICI rule are feasible, i.e., the above-explained procedure may be applied to only one side of the considered sample (left or right) or to both sides (symmetrical or asymmetrical filter supports). In this paper, we chose the asymmetrical filter support selection where the RICI algorithm is applied independently to the left-hand and right-hand side of the considered sample k [72].
Calculations are done for each signal sample k, resulting in two sets of confidence intervals ∆ l (k, w) and ∆ r (k, w), one for the left-hand side and one for the right-hand side of the considered sample [72]: According to confidence intervals definition, given in (22), their widths decrease as the number of samples k used for their calculation increases: . In accordance with (22) and (23), the confidence limits of the confidence intervals determined by the asymmetrical RICI procedure are calculated as [72]: where ∆ l l,m (k, w) (∆ l u,m (k, w)) and ∆ r l,m (k, w) (∆ r u,m (k, w)) are the lower (upper) confidence limits of the confidence interval calculated using m adjacent noisy signal sample values (including the considered signal sample) to the left-hand side and to the right-hand side of the considered sample k, respectively; s l m (k, w) andŝ r m (k, w) are the estimates of the considered signal sample value, calculated using sample values to its left-and right-hand side, respectively; σ l m (k, w) and σ r m (k, w) are the standard deviations of the estimation error for the left-and right-hand side calculations, respectively.
The algorithm operates on the each side of the considered kth sample independently by tracking the intersection of the currently calculated mth confidence interval for the right-hand side) with the intersection of all previous m − 1 confidence intervals (denoted as ∆ l m−1 (k, w) for the left-and ∆ r m−1 (k, w) for the right-hand side). If the confidence intervals' intersection is non-empty, the algorithm marks w l m (k) and w r m (k) as the current candidate for the optimal filter support to the left-and to the right-hand side of the considered sample, respectively. This intersection condition is checked by comparing the values of the smallest upper ∆ l u,m (k, w) and the largest lower ∆ l l,m (k, w) confidence limit on the left side [72]: and the values of the smallest upper ∆ r u,m (k, w) and the largest lower ∆ r l,m (k, w) confidence limit on the right side: The above described first stage of the RICI algorithm (the ICI stage) results in w l− (k) for the left-hand side calculations and w r− (k) for the right-hand side calculations, as the largest filter supports satisfying (35) and (36), respectively. Finally, the largest candidate for the optimal filter support w − (k) is obtained as [72]: where the fact that both supports (left and right) contain the considered sample k is taken into account. The thus obtained filter support w − (k) is close to the optimal value w o (k), defined in (11) [70,73]. The performance of the described ICI stage relies heavily on the proper selection of the threshold parameter Γ. Namely, selecting too small Γ values leads to w − (k) < w o (k), thus resulting in signal undersmoothing, while selecting Γ too large results in signal oversmoothing, as w − (k) > w o (k) [65,66]. Moreover, the ICI stage shows weaker performance when applied to wide regions of nearly constant signal values in which sudden changes occur [72].
Therefore, the second stage of the algorithm, the RICI rule stage, introduces the additional criterion for the adaptive filter support selection and applies it to the filter support candidates obtained in the first stage (the ICI rule stage) of the algorithm. The RICI rule method significantly improves the estimation accuracy of the ICI rule method for the same values of parameter Γ, while simultaneously being more robust to suboptimal Γ values [72,74,75]. The study in [74] considered only high SNR scenarios of the synthetic data corrupted by the additive white Gaussian noise. In this paper, the approach is extended to the real-life problem of gravitational-wave denoising using different orders of the LPA in the intensive noise scenarios (low SNRs), characterized by the non-stationary and non-Gaussian noise.
The RICI criterion is defined with respect to the ratio of the intersection of the confidence intervals' width and the current confidence interval's width, thus taking into account the relative amount of confidence intervals overlapping [72,74]. As in the previous algorithm stage, this ratio is also calculated for both sides of the considered sample, thus obtaining R l m (k, w) for the left-hand side [72]: and R r m (k, w) for the right-hand side: The RICI criterion is defined for the left-hand side of the considered sample as [72]: and for the right-hand side as: where R c is a preset threshold value (0 ≤ R c ≤ 1). The RICI stage results in w l+ (k) as the largest filter support satisfying (35) and (40) for the left-hand side calculations, while w r+ (k) is obtained by the right-hand side calculations, as the largest filter support satisfying (36) and (41) [72].
Finally, the filter support w + (k) obtained by the RICI algorithm is calculated as:

Data Conditioning
In order to test whether the proposed LPA-RICI-based denoising algorithm can efficiently suppress noise in gravitational-wave data in real-life conditions, numerically generated CCSN burst signal templates are injected into the real-life noise data collected by the Advanced LIGO Livingston detector. Burst signals are obtained from the Dimmelmeier catalog [76], which contains waveforms generated by general-relativistic simulations of rotating stellar core collapse neutron stars. We assess the denoising performance of the LPA-RICI method on three CCSN template signals: s20a1o05, s20a2o09, and s20a3o15 (data publicly available at https://zenodo.org/record/4108838# .X48R7pxR1Pa), as in [64]. Moreover, the denoising performance for each signal is analyzed when the signal source is located at three different distances (5, 10, and 20 kpc), which correspond to different SNR levels.
As the Advanced LIGO detector noise is non-Gaussian and non-stationary, the data are first preprocessed using the autoregressive model developed in [77,78], in order to whiten the data, i.e., to transform the colored noise into the white noise that is flat in frequency. Afterward, the LPA-RICI denoising algorithm is applied.

Whitening Procedure
The data whitening procedure is applied to make the data delta-correlated. This means that the sequence of data x[n], after the whitening procedure, will be uncorrelated at each lag different from zero, i.e., the autocorrelation r xx [n] is a delta function. There are several strategies to perform this process: some are based on techniques in the frequency domain, others on techniques in the time domain. The result of this operation is that the contribution of the statistics up to the second order will be removed from the data. In this work, we used the time domain technique, developed in [77,79], which is based on the time domain procedure, using an autoregressive (AR) fit to the data. An autoregressive process x[n] of order P with parameter a k , hereafter AR(P), is characterized by the relation: where w[n] is a white Gaussian process. The problem of determining the AR parameters is the same as that of finding the optimal "weights vector" w = w k , for k = 1, ...P, for the linear prediction problem [80]. In the linear prediction, we would predict the sample x[n] using the P previous observed data building the estimate as a transversal filter: We can find the coefficients of the linear predictor by minimizing a cost function that is the mean squared error = E [e[n] 2 ] (E is the ensemble average operator), with: Being the error we make in this prediction and obtaining the so-called normal or Wiener-Hopf equations: which are identical to the Yule-Walker equations [80] used to estimate the AR parameters a k from the autocorrelation function, with w k = − a k and min = σ 2 . This is the key relationship between the AR model and the linear prediction, which assures obtaining a filter that is stable and causal [80]. In this way, we can use the AR model to reproduce stable processes in the time domain.
What we have to do is to simply find the P parameters that fit the power spectral density of our process, and at the same time, we find the optimal weights vector that allows us to reproduce the process at time n knowing the process at the previous time P. The method that uses this estimation tries to make the error signal (45) a white process. This is the reason we call this a whitening procedure.
Using a lattice structure [78], we can implement the whitening filter in the time domain. This is a procedure that is used for other pipelines [50,81] and that can eventually be also implemented in an adaptive way [77], taking care of the non-stationary noise.

Data Denoising
The results obtained by the LPA-RICI denoising of gravitational-wave data are also compared to the results obtained by several conventionally applied signal denoising methods, including the LPA-ICI method [65], the TV-L1 denoising method with the primal-dual algorithm [56,82,83], the method based on the neighboring thresholding in the short-time Fourier transform (STFT) domain (Neigh STFT) [84], and three wavelet denoising methods. The data-driven Neigh STFT method represents an adaptive noise level estimation and denoising algorithm based on the minimal controlled recursive averaging estimator and neighboring block thresholding in the STFT domain, where the optimal threshold and block size are automatically adjusted by minimizing Stein's unbiased risk estimator (SURE) [84]. An analysis of the wavelet-based denoising techniques was conducted by inspecting the denoising performances of three different wavelets-symlet wavelets, Daubechies wavelets, and coiflet wavelets. For each wavelet, a range of the numbers of vanishing moments was considered (sym2-sym45, db1-db45, and coif1-coif5), as well as different levels of wavelet decomposition and threshold selection rules applied to the wavelet coefficients (including SURE [85] and minimax [86] thresholding). These wavelet-based methods employ hard thresholding and multiplicative threshold rescaling using a level-dependent estimation of level noise.
For each tested denoising method, the optimal algorithm parameters are selected by conducting an extensive search in the parameter space and choosing parameter values that minimize the estimated root mean squared error (RMSE). In order to quantify the performance of the proposed LPA-RICI-based denoising algorithm in comparison to the alternative methods, a set of performance indices is used.

Performance Indices
The estimation efficiency of the LPA-RICI-based gravitational-wave denoising is assessed using the following performance indices: • Improvement in the signal-to-noise ratio (ISNR): • Peak signal-to-noise ratio (PSNR): • Root mean squared error (RMSE): • Mean absolute error (MAE): • Maximum absolute error (MAX): Performance indices RMSE, MAE, and MAX are given as normalized values in the following subsection, i.e., they are calculated based on the normalized signals obtained by dividing the signals by their maximum values. This normalized representation facilitates the comparison of the performance indices' values between different signals and corresponding SNR values.

Case Study-Signal s20a1o05
The denoising accuracy of the LPA-RICI method with the signal s20a1o05 was analyzed for three different signal source distances, namely, 5, 10, and 20 kpc, corresponding to the SNR levels of 3.9 dB, −2.11 dB, and −8.13 dB, respectively. CCSN burst template signal s20a1o05 at a distance of 5 kpc is shown in Figure 1a, while Figure 1b shows the noise-corrupted version of this signal. As seen in Figure 1, signal s20a1o05 and other signals of this type, including signal s20a2o09 and signal s20a3o15, are characterized by a negative peak whose occurrence is associated with the core bounce and is followed by damped oscillations of the proto-neutron star. The results obtained by applying the LPA-RICI denoising algorithm to the noisy signal s20a1o05 at a distance of 5 kpc are presented in Figure 2. Figure 2a,c,e shows the comparison between the original template signal and the signal obtained by the LPA-RICI denoising procedure with the LPA order set to the values of n = 0, n = 1, and n = 2, respectively. Figure 2b,d,f shows the respective estimation errors. The results presented in Figure 2 suggest that the LPA-RICI method successfully removes the noise from the noisy signal s20a1o05 at a distance of 5 kpc, as the denoised signals fit the templates almost perfectly. All three LPA-RICI variants provide excellent denoising performance, with the second-order LPA variant producing the best results visually.  The quantitative comparisons of the denoising results obtained by the LPA-RICI method, the LPA-ICI technique, the TV method, the Neigh STFT technique, the symlet, the Daubechies, and the coiflet wavelet-based methods, for the signal s20a1o05 at distances of 5, 10, and 20 kpc, are given in Tables 1-3 Tables 1-3 confirm that all three variants of the proposed LPA-RICI method provide excellent denoising performance, improving the SNR of the signal and reducing the estimation errors. As the signal source distance increases (and the SNR of the noisy signal decreases), the performance of the LPA-RICI method deteriorates slightly, but still provides satisfactory denoising results. At all three distances, the best results are obtained when the second-order LPA is applied.
The relative performance improvement of the second-order LPA-RICI method over the other tested methods for the signal s20a1o05 at distances of 5, 10, and 20 kpc is calculated, and the percentage values are given in Tables Table 2. Denoising results for the CCSN signal s20a1o05 at a distance of 10 kpc (SNR = −2.11 dB). The best performance indices are marked in bold.   Table 4. Relative performance improvement of the LPA-RICI-based (n = 2, Γ = 11, R c = 1) denoising over other tested methods, for the CCSN signal s20a1o05 at a distance of 5 kpc (SNR = 3.9 dB).   Table 6. Relative performance improvement of the LPA-RICI-based (n = 2, Γ = 28, R c = 1) denoising over other tested methods, for the CCSN signal s20a1o05 at a distance of 20 kpc (SNR = −8.13 dB). The execution times were also calculated for each tested denoising method. The algorithm execution times were obtained on a computer with the Intel Core i7-4720HQ CPU @ 2.60 GHz, and 8 GB of RAM. The results were averaged over 1000 algorithm runs. The algorithm execution times of each method applied to the denoising of the signal s20a1o05, at distances of 5, 10, and 20 kpc, are given in Table 7. The presented results suggest that the proposed LPA-RICI method outperforms, in terms of algorithm execution times, the original LPA-ICI method in all considered cases, and the Neigh STFT method in most cases. However, it exhibits weaker performance when compared to the TV-L1 method and wavelet-based techniques. The denoising of the CCSN burst signal s20a2o09 was assessed for three different cases obtained by placing the signal source at three different distances. Distances of 5, 10, and 20 kpc correspond to the SNR levels of −4.54 dB, −10.09 dB, and −15.98 dB, respectively. Figure 3a shows the template signal s20a2o09 at a distance of 5 kpc, while Figure 3b shows the same signal embedded in the detector noise.   Figure 4a,c,e displays the comparison between the template signal and the signal obtained by the LPA-RICI denoising procedure with the LPA order set to the values of n = 0, n = 1, and n = 2, respectively. The corresponding estimation errors are given in Figure 4b,d,f. As seen in Figure 4, the denoised signals obtained by all three LPA-RICI variants are very close to the templates. The main positive and negative peaks are well reconstructed, as well as subsequent oscillations. Therefore, the presented results suggest that the LPA-RICI method efficiently removes the noise from the noisy signal s20a2o09 at a distance of 5 kpc, while simultaneously preserving the signal morphology. Tables 8-10 provide the values of denoising performance indices obtained by the LPA-RICI method and other tested methods applied to the noisy signal s20a2o09 at distances of 5, 10, and 20 kpc, respectively. The presented results indicate that all three variants of the applied LPA-RICI method provide good denoising performance, based on improving the noisy signal SNR and reducing the estimation errors. The LPA-RICI method's denoising performance somewhat deteriorates as the distance is increased due to the very low SNR, but the obtained estimation results are still satisfactory.  (f) Figure 4. Results of applying the LPA-RICI denoising method to the noisy CCSN signal s20a2o09 at a distance of 5 kpc (SNR = −4.54 dB): (a) template and LPA-RICI denoised signal (n = 0, Γ = 9, R c = 1); (b) LPA-RICI estimation error (n = 0, Γ = 9, R c = 1); (c) template and LPA-RICI denoised signal (n = 1, Γ = 13, R c = 1); (d) LPA-RICI estimation error (n = 1, Γ = 13, R c = 1); (e) template and LPA-RICI denoised signal (n = 2, Γ = 16, R c = 1); (f) LPA-RICI estimation error (n = 2, Γ = 16, R c = 1). Table 11 gives the relative change of the performance indices obtained by the second-order LPA-RICI method compared to those obtained by other denoising methods applied to the signal s20a2o09 at a distance of 5 kpc. As seen in Table 11, the LPA-RICI method outperforms the other tested denoising methods, except the TV method and the coif1 wavelet method. However, the performances of these three methods are very close. The results presented in Table 12 suggest that the second-order LPA-RICI method, applied to the denoising of the signal s20a2o09 at a distance of 10 kpc, outperforms the other tested methods, with the exception of the performance index MAE, which is slightly worse than those obtained by the LPA-ICI method and the TV method. Table 13 shows the comparison of performance indices obtained by the zero-order LPA-RICI method and performance indices obtained by other methods applied to the denoising of the signal s20a2o09 at a distance of 20 kpc. In this case, the LPA-RICI method outperforms the other methods, except the TV method, which is slightly better for performance indices ISNR, PSNR, and RMSE, but only by a very small margin.

LPA-ICI
To sum up, the LPA-RICI method applied to the denoising of the signal s20a2o09 at all three distances provides a denoising performance close to the one obtained by the TV method. It outperforms the other methods by increasing ISNR by up to 95.56% and PSNR by up to 132.60%, as well as by reducing RMSE by up to 62.13%, MAE by up to 55.60%, and MAX by up to 83.31%. Table 14 presents the algorithm execution times of each method applied to the denoising of the signal s20a2o09, at distances of 5, 10, and 20 kpc. These results indicate that the LPA-RICI method is outperformed by the TV-L1 method and wavelet-based techniques in terms of algorithm execution times. However, it runs competitively with the Neigh STFT denoising method and significantly reduces the execution time when compared to the original LPA-ICI approach. Table 8. Denoising results for the CCSN signal s20a2o09 at a distance of 5 kpc (SNR = −4.54 dB). The best performance indices are marked in bold.  Table 9. Denoising results for the CCSN signal s20a2o09 at a distance of 10 kpc (SNR = −10.09 dB). The best performance indices are marked in bold.  Table 11. Relative performance improvement of the LPA-RICI-based (n = 2, Γ = 16, R c = 1) denoising over other tested methods, for the CCSN signal s20a2o09 at a distance of 5 kpc (SNR = −4.54 dB).

Case Study-Signal s20a3o15
The denoising procedure for the signal s20a3o15 was conducted at three different signal source distances: 5, 10, and 20 kpc, corresponding to the SNR levels of −2.18 dB, −8.17 dB, and −14.19 dB, respectively. The considered template signal s20a3o15 at a distance of 5 kpc is shown in Figure 5a, while Figure 5b shows the template signal corrupted by the LIGO detector noise.
The results obtained by applying the LPA-RICI method to the noisy signal s20a3o15 at a distance of 5 kpc are displayed in Figure 6. Figure 6a,c,e provides the comparison between the template signal and the denoised signal obtained by the LPA-RICI procedure with the LPA order set to the values of n = 0, n = 1, and n = 2, respectively. Figure 6b,d,f shows the corresponding estimation errors.
The denoised signals obtained by all three LPA-RICI variants fit the templates very well, indicating successful removal of the noise from the noisy signal s20a3o15 at a 5 kpc distance.  The comparisons of the results obtained by the LPA-RICI method and other tested methods for the denoising of the signal s20a3o15 at distances of 5, 10, and 20 kpc are given in Tables 15-17, respectively. The results presented in these three tables suggest that all three variants of the LPA-RICI method provide excellent denoising performance, significantly improving the SNR of the noisy signal and reducing the estimation errors. The denoising performance of the LPA-RICI method deteriorates slightly with the increasing distance due to the increased noise intensity, but this decline in estimation accuracy is less pronounced than with other methods, and the LPA-RICI still provides good denoising results even for a very low SNR value.
The percentage values of the relative estimation improvement of the LPA-RICI method over the other tested techniques for the signal s20a3o15 at distances of 5, 10, and 20 kpc are given in Table 18 (the first-order LPA), Table 19 (the zero-order LPA), and Table 20 (the zero-order LPA), respectively. The results presented in these tables suggest that the LPA-RICI method significantly outperforms the other tested methods at each considered distance and SNR level, by increasing ISNR by up to 100.07% and PSNR by up to 138.52%, as well as by reducing RMSE by up to 64.59%, MAE by up to 44.13%, and MAX by up to 84.79%.      The algorithm execution times of each technique applied to the denoising of the signal s20a3o15, at distances of 5, 10, and 20 kpc, are provided in Table 21. The presented results show that the wavelet-based techniques provide the best performance in terms of algorithm execution times, followed by the TV-L1 method. However, the proposed LPA-RICI method provides execution time performance that is in most cases competitive with the one obtained by the Neigh STFT denoising technique and significantly reduces the algorithm execution time when compared to the LPA-ICI approach. To sum up the above discussion, we may point out that the proposed LPA-RICI method outperforms other tested competitive methods (the approach combining LPA and the original ICI rule, the TV-L1 method, the method based on the neighboring thresholding in the STFT domain, and the three wavelet-based denoising techniques) in gravitational-wave denoising in low SNR scenarios.

Conclusions
In this paper, the LPA-RICI algorithm is proposed to denoise the gravitational-wave burst signals from the CCSN. This data-driven, locally adaptive, and easy-to-implement method is applied to denoising numerically generated burst signals injected into the real-life non-Gaussian and non-stationary noise data obtained by the Advanced LIGO detector. The estimation accuracy of the LPA-RICI denoising method is assessed for three different burst signals, with each being placed at different distances corresponding to the different values of SNR. The analysis of the experimental results obtained by these case studies indicates that the LPA-RICI method efficiently suppresses the noise and simultaneously preserves the critical features (morphology) of the gravitational-wave burst signals. The approach provides good denoising performance even for signals corrupted by the intensive noise (very low SNR values). Moreover, the comparative analysis shows that the LPA-RICI method outperforms several conventionally applied denoising techniques of similar complexity (the original LPA-ICI approach, the TV-L1 method, the method based on the neighboring thresholding in the STFT domain, and the three wavelet-based denoising techniques). Therefore, the results obtained in this work suggest that the LPA-RICI method may be successfully applied in the preprocessing of the noisy gravitational-wave data collected by the Advanced LIGO and Advanced Virgo detectors and included as a part of the algorithm pipeline for the advanced detection and identification of gravitational-wave events.
The main challenges in this research field include detecting useful information and burst signals in intensive noise environments (low SNRs). This problem may also be approached as a classification problem, which can be solved by machine learning techniques. We plan to research this in the near future; namely, instead of analyzing noisy time-series, we want to implement Cohen's class of time-frequency distributions and apply machine learning techniques to distinguish between signal and noise in the quadratic time-frequency representations. According to the literature review, this approach has not been used in the gravitational-waves field up to now.