# 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 |

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**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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## Share and Cite

**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