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

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. The LPA Filter Design Method

#### 2.2. The RICI Algorithm

## 3. Results and Discussion

#### 3.1. Data Conditioning

#### Whitening Procedure

#### 3.2. Data Denoising

#### 3.3. Performance Indices

- Improvement in the signal-to-noise ratio (ISNR):$$\mathrm{ISNR}=10{log}_{10}\left(\frac{{\sum}_{k=1}^{{N}_{k}}{\left(s\left(k\right)-x\left(k\right)\right)}^{2}}{{\sum}_{k=1}^{{N}_{k}}{\left(s\left(k\right)-{\widehat{s}}_{m}\left(k\right)\right)}^{2}}\right)$$
- Peak signal-to-noise ratio (PSNR):$$\mathrm{PSNR}=20{log}_{10}\left(\frac{{max}_{k=1,\cdots ,{N}_{k}}s\left(k\right)}{\sqrt{\frac{1}{{N}_{k}}{\sum}_{k=1}^{{N}_{k}}{\left(s\left(k\right)-{\widehat{s}}_{m}\left(k\right)\right)}^{2}}}\right)$$
- Root mean squared error (RMSE):$$\mathrm{RMSE}=\sqrt{\frac{1}{{N}_{k}}\sum _{k=1}^{{N}_{k}}{\left(s\left(k\right)-{\widehat{s}}_{m}\left(k\right)\right)}^{2}}$$
- Mean absolute error (MAE):$$\mathrm{MAE}=\frac{1}{{N}_{k}}\sum _{k=1}^{{N}_{k}}\left|s\left(k\right)-{\widehat{s}}_{m}\left(k\right)\right|$$
- Maximum absolute error (MAX):$$\mathrm{MAX}=\underset{k=1,\cdots ,{N}_{k}}{max}\left|s\left(k\right)-{\widehat{s}}_{m}\left(k\right)\right|$$

#### 3.4. Case Studies

#### 3.4.1. Case Study—Signal s20a1o05

#### 3.4.2. Case Study—Signal s20a2o09

#### 3.4.3. Case Study—Signal s20a3o15

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AR | autoregressive |

BBH | binary black holes |

BNS | binary neutron star |

CBC | compact binary coalescence |

CCSN | core collapse supernova |

CNN | convolutional neural network |

EGO | European Gravitational Observatory |

ICI | intersection of confidence intervals |

ISI | internal seismic isolation |

ISNR | improvement in the signal-to-noise ratio |

LIGO | Laser Interferometer Gravitational-Wave Observatory |

LPA | local polynomial approximation |

MAE | mean absolute error |

MAX | maximum absolute error |

MSE | mean squared error |

NSBH | neutron star-black hole |

PCA | principal component analysis |

PSNR | peak signal-to-noise ratio |

RICI | relative intersection of confidence intervals |

RMSE | root mean squared error |

ROF | Rudin–Osher–Fatemi |

SNR | signal-to-noise ratio |

STFT | Short-time Fourier transform |

SURE | Stein’s unbiased risk estimator |

TV | total variation |

WLS | weighted least squares |

## References

- Abbott, B.P.; Abbott, R.; Abbott, T.; Abernathy, M.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.; et al. Observation of gravitational-waves from a binary black hole merger. Phys. Rev. Lett.
**2016**, 116, 061102. [Google Scholar] [CrossRef] [PubMed] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.; et al. GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence. Phys. Rev. Lett.
**2016**, 116, 241103. [Google Scholar] [CrossRef] [PubMed] - Aasi, J.; Abbott, B.; Abbott, R.; Abbott, T.; Abernathy, M.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.; et al. Advanced LIGO. Class. Quantum Gravity
**2015**, 32, 074001. [Google Scholar] [CrossRef] - Acernese, F.; Agathos, M.; Agatsuma, K.; Aisa, D.; Allemandou, N.; Allocca, A.; Amarni, J.; Astone, P.; Balestri, G.; Ballardin, G.; et al. Advanced Virgo: A second-generation interferometric gravitational wave detector. Class. Quantum Gravity
**2014**, 32, 024001. [Google Scholar] [CrossRef] [Green Version] - Acernese, F.; Adams, T.; Agatsuma, K.; Aiello, L.; Allocca, A.; Amato, A.; Antier, S.; Arnaud, N.; Ascenzi, S.; Astone, P.; et al. Advanced Virgo Status. J. Phys. Conf. Ser.
**2020**, 1342, 012010. [Google Scholar] [CrossRef] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, C.; Adhikari, R.; Adya, V.B.; Affeldt, C.; et al. GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs. Phys. Rev. X
**2019**, 9, 031040. [Google Scholar] [CrossRef] [Green Version] - Akutsu, T.; Ando, M.; Arai, K.; Arai, Y.; Araki, S.; Araya, A.; Aritomi, N.; Asada, H.; Aso, Y.; Atsuta, S.; et al. KAGRA: 2.5 generation interferometric gravitational-wave detector. Nat. Astron.
**2019**, 3, 35–40. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.; Abernathy, M.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.; et al. Prospects for observing and localizing gravitational-wave transients with Advanced LIGO, Advanced Virgo and KAGRA. Living Rev. Relativ.
**2018**, 21, 3. [Google Scholar] [CrossRef] [Green Version] - The Virgo Collaboration. Advanced Virgo. 2020. Available online: http://public.virgo-gw.eu/advanced-virgo/ (accessed on 21 October 2020).
- LIGO’s Interferometer. Available online: https://www.ligo.caltech.edu/page/ligos-ifo (accessed on 7 September 2020).
- Blair, D.G. The Detection of Gravitational Waves; Cambridge University Press: Cambridge, MA, USA, 2005. [Google Scholar]
- Meystre, P.; Scully, M.O. Quantum Optics, Experimental Gravity, and Measurement Theory; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012; Volume 94. [Google Scholar]
- Kwee, P.; Bogan, C.; Danzmann, K.; Frede, M.; Kim, H.; King, P.; Pöld, J.; Puncken, O.; Savage, R.L.; Seifert, F.; et al. Stabilized high-power laser system for the gravitational-wave detector advanced LIGO. Opt. Express
**2012**, 20, 10617–10634. [Google Scholar] [CrossRef] [Green Version] - Mueller, C.L.; Arain, M.A.; Ciani, G.; DeRosa, R.T.; Effler, A.; Feldbaum, D.; Frolov, V.V.; Fulda, P.; Gleason, J.; Heintze, M.; et al. The advanced LIGO input optics. Rev. Sci. Instrum.
**2016**, 87, 014502. [Google Scholar] [CrossRef] [Green Version] - LIGO Technology. Available online: https://www.ligo.caltech.edu/page/ligo-technology (accessed on 7 September 2020).
- Matichard, F.; Lantz, B.; Mittleman, R.; Mason, K.; Kissel, J.; Abbott, B.; Biscans, S.; McIver, J.; Abbott, R.; Abbott, S.; et al. Seismic isolation of Advanced LIGO: Review of strategy, instrumentation and performance. Class. Quantum Gravity
**2015**, 32, 185003. [Google Scholar] [CrossRef] - Aston, S.; Barton, M.; Bell, A.; Beveridge, N.; Bland, B.; Brummitt, A.; Cagnoli, G.; Cantley, C.; Carbone, L.; Cumming, A.; et al. Update on quadruple suspension design for Advanced LIGO. Class. Quantum Gravity
**2012**, 29, 235004. [Google Scholar] [CrossRef] - Cumming, A.; Bell, A.; Barsotti, L.; Barton, M.; Cagnoli, G.; Cook, D.; Cunningham, L.; Evans, M.; Hammond, G.; Harry, G.; et al. Design and development of the advanced LIGO monolithic fused silica suspension. Class. Quantum Gravity
**2012**, 29, 035003. [Google Scholar] [CrossRef] - Harry, G.M.; Abernathy, M.R.; Becerra-Toledo, A.E.; Armandula, H.; Black, E.; Dooley, K.; Eichenfield, M.; Nwabugwu, C.; Villar, A.; Crooks, D.; et al. Titania-doped tantala/silica coatings for gravitational-wave detection. Class. Quantum Gravity
**2006**, 24, 405. [Google Scholar] [CrossRef] - Granata, M.; Saracco, E.; Morgado, N.; Cajgfinger, A.; Cagnoli, G.; Degallaix, J.; Dolique, V.; Forest, D.; Franc, J.; Michel, C.; et al. Mechanical loss in state-of-the-art amorphous optical coatings. Phys. Rev. D
**2016**, 93, 012007. [Google Scholar] [CrossRef] [Green Version] - Martynov, D.V.; Hall, E.D.; Abbott, B.P.; Abbott, R.; Abbott, T.D.; Adams, C.; Adhikari, R.; Anderson, R.A.; Anderson, S.B.; Arai, K.; et al. Sensitivity of the Advanced LIGO detectors at the beginning of gravitational-wave astronomy. Phys. Rev. D
**2016**, 93, 112004. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.; Abernathy, M.; Acernese, F.; Ackley, K.; Adamo, M.; Adams, C.; Adams, T.; Addesso, P.; et al. Characterization of transient noise in Advanced LIGO relevant to gravitational-wave signal GW150914. Class. Quantum Gravity
**2016**, 33, 134001. [Google Scholar] [CrossRef] - Effler, A.; Schofield, R.; Frolov, V.; González, G.; Kawabe, K.; Smith, J.; Birch, J.; McCarthy, R. Environmental influences on the LIGO gravitational-wave detectors during the 6th science run. Class. Quantum Gravity
**2015**, 32, 035017. [Google Scholar] [CrossRef] [Green Version] - Sathyaprakash, B.S.; Schutz, B.F. Physics, astrophysics and cosmology with gravitational-waves. Living Rev. Relativ.
**2009**, 12, 2. [Google Scholar] [CrossRef] [Green Version] - Usman, S.A.; Nitz, A.H.; Harry, I.W.; Biwer, C.M.; Brown, D.A.; Cabero, M.; Capano, C.D.; Dal Canton, T.; Dent, T.; Fairhurst, S.; et al. The PyCBC search for gravitational-waves from compact binary coalescence. Class. Quantum Gravity
**2016**, 33, 215004. [Google Scholar] [CrossRef] [Green Version] - Nitz, A.H.; Dal Canton, T.; Davis, D.; Reyes, S. Rapid detection of gravitational-waves from compact binary mergers with PyCBC Live. Phys. Rev. D
**2018**, 98, 024050. [Google Scholar] [CrossRef] [Green Version] - Bose, S.; Pai, A.; Dhurandhar, S. Detection of gravitational-waves from inspiraling, compact binaries using a network of interferometric detectors. Int. J. Mod. Phys. D
**2000**, 9, 325–329. [Google Scholar] [CrossRef] [Green Version] - Riles, K. Recent searches for continuous gravitational-waves. Mod. Phys. Lett. A
**2017**, 32, 1730035. [Google Scholar] [CrossRef] - Edwards, M.C.; Meyer, R.; Christensen, N. Bayesian parameter estimation of core collapse supernovae using gravitational-wave simulations. Inverse Probl.
**2014**, 30, 114008. [Google Scholar] [CrossRef] [Green Version] - Engels, W.J.; Frey, R.; Ott, C.D. Multivariate regression analysis of gravitational-waves from rotating core collapse. Phys. Rev. D
**2014**, 90, 124026. [Google Scholar] [CrossRef] [Green Version] - Edwards, M.C.; Meyer, R.; Christensen, N. Bayesian semiparametric power spectral density estimation with applications in gravitational-wave data analysis. Phys. Rev. D
**2015**, 92, 064011. [Google Scholar] [CrossRef] [Green Version] - Powell, J.; Gossan, S.E.; Logue, J.; Heng, I.S. Inferring the core-collapse supernova explosion mechanism with gravitational-waves. Phys. Rev. D
**2016**, 94, 123012. [Google Scholar] [CrossRef] [Green Version] - Powell, J.; Szczepanczyk, M.; Heng, I.S. Inferring the core-collapse supernova explosion mechanism with three-dimensional gravitational-wave simulations. Phys. Rev. D
**2017**, 96, 123013. [Google Scholar] [CrossRef] [Green Version] - Thrane, E.; Coughlin, M. Detecting Gravitational-Wave Transients at 5σ: A Hierarchical Approach. Phys. Rev. Lett.
**2015**, 115, 181102. [Google Scholar] [CrossRef] [Green Version] - Klimenko, S.; Vedovato, G.; Drago, M.; Salemi, F.; Tiwari, V.; Prodi, G.A.; Lazzaro, C.; Ackley, K.; Tiwari, S.; Da Silva, C.F.; et al. Method for detection and reconstruction of gravitational-wave transients with networks of advanced detectors. Phys. Rev. D
**2016**, 93, 042004. [Google Scholar] [CrossRef] [Green Version] - Littenberg, T.B.; Kanner, J.B.; Cornish, N.J.; Millhouse, M. Enabling high confidence detections of gravitational-wave bursts. Phys. Rev. D
**2016**, 94, 044050. [Google Scholar] [CrossRef] [Green Version] - Kanner, J.B.; Littenberg, T.B.; Cornish, N.; Millhouse, M.; Xhakaj, E.; Salemi, F.; Drago, M.; Vedovato, G.; Klimenko, S. Leveraging waveform complexity for confident detection of gravitational-waves. Phys. Rev. D
**2016**, 93, 022002. [Google Scholar] [CrossRef] [Green Version] - Lynch, R.; Vitale, S.; Essick, R.; Katsavounidis, E.; Robinet, F. Information-theoretic approach to the gravitational-wave burst detection problem. Phys. Rev. D
**2017**, 95, 104046. [Google Scholar] [CrossRef] [Green Version] - Cuoco, E.; Powell, J.; Cavaglià, M.; Ackley, K.; Bejger, M.; Chatterjee, C.; Coughlin, M.; Coughlin, S.; Easter, P.; Essick, R.; et al. Enhancing Gravitational-Wave Science with Machine Learning. arXiv
**2020**, arXiv:2005.03745. [Google Scholar] - Torres-Forné, A.; Marquina, A.; Font, J.A.; Ibáñez, J.M. Denoising of gravitational-wave signals via dictionary learning algorithms. Phys. Rev. D
**2016**, 94, 124040. [Google Scholar] [CrossRef] [Green Version] - Ormiston, R.; Nguyen, T.; Coughlin, M.; Adhikari, R.X.; Katsavounidis, E. Noise reduction in gravitational-wave data via deep learning. Phys. Rev. Res.
**2020**, 2, 033066. [Google Scholar] [CrossRef] - Wei, W.; Huerta, E. Gravitational-wave denoising of binary black hole mergers with deep learning. Phys. Lett. B
**2020**, 800, 135081. [Google Scholar] [CrossRef] - George, D.; Huerta, E. Deep Learning for real-time gravitational-wave detection and parameter estimation: Results with Advanced LIGO data. Phys. Lett. B
**2018**, 778, 64–70. [Google Scholar] [CrossRef] - Kim, K.; Harry, I.W.; Hodge, K.A.; Kim, Y.M.; Lee, C.H.; Lee, H.K.; Oh, J.J.; Oh, S.H.; Son, E.J. Application of artificial neural network to search for gravitational-wave signals associated with short gamma-ray bursts. Class. Quantum Gravity
**2015**, 32, 245002. [Google Scholar] [CrossRef] [Green Version] - Gebhard, T.; Kilbertus, N.; Parascandolo, G.; Harry, I.; Schölkopf, B. CONVWAVE: Searching for gravitational-waves with fully convolutional neural nets. In Proceedings of the Workshop on Deep Learning for Physical Sciences (DLPS) at the 31st Conference on Neural Information Processing Systems (NIPS), Long Beach, CA, USA, 4–7 December 2017; pp. 1–6. [Google Scholar]
- Gabbard, H.; Williams, M.; Hayes, F.; Messenger, C. Matching Matched Filtering with Deep Networks for Gravitational-Wave Astronomy. Phys. Rev. Lett.
**2018**, 120, 141103. [Google Scholar] [CrossRef] [Green Version] - Kim, K.; Li, T.G.F.; Lo, R.K.L.; Sachdev, S.; Yuen, R.S.H. Ranking candidate signals with machine learning in low-latency searches for gravitational-waves from compact binary mergers. Phys. Rev. D
**2020**, 101, 083006. [Google Scholar] [CrossRef] [Green Version] - George, D.; Shen, H.; Huerta, E.A. Classification and unsupervised clustering of LIGO data with Deep Transfer Learning. Phys. Rev. D
**2018**, 97, 101501. [Google Scholar] [CrossRef] [Green Version] - Razzano, M.; Cuoco, E. Image-based deep learning for classification of noise transients in gravitational-wave detectors. Class. Quantum Gravity
**2018**, 35, 095016. [Google Scholar] [CrossRef] [Green Version] - Powell, J.; Trifirò, D.; Cuoco, E.; Heng, I.S.; Cavaglià, M. Classification methods for noise transients in advanced gravitational-wave detectors. Class. Quantum Gravity
**2015**, 32, 215012. [Google Scholar] [CrossRef] - Mukund, N.; Abraham, S.; Kandhasamy, S.; Mitra, S.; Philip, N.S. Transient classification in LIGO data using difference boosting neural network. Phys. Rev. D
**2017**, 95, 104059. [Google Scholar] [CrossRef] [Green Version] - Powell, J.; Torres-Forné, A.; Lynch, R.; Trifirò, D.; Cuoco, E.; Cavaglià, M.; Heng, I.S.; Font, J.A. Classification methods for noise transients in advanced gravitational-wave detectors II: Performance tests on Advanced LIGO data. Class. Quantum Gravity
**2017**, 34, 034002. [Google Scholar] [CrossRef] [Green Version] - Huerta, E.A.; Allen, G.; Andreoni, I.; Antelis, J.M.; Bachelet, E.; Berriman, G.B.; Bianco, F.B.; Biswas, R.; Kind, M.C.; Chard, K.; et al. Enabling real-time multi-messenger astrophysics discoveries with deep learning. Nat. Rev. Phys.
**2019**, 1, 600–608. [Google Scholar] [CrossRef] [Green Version] - Cavaglia, M.; Gaudio, S.; Hansen, T.; Staats, K.; Szczepańczyk, M.; Zanolin, M. Improving the background of gravitational-wave searches for core collapse supernovae: A machine learning approach. Mach. Learn. Sci. Technol.
**2020**, 1, 015005. [Google Scholar] [CrossRef] - Torres, A.; Marquina, A.; Font, J.A.; Ibáñez, J.M. Total-variation-based methods for gravitational-wave denoising. Phys. Rev. D
**2014**, 90, 084029. [Google Scholar] [CrossRef] [Green Version] - Rudin, L.I.; Osher, S.; Fatemi, E. Nonlinear total variation based noise removal algorithms. Physica
**1992**, 60, 259–268. [Google Scholar] [CrossRef] - Chambolle, A. An algorithm for total variation minimization and applications. J. Math. Imaging Vis.
**2004**, 20, 89–97. [Google Scholar] [CrossRef] - Goldstein, T.; Osher, S. The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci.
**2009**, 2, 323–343. [Google Scholar] [CrossRef] - Lee, K.J.; Lee, B. Sequential Total Variation Denoising for the Extraction of Fetal ECG from Single-Channel Maternal Abdominal ECG. Sensors
**2016**, 16, 1020. [Google Scholar] [CrossRef] [Green Version] - Xiong, N.; Liu, R.W.; Liang, M.; Wu, D.; Liu, Z.; Wu, H. Effective Alternating Direction Optimization Methods for Sparsity-Constrained Blind Image Deblurring. Sensors
**2017**, 17, 174. [Google Scholar] [CrossRef] [Green Version] - Boutemedjet, A.; Deng, C.; Zhao, B. Edge-Aware Unidirectional Total Variation Model for Stripe Non-Uniformity Correction. Sensors
**2018**, 18, 1164. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Du, Q.; Xu, H.; Ma, Y.; Huang, J.; Fan, F. Fusing Infrared and Visible Images of Different Resolutions via Total Variation Model. Sensors
**2018**, 18, 3827. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, B.; Abascal, J.F.P.J.; Soleimani, M. Electrical Resistance Tomography for Visualization of Moving Objects Using a Spatiotemporal Total Variation Regularization Algorithm. Sensors
**2018**, 18, 1704. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Torres-Forné, A.; Cuoco, E.; Marquina, A.; Font, J.A.; Ibáñez, J.M. Total-variation methods for gravitational-wave denoising: Performance tests on Advanced LIGO data. Phys. Rev. D
**2018**, 98, 084013. [Google Scholar] [CrossRef] [Green Version] - Katkovnik, V. A new method for varying adaptive bandwidth selection. IEEE Trans. Signal Process.
**1999**, 47, 2567–2571. [Google Scholar] [CrossRef] [Green Version] - Katkovnik, V.; Egiazarian, K.; Astola, J. Adaptive window size image de-noising based on intersection of confidence intervals (ICI) rule. J. Math. Imaging Vis.
**2002**, 16, 223–235. [Google Scholar] [CrossRef] - Katkovnik, V.; Egiazarian, K.; Astola, J. Adaptive Varying Scale Methods in Image Processing; TTY Monistamo: Tampere, Finland, 2003; Volume 19. [Google Scholar]
- Katkovnik, V. Multiresolution local polynomial regression: A new approach to pointwise spatial adaptation. Digit. Signal Process.
**2005**, 15, 73–116. [Google Scholar] [CrossRef] [Green Version] - Katkovnik, V.; Egiazarian, K.; Astola, J. Local Approximation Techniques in Signal and Image Processing; The International Society for Optical Engineering: Washington, DC, USA, 2006. [Google Scholar]
- Katkovnik, V.; Shmulevich, I. Kernel density estimation with adaptive varying window size. Pattern Recognit. Lett.
**2002**, 23, 1641–1648. [Google Scholar] [CrossRef] - Goldenshluger, A.; Nemirovski, A. On spatially adaptive estimation of nonparametric regression. Math. Meth. Stat.
**1997**, 6, 135–170. [Google Scholar] - Sucic, V.; Lerga, J.; Vrankic, M. Adaptive filter support selection for signal denoising based on the improved ICI rule. Digit. Signal Process.
**2013**, 23, 65–74. [Google Scholar] [CrossRef] - Stankovic, L. Performance analysis of the adaptive algorithm for bias-to-variance tradeoff. IEEE Trans. Signal Process.
**2004**, 52, 1228–1234. [Google Scholar] [CrossRef] - Lerga, J.; Vrankic, M.; Sucic, V. A Signal Denoising Method Based on the Improved ICI Rule. IEEE Signal Process. Lett.
**2008**, 15, 601–604. [Google Scholar] [CrossRef] - Lerga, J.; Sucic, V.; Sersic, D. Performance analysis of the LPA-RICI denoising method. In Proceedings of the 6th International Symposium on Image and Signal Processing and Analysis, Salzburg, Austria, 16–18 September 2009; pp. 28–33. [Google Scholar] [CrossRef]
- Dimmelmeier, H.; Ott, C.D.; Marek, A.; Janka, H.T. Gravitational-wave burst signal from core collapse of rotating stars. Phys. Rev. D
**2008**, 78, 064056. [Google Scholar] [CrossRef] [Green Version] - Cuoco, E.; Calamai, G.; Fabbroni, L.; Losurdo, G.; Mazzoni, M.; Stanga, R.; Vetrano, F. On-line power spectra identification and whitening for the noise in interferometric gravitational-wave detectors. Class. Quantum Gravity
**2001**, 18, 1727–1751. [Google Scholar] [CrossRef] [Green Version] - Cuoco, E.; Losurdo, G.; Calamai, G.; Fabbroni, L.; Mazzoni, M.; Stanga, R.; Guidi, G.; Vetrano, F. Noise parametric identification and whitening for LIGO 40-m interferometer data. Phys. Rev. D
**2001**, 64, 122002. [Google Scholar] [CrossRef] [Green Version] - Cuoco, E.; Cella, G.; Guidi, G.M. Whitening of non-stationary noise from gravitational-wave detectors. Class. Quantum Gravity
**2004**, 21, S801–S806. [Google Scholar] [CrossRef] - Kay, S.M. Modern Spectral Estimation: Theory and Application; Prentice-Hall: Englewood Cliffs, NJ, USA, 1988. [Google Scholar]
- Acernese, F.; Amico, P.; Alshourbagy, M.; Antonucci, F.; Aoudia, S.; Astone, P.; Avino, S.; Babusci, D.; Ballardin, G.; Barone, F.; et al. Gravitational-waves by gamma-ray bursts and the Virgo detector: The case of GRB 050915a. Class. Quantum Gravity
**2007**, 24, S671. [Google Scholar] [CrossRef] - Chambolle, A.; Caselles, V.; Cremers, D.; Novaga, M.; Pock, T. An introduction to total variation for image analysis. Theor. Found. Numer. Methods Sparse Recovery
**2010**, 9, 227. [Google Scholar] - Chambolle, A.; Pock, T. A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis.
**2011**, 40, 120–145. [Google Scholar] [CrossRef] [Green Version] - Mousavi, S.M.; Langston, C.A. Adaptive noise estimation and suppression for improving microseismic event detection. J. Appl. Geophy.
**2016**, 132, 116–124. [Google Scholar] [CrossRef] - Donoho, D.L.; Johnstone, J.M. Ideal spatial adaptation by wavelet shrinkage. Biometrika
**1994**, 81, 425–455. [Google Scholar] [CrossRef] - Donoho, D.L.; Johnstone, I.M. Minimax estimation via wavelet shrinkage. Ann. Stat.
**1998**, 26, 879–921. [Google Scholar] [CrossRef]

**Figure 1.**CCSN signal s20a1o05 at a distance of 5 kpc: (

**a**) template signal; (

**b**) noisy signal (SNR = 3.9 dB).

**Figure 2.**Results of applying the LPA-RICI denoising method to the noisy CCSN signal s20a1o05 at a distance of 5 kpc (SNR = 3.9 dB): (

**a**) template and LPA-RICI denoised signal (n = 0, Γ = 5.5, R

_{c}= 1); (

**b**) LPA-RICI estimation error (n = 0, Γ = 5.5, R

_{c}= 1); (

**c**) template and LPA-RICI denoised signal (n = 1, Γ = 7, R

_{c}= 1); (

**d**) LPA-RICI estimation error (n = 1, Γ = 7, R

_{c}= 1); (

**e**) template and LPA-RICI denoised signal (n = 2, Γ = 11, R

_{c}= 1); (

**f**) LPA-RICI estimation error (n = 2, Γ = 11, R

_{c}= 1).

**Figure 3.**CCSN signal s20a2o09 at a distance of 5 kpc: (

**a**) template signal; (

**b**) noisy signal (SNR = −4.54 dB).

**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).

**Figure 5.**CCSNsignal s20a3o15 at a distance of 5 kpc: (

**a**) template signal; (

**b**) noisy signal (SNR = −2.18 dB).

**Figure 6.**Results of applying the LPA-RICI denoising method to the noisy CCSN signal s20a3o15 at a distance of 5 kpc (SNR = −2.18 dB): (

**a**) template and LPA-RICI denoised signal (n = 0, Γ = 10.75, R

_{c}= 1); (

**b**) LPA-RICI estimation error (n = 0, Γ = 10.75, R

_{c}= 1); (

**c**) template and LPA-RICI denoised signal (n = 1, Γ = 16, R

_{c}= 1); (

**d**) LPA-RICI estimation error (n = 1, Γ = 16, R

_{c}= 1); (

**e**) template and LPA-RICI denoised signal (n = 2, Γ = 20, R

_{c}= 1); (

**f**) LPA-RICI estimation error (n = 2, Γ = 20, R

_{c}= 1).

**Table 1.**Denoising results for the CCSN signal s20a1o05 at a distance of 5 kpc (SNR = 3.9 dB). The best performance indices are marked in bold.

Perform. Index | LPA-RICI $\mathit{n}=0$ $\mathbf{\Gamma}=5.5,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=1$ $\mathbf{\Gamma}=7,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=2$ $\mathbf{\Gamma}=11,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-ICI $\mathbf{\Gamma}=0.75$ | TV $\mathit{\mu}=0.49$ | Neigh STFT | sym5 Wavelet SURE, Level 6 | db13 Wavelet SURE, Level 5 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|---|---|---|

ISNR (db) | 11.1639 | 11.8899 | 12.6307 | 8.9827 | 7.3205 | 5.7691 | 10.1766 | 12.1502 | 9.7551 |

PSNR (db) | 30.3510 | 31.0770 | 31.8178 | 28.1813 | 26.5192 | 24.9562 | 29.3636 | 31.3373 | 28.9421 |

RMSE | 0.0304 | 0.0279 | 0.0257 | 0.0390 | 0.0472 | 0.0565 | 0.0340 | 0.0271 | 0.0357 |

MAE | 0.0225 | 0.0212 | 0.0195 | 0.0258 | 0.0310 | 0.0377 | 0.0246 | 0.0193 | 0.0255 |

MAX | 0.1642 | 0.1293 | 0.1100 | 0.2377 | 0.3011 | 0.2972 | 0.3372 | 0.1719 | 0.4001 |

**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.

Perform. Index | LPA-RICI $\mathit{n}=0$ $\mathbf{\Gamma}=11.25,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=1$ $\mathbf{\Gamma}=13,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=2$ $\mathbf{\Gamma}=24,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-ICI $\mathbf{\Gamma}=1.25$ | TV $\mathit{\mu}=0.26$ | Neigh STFT | sym4 Wavelet SURE, Level 6 | db13 Wavelet SURE, Level 5 | coif4 Wavelet SURE, Level 5 |
---|---|---|---|---|---|---|---|---|---|

ISNR (db) | 13.3754 | 13.4002 | 14.0397 | 11.6172 | 9.4346 | 10.2554 | 11.4397 | 12.6607 | 10.4296 |

PSNR (db) | 26.5485 | 26.5733 | 27.2127 | 24.7953 | 22.6127 | 23.4284 | 24.6127 | 25.8338 | 23.6026 |

RMSE | 0.0471 | 0.0469 | 0.0436 | 0.0576 | 0.0740 | 0.0674 | 0.0588 | 0.0511 | 0.0660 |

MAE | 0.0364 | 0.0357 | 0.0326 | 0.0388 | 0.0439 | 0.0419 | 0.0379 | 0.0369 | 0.0421 |

MAX | 0.2309 | 0.2200 | 0.2200 | 0.4277 | 0.4876 | 0.5204 | 0.7805 | 0.3323 | 0.8102 |

**Table 3.**Denoising results for the CCSN signal s20a1o05 at a distance of 20 kpc (SNR = −8.13 dB). The best performance indices are marked in bold.

Perform. Index | LPA-RICI $\mathit{n}=0$ $\mathbf{\Gamma}=20.5,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=1$ $\mathbf{\Gamma}=20,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=2$ $\mathbf{\Gamma}=28,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-ICI $\mathbf{\Gamma}=1.5$ | TV $\mathit{\mu}=0.31$ | Neigh STFT | sym3 Wavelet SURE, Level 7 | db3 Wavelet SURE, Level 7 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|---|---|---|

ISNR (db) | 15.5200 | 15.3378 | 15.6755 | 14.2550 | 12.5169 | 13.8119 | 13.1240 | 13.1240 | 11.4983 |

PSNR (db) | 22.6734 | 22.4912 | 22.8289 | 21.4124 | 19.6743 | 20.9653 | 20.2774 | 20.2774 | 18.6518 |

RMSE | 0.0735 | 0.0751 | 0.0722 | 0.0850 | 0.1038 | 0.0895 | 0.0969 | 0.0969 | 0.1168 |

MAE | 0.0537 | 0.0570 | 0.0551 | 0.0547 | 0.0618 | 0.0471 | 0.0644 | 0.0644 | 0.0775 |

MAX | 0.4227 | 0.4399 | 0.4399 | 0.6901 | 0.6482 | 1.0478 | 1.4032 | 1.4032 | 1.5769 |

**Table 4.**Relative performance improvement of the LPA-RICI-based ($n=2,\mathsf{\Gamma}=11,{R}_{c}=1$) denoising over other tested methods, for the CCSN signal s20a1o05 at a distance of 5 kpc (SNR = 3.9 dB).

Perform. Index | LPA-ICI $\mathbf{\Gamma}=0.75$ | TV $\mathit{\mu}=0.49$ | Neigh STFT | sym5 Wavelet SURE, Level 6 | db13 Wavelet SURE, Level 5 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|

ISNR | 40.61% | 72.54% | 118.94% | 24.12% | 3.95% | 29.48% |

PSNR | 12.90% | 19.98% | 27.49% | 8.36% | 1.53% | 9.94% |

RMSE | 34.10% | 45.55% | 54.51% | 24.41% | 5.17% | 28.01% |

MAE | 24.42% | 37.10% | 48.28% | 20.73% | −1.04% | 23.53% |

MAX | 53.72% | 63.47% | 62.99% | 67.38% | 36.01% | 72.51% |

**Table 5.**Relative performance improvement of the LPA-RICI-based ($n=2,\mathsf{\Gamma}=24,{R}_{c}=1$) denoising over other tested methods, for the CCSN signal s20a1o05 at a distance of 10 kpc (SNR = −2.11 dB).

Perform. Index | LPA-ICI $\mathbf{\Gamma}=1.25$ | TV $\mathit{\mu}=0.26$ | Neigh STFT | sym4 Wavelet SURE, Level 6 | db13 Wavelet SURE, Level 5 | coif4 Wavelet SURE, Level 5 |
---|---|---|---|---|---|---|

ISNR | 20.85% | 48.81% | 36.90% | 22.73% | 10.89% | 34.61% |

PSNR | 9.75% | 20.34% | 16.15% | 10.56% | 5.34% | 15.30% |

RMSE | 24.31% | 41.08% | 35.31% | 25.85% | 14.68% | 33.94% |

MAE | 15.98% | 25.74% | 22.20% | 13.98% | 11.65% | 22.57% |

MAX | 48.56% | 54.88% | 57.72% | 71.81% | 33.79% | 72.85% |

**Table 6.**Relative performance improvement of the LPA-RICI-based ($n=2,\mathsf{\Gamma}=28,{R}_{c}=1$) denoising over other tested methods, for the CCSN signal s20a1o05 at a distance of 20 kpc (SNR = −8.13 dB).

Perform. Index | LPA-ICI $\mathbf{\Gamma}=1.5$ | TV $\mathit{\mu}=0.31$ | Neigh STFT | sym3 Wavelet SURE, Level 7 | db3 Wavelet SURE, Level 7 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|

ISNR | 9.96% | 25.23% | 13.49% | 19.44% | 19.44% | 36.33% |

PSNR | 6.62% | 16.03% | 8.89% | 12.58% | 12.58% | 22.40% |

RMSE | 15.06% | 30.44% | 19.33% | 25.49% | 25.49% | 38.18% |

MAE | −0.73% | 10.84% | −16.99% | 14.44% | 14.44% | 28.90% |

MAX | 36.26% | 32.14% | 58.02% | 68.65% | 68.65% | 72.10% |

**Table 7.**Algorithm execution times of the tested denoising methods, for the CCSN signal s20a1o05 at distances of 5, 10, and 20 kpc.

Execution Time (s) | |||||||||
---|---|---|---|---|---|---|---|---|---|

Distance(kpc) | LPA-RICI$\mathit{n}=\mathbf{0}$ | LPA-RICI$\mathit{n}=\mathbf{1}$ | LPA-RICI$\mathit{n}=\mathbf{2}$ | LPA-ICI | TV | NeighSTFT | SymletWavelet | DaubechiesWavelet | CoifletWavelet |

5 | 0.3245 | 0.4239 | 0.7288 | 3.0619 | 0.0154 | 0.7661 | 0.0061 | 0.0059 | 0.0046 |

10 | 0.4839 | 0.5923 | 1.3046 | 4.6105 | 0.0162 | 0.7967 | 0.0073 | 0.0086 | 0.0055 |

20 | 0.8352 | 0.8296 | 1.3178 | 7.3022 | 0.0160 | 0.7457 | 0.0058 | 0.0059 | 0.0058 |

**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.

Perform. Index | LPA-RICI $\mathit{n}=0$ $\mathbf{\Gamma}=9,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=1$ $\mathbf{\Gamma}=13,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=2$ $\mathbf{\Gamma}=16,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-ICI $\mathbf{\Gamma}=1.25$ | TV $\mathit{\mu}=0.24$ | Neigh STFT | sym4 Wavelet SURE, Level 6 | db6 Wavelet SURE, Level 6 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|---|---|---|

ISNR (db) | 12.1897 | 11.8699 | 12.2523 | 11.0513 | 12.4081 | 10.4299 | 11.2266 | 11.6597 | 12.2478 |

PSNR (db) | 21.1651 | 20.8453 | 21.2277 | 20.0267 | 21.3835 | 19.4053 | 20.2021 | 20.6351 | 21.2232 |

RMSE | 0.0596 | 0.0619 | 0.0592 | 0.0680 | 0.0582 | 0.0730 | 0.0666 | 0.0634 | 0.0592 |

MAE | 0.0443 | 0.0428 | 0.0424 | 0.0416 | 0.0412 | 0.0438 | 0.0470 | 0.0439 | 0.0405 |

MAX | 0.3201 | 0.3030 | 0.2913 | 0.3627 | 0.2985 | 0.5463 | 0.4402 | 0.3855 | 0.2391 |

**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.

Perform. Index | LPA-RICI $\mathit{n}=0$ $\mathbf{\Gamma}=13.5,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=1$ $\mathbf{\Gamma}=20,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=2$ $\mathbf{\Gamma}=28,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-ICI $\mathbf{\Gamma}=1.5$ | TV $\mathit{\mu}=0.27$ | Neigh STFT | sym4 Wavelet SURE, Level 6 | db13 Wavelet SURE, Level 5 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|---|---|---|

ISNR (db) | 13.7334 | 13.5924 | 14.2997 | 13.1394 | 13.9226 | 7.8194 | 11.1812 | 12.0025 | 12.5731 |

PSNR (db) | 17.1611 | 17.0201 | 17.7275 | 16.5671 | 17.3504 | 11.2472 | 14.6089 | 15.4303 | 16.0009 |

RMSE | 0.0946 | 0.0961 | 0.0886 | 0.1012 | 0.0925 | 0.1868 | 0.1269 | 0.1154 | 0.1081 |

MAE | 0.0704 | 0.0667 | 0.0636 | 0.0581 | 0.0595 | 0.0802 | 0.0887 | 0.0870 | 0.0735 |

MAX | 0.5082 | 0.5237 | 0.4489 | 0.5831 | 0.6238 | 2.3191 | 0.8801 | 0.6794 | 0.6437 |

**Table 10.**Denoising results for the CCSN signal s20a2o09 at a distance of 20 kpc (SNR = −15.98 dB). The best performance indices are marked in bold.

Perform. Index | LPA-RICI $\mathit{n}=0$ $\mathbf{\Gamma}=5,$ ${\mathit{R}}_{\mathit{c}}=0.9$ | LPA-RICI $\mathit{n}=1$ $\mathbf{\Gamma}=26,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=2$ $\mathbf{\Gamma}=30,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-ICI $\mathbf{\Gamma}=1.5$ | TV $\mathit{\mu}=0.28$ | Neigh STFT | sym4 Wavelet SURE, Level 6 | db13 Wavelet SURE, Level 5 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|---|---|---|

ISNR (db) | 17.2693 | 15.9216 | 15.8755 | 16.1431 | 17.3306 | 8.8306 | 13.2998 | 12.4740 | 13.8150 |

PSNR (db) | 14.8028 | 13.4551 | 13.4090 | 13.6766 | 14.8641 | 6.3641 | 10.8333 | 10.0075 | 11.3485 |

RMSE | 0.1241 | 0.1449 | 0.1456 | 0.1412 | 0.1232 | 0.3277 | 0.1959 | 0.2155 | 0.1846 |

MAE | 0.0698 | 0.1012 | 0.1084 | 0.0769 | 0.0759 | 0.1072 | 0.1145 | 0.1572 | 0.1059 |

MAX | 0.6685 | 0.9006 | 0.9006 | 0.9036 | 0.7929 | 4.0053 | 3.0110 | 1.3596 | 2.7521 |

**Table 11.**Relative performance improvement of the LPA-RICI-based ($n=2,\mathsf{\Gamma}=16,{R}_{c}=1$) denoising over other tested methods, for the CCSN signal s20a2o09 at a distance of 5 kpc (SNR = −4.54 dB).

Perform. Index | LPA-ICI $\mathbf{\Gamma}=1.25$ | TV $\mathit{\mu}=0.24$ | Neigh STFT | sym4 Wavelet SURE, Level 6 | db6 Wavelet SURE, Level 6 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|

ISNR | 10.87% | −1.26% | 17.47% | 9.14% | 5.08% | 0.04% |

PSNR | 6.00% | −0.73% | 9.39% | 5.08% | 2.87% | 0.02% |

RMSE | 12.94% | −1.72% | 18.90% | 11.11% | 6.62% | 0.00% |

MAE | −1.92% | −2.91% | 3.20% | 9.79% | 3.42% | −4.69% |

MAX | 19.69% | 2.41% | 46.68% | 33.83% | 24.44% | −21.83% |

**Table 12.**Relative performance improvement of the LPA-RICI-based ($n=2,\mathsf{\Gamma}=28,{R}_{c}=1$) denoising over other tested methods, for the CCSN signal s20a2o09 at a distance of 10 kpc (SNR = −10.09 dB).

Perform. Index | LPA-ICI $\mathbf{\Gamma}=1.5$ | TV $\mathit{\mu}=0.27$ | Neigh STFT | sym4 Wavelet SURE, Level 6 | db13 Wavelet SURE, Level 5 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|

ISNR | 8.83% | 2.71% | 82.87% | 27.89% | 19.14% | 13.73% |

PSNR | 7.00% | 2.17% | 57.62% | 21.35% | 14.89% | 10.79% |

RMSE | 12.45% | 4.22% | 52.57% | 30.18% | 23.22% | 18.04% |

MAE | −9.47% | −6.89% | 20.70% | 28.30% | 26.90% | 13.47% |

MAX | 23.01% | 28.04% | 80.64% | 48.99% | 33.93% | 30.26% |

**Table 13.**Relative performance improvement of the LPA-RICI-based ($n=0,\mathsf{\Gamma}=5,{R}_{c}=0.9$) denoising over other tested methods, for the CCSN signal s20a2o09 at a distance of 20 kpc (SNR = −15.98 dB).

Perform. Index | LPA-ICI $\mathbf{\Gamma}=1.5$ | TV $\mathit{\mu}=0.28$ | Neigh STFT | sym4 Wavelet SURE, Level 6 | db13 Wavelet SURE, Level 5 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|

ISNR | 6.98% | −0.35% | 95.56% | 29.85% | 38.44% | 25.00% |

PSNR | 8.23% | −0.41% | 132.60% | 36.64% | 47.92% | 30.44% |

RMSE | 12.11% | −0.73% | 62.13% | 36.65% | 42.41% | 32.77% |

MAE | 9.23% | 8.04% | 34.89% | 39.04% | 55.60% | 34.09% |

MAX | 26.02% | 15.69% | 83.31% | 77.80% | 50.83% | 75.71% |

**Table 14.**Algorithm execution times of the tested denoising methods, for the CCSN signal s20a2o09 at distances of 5, 10, and 20 kpc.

Execution Time (s) | |||||||||
---|---|---|---|---|---|---|---|---|---|

Distance(kpc) | LPA-RICI$\mathit{n}=\mathbf{0}$ | LPA-RICI$\mathit{n}=\mathbf{1}$ | LPA-RICI$\mathit{n}=\mathbf{2}$ | LPA-ICI | TV | NeighSTFT | SymletWavelet | DaubechiesWavelet | CoifletWavelet |

5 | 0.3427 | 0.5322 | 0.6698 | 4.7678 | 0.0231 | 0.7854 | 0.0054 | 0.0054 | 0.0058 |

10 | 0.4770 | 0.7887 | 1.2294 | 9.5808 | 0.0218 | 0.7553 | 0.0054 | 0.0071 | 0.0059 |

20 | 0.1670 | 1.0763 | 1.3092 | 12.6891 | 0.0231 | 0.7768 | 0.0042 | 0.0071 | 0.0058 |

**Table 15.**Denoising results for the CCSN signal s20a3o15 at a distance of 5 kpc (SNR = −2.18 dB). The best performance indices are marked in bold.

Perform. Index | LPA-RICI $\mathit{n}=0$ $\mathbf{\Gamma}=10.75,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=1$ $\mathbf{\Gamma}=16,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=2$ $\mathbf{\Gamma}=20,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-ICI $\mathbf{\Gamma}=1$ | TV $\mathit{\mu}=0.38$ | Neigh STFT | sym4 Wavelet SURE, Level 6 | db25 Wavelet SURE, Level 4 | coif4 Wavelet SURE, Level 5 |
---|---|---|---|---|---|---|---|---|---|

ISNR (db) | 12.9255 | 12.9800 | 12.8344 | 10.7381 | 9.6876 | 7.2057 | 10.5605 | 11.8525 | 10.0418 |

PSNR (db) | 22.4275 | 22.4820 | 22.3364 | 20.2401 | 19.1895 | 16.7077 | 20.0624 | 21.3545 | 19.5438 |

RMSE | 0.0756 | 0.0751 | 0.0764 | 0.0973 | 0.1098 | 0.1461 | 0.0993 | 0.0856 | 0.1054 |

MAE | 0.0590 | 0.0557 | 0.0546 | 0.0641 | 0.0774 | 0.0877 | 0.0679 | 0.0676 | 0.0671 |

MAX | 0.3054 | 0.4222 | 0.3502 | 0.4744 | 0.4466 | 0.7650 | 1.0834 | 0.3646 | 1.2784 |

**Table 16.**Denoising results for the CCSN signal s20a3o15 at a distance of 10 kpc (SNR = −8.17 dB). The best performance indices are marked in bold.

Perform. Index | LPA-RICI $\mathit{n}=0$ $\mathbf{\Gamma}=17.5,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=1$ $\mathbf{\Gamma}=20,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=2$ $\mathbf{\Gamma}=26,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-ICI $\mathbf{\Gamma}=1.25$ | TV $\mathit{\mu}=0.3$ | Neigh STFT | sym8 Wavelet SURE, Level 5 | db4 Wavelet SURE, Level 6 | coif4 Wavelet SURE, Level 5 |
---|---|---|---|---|---|---|---|---|---|

ISNR (db) | 15.6555 | 15.4472 | 14.9058 | 12.9291 | 11.9438 | 10.3813 | 11.3717 | 12.6203 | 10.7677 |

PSNR (db) | 19.1655 | 18.9571 | 18.4157 | 16.4390 | 15.4537 | 13.8912 | 14.8816 | 16.1303 | 14.2776 |

RMSE | 0.1101 | 0.1128 | 0.1200 | 0.1507 | 0.1688 | 0.2020 | 0.1803 | 0.1561 | 0.1932 |

MAE | 0.0845 | 0.0850 | 0.0890 | 0.1002 | 0.1123 | 0.1223 | 0.1102 | 0.1044 | 0.1194 |

MAX | 0.6060 | 0.6680 | 0.6680 | 0.9034 | 0.6923 | 1.2354 | 2.1975 | 0.9295 | 2.3395 |

**Table 17.**Denoising results for the CCSN signal s20a3o15 at a distance of 20 kpc (SNR = −14.19 dB). The best performance indices are marked in bold.

Perform. Index | LPA-RICI $\mathit{n}=0$ $\mathbf{\Gamma}=22,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=1$ $\mathbf{\Gamma}=25,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-RICI $\mathit{n}=2$ $\mathbf{\Gamma}=44,$ ${\mathit{R}}_{\mathit{c}}=1$ | LPA-ICI $\mathbf{\Gamma}=1.5$ | TV $\mathit{\mu}=0.29$ | Neigh STFT | sym5 Wavelet SURE, Level 6 | db6 Wavelet SURE, Level 6 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|---|---|---|

ISNR (db) | 18.0346 | 17.6215 | 17.6932 | 15.4138 | 15.7267 | 9.0143 | 12.0204 | 12.6513 | 11.3468 |

PSNR (db) | 15.5324 | 15.1193 | 15.1910 | 12.9116 | 13.2245 | 6.5121 | 9.5182 | 10.1491 | 8.8446 |

RMSE | 0.1673 | 0.1754 | 0.1740 | 0.2262 | 0.2182 | 0.4725 | 0.3343 | 0.3108 | 0.3612 |

MAE | 0.1299 | 0.1335 | 0.1284 | 0.1415 | 0.1411 | 0.2302 | 0.2236 | 0.1835 | 0.2325 |

MAX | 0.7221 | 1.3360 | 1.3360 | 1.4224 | 0.9131 | 3.1182 | 3.8350 | 3.7716 | 4.7470 |

**Table 18.**Relative performance improvement of the LPA-RICI-based ($n=1,\mathsf{\Gamma}=16,{R}_{c}=1$) denoising over other tested methods, for the CCSN signal s20a3o15 at a distance of 5 kpc (SNR = −2.18 dB).

Perform. Index | LPA-ICI $\mathbf{\Gamma}=1$ | TV $\mathit{\mu}=0.38$ | Neigh STFT | sym4 Wavelet SURE, Level 6 | db25 Wavelet SURE, Level 4 | coif4 Wavelet SURE, Level 5 |
---|---|---|---|---|---|---|

ISNR | 20.88% | 33.99% | 80.14% | 22.91% | 9.51% | 29.26% |

PSNR | 11.08% | 17.16% | 34.56% | 12.06% | 5.28% | 15.03% |

RMSE | 22.82% | 31.60% | 48.60% | 24.37% | 12.27% | 28.75% |

MAE | 13.10% | 28.04% | 36.49% | 17.97% | 17.60% | 16.99% |

MAX | 11.00% | 5.46% | 44.81% | 61.03% | −15.80% | 66.97% |

**Table 19.**Relative performance improvement of the LPA-RICI-based ($n=0,\mathsf{\Gamma}=17.5,{R}_{c}=1$) denoising over other tested methods, for the CCSN signal s20a3o15 at a distance of 10 kpc (SNR = −8.17 dB).

Perform. Index | LPA-ICI $\mathbf{\Gamma}=1.25$ | TV $\mathit{\mu}=0.3$ | Neigh STFT | sym8 Wavelet SURE, Level 5 | db4 Wavelet SURE, Level 6 | coif4 Wavelet SURE, Level 5 |
---|---|---|---|---|---|---|

ISNR | 21.09% | 31.08% | 50.80% | 37.67% | 24.05% | 45.39% |

PSNR | 16.59% | 24.02% | 37.97% | 28.79% | 18.82% | 34.23% |

RMSE | 26.94% | 34.77% | 45.50% | 38.94% | 29.47% | 43.01% |

MAE | 15.67% | 24.76% | 30.91% | 23.32% | 19.06% | 29.23% |

MAX | 32.92% | 12.47% | 50.95% | 72.42% | 34.80% | 74.10% |

**Table 20.**Relative performance improvement of the LPA-RICI-based ($n=0,\mathsf{\Gamma}=22,{R}_{c}=1$) denoising over other tested methods, for the CCSN signal s20a3o15 at a distance of 20 kpc (SNR = −14.19 dB).

Perform. Index | LPA-ICI $\mathbf{\Gamma}=1.5$ | TV $\mathit{\mu}=0.29$ | Neigh STFT | sym5 Wavelet SURE, Level 6 | db6 Wavelet SURE, Level 6 | coif1 Wavelet SURE, Level 7 |
---|---|---|---|---|---|---|

ISNR | 17.00% | 14.68% | 100.07% | 50.03% | 42.55% | 58.94% |

PSNR | 20.30% | 17.45% | 138.52% | 63.19% | 53.04% | 75.61% |

RMSE | 26.04% | 23.33% | 64.59% | 49.96% | 46.17% | 53.68% |

MAE | 8.20% | 7.94% | 43.57% | 41.91% | 29.21% | 44.13% |

MAX | 49.23% | 20.92% | 76.84% | 81.17% | 80.85% | 84.79% |

**Table 21.**Algorithm execution times of the tested denoising methods, for the CCSN signal s20a3o15 at distances of 5, 10, and 20 kpc.

Execution Time (s) | |||||||||
---|---|---|---|---|---|---|---|---|---|

Distance(kpc) | LPA-RICI$\mathit{n}=\mathbf{0}$ | LPA-RICI$\mathit{n}=\mathbf{1}$ | LPA-RICI$\mathit{n}=\mathbf{2}$ | LPA-ICI | TV | NeighSTFT | SymletWavelet | DaubechiesWavelet | CoifletWavelet |

5 | 0.4498 | 0.7315 | 0.9687 | 3.8876 | 0.0163 | 0.7809 | 0.0054 | 0.0162 | 0.0051 |

10 | 0.6978 | 0.8143 | 1.1634 | 4.9161 | 0.0169 | 0.7755 | 0.0050 | 0.0054 | 0.0050 |

20 | 0.8756 | 1.0480 | 2.3415 | 10.5730 | 0.0169 | 0.7700 | 0.0054 | 0.0055 | 0.0058 |

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**MDPI and ACS Style**

Lopac, N.; Lerga, J.; Cuoco, E.
Gravitational-Wave Burst Signals Denoising Based on the Adaptive Modification of the Intersection of Confidence Intervals Rule. *Sensors* **2020**, *20*, 6920.
https://doi.org/10.3390/s20236920

**AMA Style**

Lopac N, Lerga J, Cuoco E.
Gravitational-Wave Burst Signals Denoising Based on the Adaptive Modification of the Intersection of Confidence Intervals Rule. *Sensors*. 2020; 20(23):6920.
https://doi.org/10.3390/s20236920

**Chicago/Turabian Style**

Lopac, Nikola, Jonatan Lerga, and Elena Cuoco.
2020. "Gravitational-Wave Burst Signals Denoising Based on the Adaptive Modification of the Intersection of Confidence Intervals Rule" *Sensors* 20, no. 23: 6920.
https://doi.org/10.3390/s20236920