Methods to use in impulse (outlier) detection and estimation of outlier-free signal segments

Speckle noise is inevitable due to the working principle of laser Doppler vibrometry, which relies on transmitting a laser beam to a target surface and detecting the changes in the phase and frequency of the reflected beam relative to the transmitted beam. These changes occur due to the laser beam’s interaction with the vibrating surface (i.e., Doppler shift) (Johansmann et al. 2005). The reflected laser beam consists of different wave fronts, which form a speckle pattern. When a surface has a high roughness, different distances between the various points on the surface and the LDV result in wave fronts with different phases. As a result, the noise level in the measurements becomes more severe. Noise level can also increase when the laser beam is reflected from dark (unreflective) spots (Dräbenstedt 2007). Such type of noise is called speckle noise and it is more problematic when the laser beam moves over the target surface. The reflection of the laser beam from a dynamically changing surface pattern causes (i) phase modulation due to the dynamic changes in the wave fronts and (ii) amplitude modulation due to the movement of the laser beam over dark spots. If the phase modulation is more than -/+ 2π, a signal cannot be constructed and drop-outs in the signal power occur. Dropouts also occur if the laser beam moves over the dark spots. Dropouts in the signal power result in artificial peaks in the recorded time-domain signals, leading to impulsive noise (i.e., IN) in the measurements.

The existence of IN in the moving LDV measurements might cause misidentification of defects when damage identification is performed using wave propagation. Therefore, for reliable damage detection, it is necessary to apply IN filters to the signals obtained from moving LDV measurements (i.e., moving LDV signals). These filters operate in two steps: (i) the detection of the IN corrupted signal segments (i.e., detection step), and (ii) the estimation of IN-free values of the IN corrupted segments (i.e., estimation step).

The detection step of IN filters that were developed for moving LDV signals uses Phase Space Thresholding (i.e., PST) method (Goring and Nikora 2002), the Acceleration Thresholding (i.e., AT) method (Goring and Nikora 2002), wavelet thresholding approach (Goring and Nikora 2002), kurtosis ratio (Vass et al. 2008), and a fuzzy logic based filter (Naso et al. 2006). In the PS method, the first and second-order derivatives of a moving LDV signal are computed. Then, three 2-dimensional plots (i.e., phase spaces) are created using combinations of the LDV signal and its first and second-order derivatives. Next, ellipsoids were created using the universal threshold, and then they are placed onto the 2-dimensional plots. The points lying out of these ellipsoids are considered as IN corrupted signal points. In AT method, the acceleration signal is obtained from the LDV signal, and IN corrupted points are detected when the points of the acceleration signal exceed a threshold. In wavelet thresholding approach, the LDV signal is transformed to the wavelet domain and IN corrupted segments are detected using the wavelet coefficients exceeding a threshold. The kurtosis ratio calculated for different segments of an LDV signal is used to determine the IN corrupted segments (Vass et al. 2008). The developed fuzzy logic filter calculates the difference of several consecutive signal points and performs IN detection using user-defined decision rules.

The estimation step of the IN filters developed for moving LDV signals compute IN-free values of IN corrupted segments through (i) interpolation using polynomials fitted to the two sides of IN corrupted segment, (ii) the smoothed signal points at IN corrupted points which are computed using the entire recorded signal, (iii) extrapolations from the proceeding one or two signal points, and (iv) the weighted difference between two consecutive signal points based on the severity of IN determined by the membership functions of the fuzzy-logic based filter.

However, although IN is a major concern in dynamic LDV measurements, only a limited number of studies were performed to tackle this concern. On the other hand, the literature is rich with IN filtering studies since IN appear also in other kinds of signals, such as biomedical, audio, speech, image, and communication signals. Most of the IN filters developed for such signals also operate using detection and estimation steps. Therefore, it is beneficial to examine the detection and estimation steps of other IN filters, as they may be applicable to the IN observed in moving LDV signals.

The detection step of IN filters developed for communication signals generally uses nonlinear functions (Barazideh et al. 2018, 2019a; b; c; Efron 1995). These nonlinear functions consist of a linearly increasing segment up to chosen thresholds, and then a rapidly decreasing segment. Signal points exceeding the thresholds are considered to be IN corrupted. In (Barazideh et al. 2018, 2019a; b; c; Efron 1995), thresholds were chosen based on the quartile of signals, the previously developed signal models, and the experience on IN corrupted communication signals.

The estimation step of IN filters developed for communication signals replace the values of detected IN corrupted segments with (i) the corresponding values located in the rapidly decreasing segment of nonlinear functions or (ii) with the weighted sum of the IN corrupted signals’ values and their estimated IN-free values (i.e., estimated through an autoregressive model) (Efron 1995).

The detection step of IN developed for images generally use methods that are based on rank-ordered mean (i.e., ROM) (Abreu and Mztra 1995), rank-ordered absolute differences (i.e., ROAD) (Dong et al. 2007; Garnett et al. 2005), Boundary Discriminative Noise Detection (i.e., BDND) (Duan and Zhang 2010; Jafar et al. 2013; Ng and Ma 2006), fuzzy logic (Aborisade 2011), linear prediction (Roy and Laskar 2017), and Artificial Neural Networks (i.e., ANN’s) (Budak et al. 2015; Khaw et al. 2019; Kong and Guan 1996; Li et al. 2020; Rezvanian et al. 2008). In ROM-based methods, ROMs (i.e., the average of two points in the middle of the sorted values of the signal points in a two-dimensional windows) is computed and IN corrupted points are detected when ROMs exceed a threshold (Abreu and Mztra 1995). Similarly, in ROAD-based methods, ROADs (i.e., the average of the smallest absolute differences between the central pixel and its neighbors in two-dimensional windows) is computed, and IN corrupted points are detected when ROADs exceed a threshold (Dong et al. 2007; Garnett et al. 2005). In BDND based methods, the pixels are classified into different noise levels using the differences between the sorted values of the signal points in a two-dimensional windows and the median of these sorted values (or using similar approaches). In fuzzy logic-based methods, decision rules are implemented to detect IN corrupted pixels (Aborisade 2011; Suganthi and Senthilmurugan 2013). In an linear prediction-based method, IN detection is performed using the difference between the value of the pixels located in the center of two-dimensional windows and their estimated IN-free values which are obtained by linear prediction (Roy and Laskar 2017). To further improve the IN detection performance, the use of ANN’s is proposed (Budak et al. 2015; Khaw et al. 2019; Li et al. 2020; Rezvanian et al. 2008). However, training samples are required for ANN’s to learn the parameters of the noise filtering process. One approach in which training is not required was using a self-organizing map (i.e., SOM) for the detection of IN (Kong and Guan 1996).

The estimation step for IN corrupted images was carried out generally by median filter (i.e., MF) (Dong and Xu 2007; Jafar et al. 2013; Khan and Lee 2017; Ng and Ma 2006; Roy and Laskar 2017; Shaik and Lorraine 2012; Vijaykumar et al. 2008), ROM (Abreu and Mztra 1995), ROAD (Garnett et al. 2005), and weighted mean and sum (Hussain et al. 2012; Lee et al. 2004; Pritamdas et al. 2016; Schulte et al. 2006, 2007; Wang et al. 2002). MF replaces the central pixel of a two-

dimensional window with the value in the middle of the sorted values of the signal points in two-dimensional windows (Suganthi and Senthilmurugan 2013). MF’s are usually combined with weights (i.e. weighted MF’s) and the window size is changed adaptively based on local noise density (i.e., adaptive MF’s) (Dong and Xu 2007; Khan and Lee 2017; Ng and Ma 2006; Roy and Laskar 2017; Shaik and Lorraine 2012; Vijaykumar et al. 2008). The weights modify the shorting of the array by replicating each element. Also, MF can be formulated to incorporate deviation of the uncorrupted pixels’ values from their median as well as from the spatial relation of pixels in the windows (Jafar et al. 2013). In the ROM-based estimation step, the detected IN corrupted pixels are changed with ROMs (Abreu and Mztra 1995). In ROAD-based estimation step, (Garnett et al. 2005). ROADs are incorporated into the weights of bilateral filter which estimates the IN-free values of the detected IN corrupted pixels. In weighted mean and sum-based methods, the IN-free values are estimated using the weighted mean of pixels in two-dimensional windows, or simply weighted sum (Hussain et al. 2012; Lee et al. 2004; Schulte et al. 2006, 2007; Wang et al. 2002). It is also possible to use an additional term to control the effect of the weighted mean on the pixels (Schulte et al. 2007). The weights are usually computed based on local noise density estimated by fuzzy logic rules (Hussain et al. 2012; Lee et al. 2004; Schulte et al. 2006, 2007; Wang et al. 2002). In one study, weights were estimated based on approximated variance of pixels’ values (Pritamdas et al. 2016).

The detection step of IN corrupted signal points in the speech and audio signals were performed by several methods based on linear prediction (Figure 1990; Janseen et al. 1986; Niedźwiecki and Ciołek 2013; Oudre 2015), template matching (Niedźwiecki and Ciołek 2015), Kalman Filter (i.e., KF) (Canazza et al. 2010; Doblinger 1998; Maciej and Krzysztof 1996), Bayesian statistics (i.e., BS) (Ávila and Biscainho 2012; Godsill and Rayner 1995; Pasteur and Koch 1998; Wan et al. 2018), wavelet transformation (i.e., WT) (Nongpiur 2008; Nongpiur and Shpak 2013; Valiere et al. 1990) and ROM (Charu Chandra 1998). In linear prediction-based methods, the detection of IN is carried out by finding the time instants when the difference between the predicted and recorded signal points exceeds a threshold (Figure 1990; Janseen et al. 1986; Niedźwiecki and Ciołek 2013; Oudre 2015). The predicted signals are obtained using autoregressive models or sparse autoregressive models of the recorded signals. The purpose of sparse autoregressive modeling is to improve results when audio or speech signals exhibit a long correlation. The model parameters were obtained using methods such as Levinson-Durbin, Exponentially-weighted Least Square and Expectation Maximization. In the Template matching-based method, observed IN patterns are transformed to the prediction error domain (Niedźwiecki and Ciołek 2015). The detection of IN is performed when the transformed IN patterns are matched with the prediction error of signals. This error is the difference between the recorded signal and the modelled signal (by an autoregressive model). In KF based methods, adaptive and extended KF’s are used to estimate the signals using autoregressive model parameters embedded in the state-space form (Canazza et al. 2010; Doblinger 1998; Maciej and Krzysztof 1996). The detection of IN is carried out using the difference between the values of estimated and recorded signal points. The model parameters were obtained through methods such as Levinson-Durbin and Least Square Lattice algorithms (Doblinger 1998). Extended KF performs IN detection and the parameter calculation simultaneously (Canazza et al. 2010; Maciej and Krzysztof 1996). In BS-based methods, the maximization of the conditional probability of IN locations conditional upon the observed signals results in the location of IN corrupted signal segments. Therefore, in BS-based methods, probabilistic models of the signal parameters and IN, the locations of IN corrupted points, and the initial estimates of the probabilistic models’ parameters are required. Accordingly, it is possible to

use specifically developed models for the amplitude, duration, and shape of observed IN (Pasteur and Koch 1998). The solution of the maximization problem is tackled by Markov Chain Monte Carlo methods (Ávila and Biscainho 2012; Godsill and Rayner 1995), or recursive estimation (Pasteur and Koch 1998). In WT-based methods, the frequency localization of IN’s energy is exploited (Nongpiur 2008; Nongpiur and Shpak 2013; Valiere et al. 1990). IN corrupted signal points are detected by finding the location of wavelet coefficients exceeding a fixed or a time-varying threshold at the decomposition level in which the signals exhibit low power. The use of a time-varying threshold facilitates the detection of the wavelet coefficients of IN corrupted points when the frequency content of IN and audio signals coincide. In the ROM-based method, the IN detection approach developed for images is applied to speech signals using one-dimensional windows (Charu Chandra 1998).

The estimation of IN-free values of IN corrupted segments in the speech and audio signals uses methods based on linear prediction (i.e., LP), Bidirectional Interpolation (i.e., BI), and Adaptive Interpolation (i.e, AI). (Figure 1990; Janseen et al. 1986; Niedźwiecki and Ciołek 2013; Oudre 2015), Kalman Filter (i.e., KF) (Canazza et al. 2010; Doblinger 1998; Maciej and Krzysztof 1996), Bayesian statistics (i.e., BS) (Ávila and Biscainho 2012; Godsill and Rayner 1995; Pasteur and Koch 1998), and wavelet transformation (i.e., WT) (Nongpiur 2008; Nongpiur and Shpak 2013). In LP and BI-based methods, first, autoregressive parameters of signals are calculated using methods such as Yule-Walker, Singular Value Decomposition and Burg’s methods. Then, the IN-free values of the IN corrupted segments are predicted using an autoregressive model or a sparse autoregressive model of the recorded signals. In AI method, autoregressive model parameters were calculated iteratively along with the estimation of IN corrupted segments' IN-free values [40], [41]. Therefore, AI’s closely resembles the EM algorithm. In KF based methods, when IN is detected, only the value that is estimated by the KF filter is used (Canazza et al. 2010; Doblinger 1998; Maciej and Krzysztof 1996). (Canazza et al. 2010; Maciej and Krzysztof 1996). In BS-based methods, the maximization of the estimated signal conditional upon IN locations results in the estimation of the corrupted signal points (Ávila and Biscainho 2012; Godsill and Rayner 1995; Pasteur and Koch 1998). In WT-based methods, the amplitude of the wavelet coefficients corresponding to IN corrupted segments was (i) modified based on thresholds (Nongpiur 2008; Nongpiur and Shpak 2013), or (ii) replaced with the wavelet coefficients corresponding to some uncorrupted segments [49].

In biomedical measurements, IN occurs in Electrocardiogram (i.e., EKG) and Electroencephalogram (i.e., ECG) signals due to electrical activity from non-target sources on the test subject. Most of the IN filters developed for such signals do not use a detection-estimation approach. Instead, they operated in a direct way that the recorded signals are modified to obtain the uncorrupted signals using methods such as Adaptive Filtering and Myriad Filter (Gupta et al. 2013; Khalili et al. 2017; Mirza et al. 2015; Pander 2004). However, an IN filter operates using the detection and estimation approach. In this method, IN corrupted segments are detected using the signal points exceeding a threshold which was obtained by low-pass filtering of the signal 5 (computed by Hilbert transform) (Melia et al. 2014). The estimation step is performed by using the values of the threshold at the detected IN corrupted segments (Melia et al. 2014).

Furthermore, the literature also consists of methods solely focusing on estimation of the missing signal points. These methods can be used as an estimation step of an IN filter if the values of signal points in the IN-corrupted segments are considered missing(Horner et al. 2019; Matarazzo and Pakzad 2016; Zgheib et al. 2006). One method is to use KF similar to the KF-based estimation

steps of IN filters developed for speech and audio signals. In this method, the missing signal values are predicted using autoregressive models (embedded in KF’s) whose parameters can points can be estimated by EM algorithm or Recursive Least Square (Horner et al. 2019; Matarazzo and Pakzad 2016; Zgheib et al. 2006). When the EM algorithm is adopted, the estimation of missing points and the calculation of the autoregressive parameters for a signal segment is repeated several times till convergence is achieved. When highly correlated signals are considered, multivariate scatter and location estimators can be adopted to find the autocorrelation estimates which are required to calculate the autoregressive parameters. (Kim and Kosko 1996). In addition to KF-based methods, other widely used methods are (i) the EM with data augmentation (i) the EM with multiple imputation, (iii) the Markov Chain Monte Carlo (i.e. MCMC) with multiple imputation, and (iv) the regression method with multiple imputations (Dong and Peng 2013; Imtiaz and Shah 2008). In the EM with data augmentation, first, the missing points are estimated using the parameters of the signal (i.e., expectation step). Next, the parameters are updated using the signal with the estimated values (i.e., maximization step). In the EM combined with MI, the expectation step is repeated to estimate different values for missing points, allowing for computing probabilistic distribution of the signals’ parameters calculated in the maximization step. The MCMC method combined with MI follows a similar approach but it is applied for data with an arbitrary missing pattern. Its another difference is that multiple values of missing points and parameters are chosen after the convergence. The approach used in regression method with multiple imputations is similar as well. In this case, the parameters belong to a regression fitted on the signal of interest. Another approach is based on principle component analysis (i.e., PCA) (Imtiaz and Shah 2008; Oliveira and Gomes 2010), where principal components of a matrix (formed by the signals with missing values or by the blocks of one signal with missing values) are estimated. Then, the estimated principal components are used to estimate the values of the missing signal points (through iterative and direct algorithms). In addition to these methods solely focused on the estimation of missing points, the estimation step of Hampel Filter, which is a widely used outlier filter, can be used as an estimation step of an IN filter as well. The detection step of Hampel Filter operates by simply replacing the detected IN corrupted points with the output of an MF (Pearson et al. 2016).

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