Research on a Noise Reduction Method Based on Multi-Resolution Singular Value Decomposition
Abstract
:1. Introduction
2. Method Overview
2.1. MRSVD
2.1.1. The Two-Division Recursion SVD
2.1.2. Multi-Division Structure MRSVD
2.2. Noise Reduction Principle Based on MRSVD
- (1)
- The signal containing noise is constructed to a matrix A that is same as Formula (1).
- (2)
- Performing the first SVD decomposition on matrix A will obtain two singular values whose reconstructed components can reflect the power concentration of original signal and the noise. Larger singular value will primarily reflect the original signal, while smaller singular value will primarily reflect the noise.
- (3)
- Zeroing the singular value that reflects the noise, that is to discard the detail signal , which is equivalent to removing some of the noise in the signal.
- (4)
- The approximation signal is retained to reconstruct the Hankel matrix for decomposition of the next layer, in such a way that decomposition of each layer separates the noise of amplitude .
2.3. Noise Reduction Process
3. Analysis
3.1. The Optimal Model of MRSVD
3.2. Comparison among Multiple Models
3.3. Instance Verification
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
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SNR/db | ||||||||
---|---|---|---|---|---|---|---|---|
−5 | 1 | 5 | 20 | −5 | 1 | 5 | 20 | |
Two-division | 9.182 | 12.621 | 14.210 | 26.718 | 7.129 | 12.526 | 15.618 | 26.458 |
Three-division(N = 1) | 0.752 | 6.253 | 8.071 | 7.107 | 2.445 | 7.375 | 8.031 | 6.997 |
Three-division(N = 2) | −2.186 | 5.734 | 7.361 | 5.585 | 1.762 | 7.322 | 8.039 | 5.867 |
Four-division(N = 1) | 8.396 | 11.735 | 13.374 | 26.135 | 7.597 | 11.662 | 14.388 | 22.725 |
Four-division(N = 2) | 5.501 | 12.611 | 14.125 | 30.321 | 7.012 | 12. 523 | 15.606 | 30.385 |
Four-division(N = 3) | 7.453 | 8.255 | 14.212 | 29.712 | 6.821 | 12.463 | 15.528 | 29.263 |
MSE | ||||||||
---|---|---|---|---|---|---|---|---|
−5 | 1 | 5 | 20 | −5 | 1 | 5 | 20 | |
Undecomposed | 3.141 | 0.777 | 0.352 | 0.009 | 3.308 | 0.827 | 0.286 | 0.013 |
Two-division | 0.184 | 0.071 | 0.022 | 0.002 | 0.138 | 0.112 | 0.039 | 0.004 |
Three-division(N = 1) | 0.389 | 0.247 | 0.156 | 0.138 | 0.399 | 0.365 | 0.291 | 0.244 |
Three-division(N = 2) | 0.552 | 0.266 | 0.183 | 0.118 | 0.518 | 0.369 | 0.272 | 0.234 |
Four-division(N = 1) | 0.199 | 0.087 | 0.033 | 1.9 × 10−3 | 0.182 | 0.129 | 0.059 | 0.010 |
Four-division(N = 2) | 0.186 | 0.073 | 0.023 | 6 × 10−4 | 0.154 | 0.113 | 0.039 | 1.9 × 10−3 |
Four-division(N = 3) | 0.265 | 0.124 | 0.036 | 9.4 × 10−4 | 0.291 | 0.117 | 0.047 | 2.1 × 10−3 |
Decomposition Layer | ||||||||
---|---|---|---|---|---|---|---|---|
−5 | 1 | 5 | 20 | −5 | 1 | 5 | 20 | |
Two-division | 78 | 48 | 29 | 7 | 22 | 9 | 8 | 1 |
Three-division(N = 1) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Three-division(N = 2) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Four-division(N = 1) | 18 | 9 | 1 | 1 | 4 | 1 | 1 | 1 |
Four-division(N = 2) | 5734 | 4144 | 2583 | 1755 | 251 | 139 | 90 | 15 |
Four-division(N = 3) | 5623 | 4994 | 3673 | 3112 | 366 | 185 | 88 | 25 |
SNR/db | ||||||||
---|---|---|---|---|---|---|---|---|
−5 db | 1 db | 5 db | 20 db | −5 db | 1 db | 5 db | 20 db | |
EMD | 1.786 | 7.273 | 10.376 | 26.005 | −0.258 | 2.957 | 5.078 | 6.329 |
EEMD | 1.850 | 8.380 | 12.854 | 26.642 | 1.745 | 8.555 | 12.305 | 27.652 |
VMD | 0.206 | 6.004 | 9.542 | 25.472 | 4.533 | 10.477 | 13.146 | 20.405 |
SVD | 8.946 | 11.842 | 13.346 | 26.685 | 6.759 | 11.031 | 14.186 | 24.919 |
MRSVD | 9.182 | 12.621 | 14.210 | 26.718 | 7.129 | 12.526 | 15.618 | 26.458 |
Wavelet decomposition | 6.191 | 5.928 | 6.266 | 7.005 | 4.078 | 4.784 | 5.033 | 5.134 |
MSE | ||||||||
---|---|---|---|---|---|---|---|---|
−5 db | 1 db | 5 db | 20 db | −5 db | 1 db | 5 db | 20 db | |
Undecomposed | 3.351 | 0.707 | 0.299 | 0.009 | 3.206 | 0.833 | 0.303 | 0.010 |
EMD | 0.662 | 0.187 | 0.916 | 0.002 | 2.162 | 1.015 | 0.697 | 0.468 |
EEMD | 0.652 | 0.146 | 0.052 | 0.002 | 1.338 | 0.298 | 0.117 | 0.004 |
VMD | 1.058 | 0.282 | 0.101 | 0.003 | 0.713 | 0.201 | 0.097 | 0.018 |
SVD | 0.242 | 0.114 | 0.103 | 0.003 | 0.428 | 0.205 | 0.066 | 0.005 |
MRSVD | 0.184 | 0.071 | 0.022 | 0.002 | 0.138 | 0.112 | 0.039 | 0.004 |
Wavelet decomposition | 0.241 | 0.256 | 0.239 | 0.201 | 0.811 | 0.654 | 0.627 | 0.612 |
Noise reduction model | EMD | EEMD | VMD | Wavelet decomposition | SVD | MRSVD |
Maximum kurtosis component | IMF1 | IMF3 | IMF3 | d1 | / | D2 |
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Zhang, G.; Xu, B.; Zhang, K.; Hou, J.; Xie, T.; Li, X.; Liu, F. Research on a Noise Reduction Method Based on Multi-Resolution Singular Value Decomposition. Appl. Sci. 2020, 10, 1409. https://doi.org/10.3390/app10041409
Zhang G, Xu B, Zhang K, Hou J, Xie T, Li X, Liu F. Research on a Noise Reduction Method Based on Multi-Resolution Singular Value Decomposition. Applied Sciences. 2020; 10(4):1409. https://doi.org/10.3390/app10041409
Chicago/Turabian StyleZhang, Gang, Benben Xu, Kaoshe Zhang, Jinwang Hou, Tuo Xie, Xin Li, and Fuchao Liu. 2020. "Research on a Noise Reduction Method Based on Multi-Resolution Singular Value Decomposition" Applied Sciences 10, no. 4: 1409. https://doi.org/10.3390/app10041409
APA StyleZhang, G., Xu, B., Zhang, K., Hou, J., Xie, T., Li, X., & Liu, F. (2020). Research on a Noise Reduction Method Based on Multi-Resolution Singular Value Decomposition. Applied Sciences, 10(4), 1409. https://doi.org/10.3390/app10041409