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A numerical renormalization method for quasi–conservative periodic attractors
Time-resolved denoising using model order reduction, dynamic mode decomposition, and kalman filter and smoother
1. | Department of Mechanical Engineering, University of Wisconsin-Milwaukee, Milwaukee, WI, USA |
2. | Department of Electrical and Computer Engineering, New York Institute of Technology, Long Island, NY, USA |
3. | Departments of Neurosurgery, Medical College of Wisconsin, Milwaukee, WI, USA |
In this research, we investigate the application of Dynamic Mode Decomposition combined with Kalman Filtering, Smoothing, and Wavelet Denoising (DMD-KF-W) for denoising time-resolved data. We also compare the performance of this technique with state-of-the-art denoising methods such as Total Variation Diminishing (TV) and Divergence-Free Wavelets (DFW), when applicable. Dynamic Mode Decomposition (DMD) is a data-driven method for finding the spatio-temporal structures in time series data. In this research, we use an autoregressive linear model resulting from applying DMD to the time-resolved data as a predictor in a Kalman Filtering-Smoothing framework for the purpose of denoising. The DMD-KF-W method is parameter-free and runs autonomously. Tests on numerical phantoms show lower error metrics when compared to TV and DFW, when applicable. In addition, DMD-KF-W runs an order of magnitude faster than DFW and TV. In the case of synthetic datasets, where the noise-free datasets were available, our method was shown to perform better than TV and DFW methods (when applicable) in terms of the defined error metric.
References:
[1] |
E. Bostan, S. Lefkimmiatis, O. Vardoulis, N. Stergiopulos and M. Unser,
Improved variational denoising of flow fields with application to phase-contrast MRI data, IEEE Signal Processing Letters, 22 (2015), 762-766.
doi: 10.1109/LSP.2014.2369212. |
[2] |
K. K. Chen, J. H. Tu and C. W. Rowley,
Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses, J. Nonlinear Sci., 22 (2012), 887-915.
doi: 10.1007/s00332-012-9130-9. |
[3] |
Y. Chen, A. Wiesel and A. O. Hero, Shrinkage Estimation of High Dimensional Covariance Matrices, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing, 2937–2940. Google Scholar |
[4] |
R. R. Coifman and D. L. Donoho, Translation-invariant de-noising, in Wavelets and Statistics, Springer, (1995), 125–150. Google Scholar |
[5] |
J. L. Crassidis and J. L. Junkins, Optimal Estimation of Dynamic Systems, Chapman and Hall/CRC, 2004.
doi: 10.1201/9780203509128. |
[6] |
S. T. M. Dawson, M. S. Hemati, M. O. Williams and C. W. Rowley,
Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition, Experiments in Fluids, 57 (2016), 1-19.
doi: 10.1007/s00348-016-2127-7. |
[7] |
M. F. Fathi, A. Bakhshinejad, A. Baghaie and R. M. D'Souza, Dynamic denoising and gappy data reconstruction based on dynamic mode decomposition and discrete cosine transform, Applied Sciences, 8 (2018), 1515.
doi: 10.3390/app8091515. |
[8] |
M. F. Fathi, A. Bakhshinejad, A. Baghaie, D. Saloner, R. H. Sacho, V. L. Rayz and R. M. D'Souza,
Denoising and spatial resolution enhancement of 4D flow MRI using proper orthogonal decomposition and Lasso regularization, Computerized Medical Imaging and Graphics, 70 (2018), 165-172.
doi: 10.1016/j.compmedimag.2018.07.003. |
[9] |
M. Gavish and D. L. Donoho,
The optimal hard threshold for singular values is $4/\sqrt 3$, IEEE Trans. Inform. Theory, 60 (2014), 5040-5053.
doi: 10.1109/TIT.2014.2323359. |
[10] |
A. Gelb, Applied Optimal Estimation, MIT press, 1974.
![]() |
[11] |
G. V. Iungo, C. Santoni-Ortiz, M. Abkar, F. Porté-Agel, M. A. Rotea and S. Leonardi, Data-driven reduced order model for prediction of wind turbine wakes, Journal of Physics: Conference Series, 625.
doi: 10.1088/1742-6596/625/1/012009. |
[12] |
K. M. Johnson and M. Markl,
Improved SNR in phase contrast velocimetry with five-point balanced flow encoding, Magnetic Resonance in Medicine, 63 (2010), 349-355.
doi: 10.1002/mrm.22202. |
[13] |
R. E. Kalman,
A new approach to linear filtering and prediction problems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), 35-45.
doi: 10.1115/1.3662552. |
[14] |
J. N. Kutz, S. L. Brunton, B. W. Brunton and J. L. Proctor, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, SIAM, 2016.
doi: 10.1137/1.9781611974508. |
[15] |
T. Nonomura, H. Shibata and R. Takaki, Dynamic mode decomposition using a kalman filter for parameter estimation, AIP Advances, 8 (2018), 105106.
doi: 10.1063/1.5031816. |
[16] |
T. Nonomura, H. Shibata and R. Takaki, Extended-kalman-filter-based dynamic mode decomposition for simultaneous system identification and denoising, PloS one, 14, e0209836.
doi: 10.1371/journal.pone.0209836. |
[17] |
F. Ong, M. Uecker, U. Tariq, A. Hsiao, M. T. Alley, S. S. Vasanawala and M. Lustig,
Robust 4D flow denoising using divergence-free wavelet transform, Magnetic Resonance in Medicine, 73 (2015), 828-842.
doi: 10.1002/mrm.25176. |
[18] |
H. E. Rauch, F. Tung and C. T. Striebel,
Maximum likelihood estimates of linear dynamic systems, AIAA J., 3 (1965), 1445-1450.
doi: 10.2514/3.3166. |
[19] |
C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter and D. S. Henningson,
Spectral analysis of nonlinear flows, J. Fluid Mech., 641 (2009), 115-127.
doi: 10.1017/S0022112009992059. |
[20] |
P. J. Schmid,
Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28.
doi: 10.1017/S0022112010001217. |
[21] |
P. J. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data, Bulletin of the American Physical Society, 53 (2008), 2008. Google Scholar |
[22] |
J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton and J. N. Kutz,
On dynamic mode decomposition: Theory and applications, Journal of Computational Dynamics, 1 (2014), 391-421.
doi: 10.3934/jcd.2014.1.391. |
[23] |
E. Wan, Sigma-point filters: An overview with applications to integrated navigation and vision assisted control, in 2006 IEEE Nonlinear Statistical Signal Processing Workshop, (2006), 201–202.
doi: 10.1109/NSSPW.2006.4378854. |
[24] |
K. Willcox and J. Peraire,
Balanced model reduction via the proper orthogonal decomposition, AIAA Journal, 40 (2002), 2323-2330.
doi: 10.2514/6.2001-2611. |
[25] |
M. O. Williams, C. W. Rowley and I. G. Kevrekidis, A kernel-based method for data-driven Koopman spectral analysis, J. Comput. Dyn., 2 (2015), 247–265. arXiv: 1411.2260.
doi: 10.3934/jcd.2015005. |
[26] |
M. O. Williams, I. G. Kevrekidis and C. W. Rowley,
A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25 (2015), 1307-1346.
doi: 10.1007/s00332-015-9258-5. |
[27] |
M. J. Zimoń, J. M. Reese and D. R. Emerson,
A novel coupling of noise reduction algorithms for particle flow simulations, J. Comput. Phys., 321 (2016), 169-190.
doi: 10.1016/j.jcp.2016.05.049. |
show all references
References:
[1] |
E. Bostan, S. Lefkimmiatis, O. Vardoulis, N. Stergiopulos and M. Unser,
Improved variational denoising of flow fields with application to phase-contrast MRI data, IEEE Signal Processing Letters, 22 (2015), 762-766.
doi: 10.1109/LSP.2014.2369212. |
[2] |
K. K. Chen, J. H. Tu and C. W. Rowley,
Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses, J. Nonlinear Sci., 22 (2012), 887-915.
doi: 10.1007/s00332-012-9130-9. |
[3] |
Y. Chen, A. Wiesel and A. O. Hero, Shrinkage Estimation of High Dimensional Covariance Matrices, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing, 2937–2940. Google Scholar |
[4] |
R. R. Coifman and D. L. Donoho, Translation-invariant de-noising, in Wavelets and Statistics, Springer, (1995), 125–150. Google Scholar |
[5] |
J. L. Crassidis and J. L. Junkins, Optimal Estimation of Dynamic Systems, Chapman and Hall/CRC, 2004.
doi: 10.1201/9780203509128. |
[6] |
S. T. M. Dawson, M. S. Hemati, M. O. Williams and C. W. Rowley,
Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition, Experiments in Fluids, 57 (2016), 1-19.
doi: 10.1007/s00348-016-2127-7. |
[7] |
M. F. Fathi, A. Bakhshinejad, A. Baghaie and R. M. D'Souza, Dynamic denoising and gappy data reconstruction based on dynamic mode decomposition and discrete cosine transform, Applied Sciences, 8 (2018), 1515.
doi: 10.3390/app8091515. |
[8] |
M. F. Fathi, A. Bakhshinejad, A. Baghaie, D. Saloner, R. H. Sacho, V. L. Rayz and R. M. D'Souza,
Denoising and spatial resolution enhancement of 4D flow MRI using proper orthogonal decomposition and Lasso regularization, Computerized Medical Imaging and Graphics, 70 (2018), 165-172.
doi: 10.1016/j.compmedimag.2018.07.003. |
[9] |
M. Gavish and D. L. Donoho,
The optimal hard threshold for singular values is $4/\sqrt 3$, IEEE Trans. Inform. Theory, 60 (2014), 5040-5053.
doi: 10.1109/TIT.2014.2323359. |
[10] |
A. Gelb, Applied Optimal Estimation, MIT press, 1974.
![]() |
[11] |
G. V. Iungo, C. Santoni-Ortiz, M. Abkar, F. Porté-Agel, M. A. Rotea and S. Leonardi, Data-driven reduced order model for prediction of wind turbine wakes, Journal of Physics: Conference Series, 625.
doi: 10.1088/1742-6596/625/1/012009. |
[12] |
K. M. Johnson and M. Markl,
Improved SNR in phase contrast velocimetry with five-point balanced flow encoding, Magnetic Resonance in Medicine, 63 (2010), 349-355.
doi: 10.1002/mrm.22202. |
[13] |
R. E. Kalman,
A new approach to linear filtering and prediction problems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), 35-45.
doi: 10.1115/1.3662552. |
[14] |
J. N. Kutz, S. L. Brunton, B. W. Brunton and J. L. Proctor, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, SIAM, 2016.
doi: 10.1137/1.9781611974508. |
[15] |
T. Nonomura, H. Shibata and R. Takaki, Dynamic mode decomposition using a kalman filter for parameter estimation, AIP Advances, 8 (2018), 105106.
doi: 10.1063/1.5031816. |
[16] |
T. Nonomura, H. Shibata and R. Takaki, Extended-kalman-filter-based dynamic mode decomposition for simultaneous system identification and denoising, PloS one, 14, e0209836.
doi: 10.1371/journal.pone.0209836. |
[17] |
F. Ong, M. Uecker, U. Tariq, A. Hsiao, M. T. Alley, S. S. Vasanawala and M. Lustig,
Robust 4D flow denoising using divergence-free wavelet transform, Magnetic Resonance in Medicine, 73 (2015), 828-842.
doi: 10.1002/mrm.25176. |
[18] |
H. E. Rauch, F. Tung and C. T. Striebel,
Maximum likelihood estimates of linear dynamic systems, AIAA J., 3 (1965), 1445-1450.
doi: 10.2514/3.3166. |
[19] |
C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter and D. S. Henningson,
Spectral analysis of nonlinear flows, J. Fluid Mech., 641 (2009), 115-127.
doi: 10.1017/S0022112009992059. |
[20] |
P. J. Schmid,
Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28.
doi: 10.1017/S0022112010001217. |
[21] |
P. J. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data, Bulletin of the American Physical Society, 53 (2008), 2008. Google Scholar |
[22] |
J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton and J. N. Kutz,
On dynamic mode decomposition: Theory and applications, Journal of Computational Dynamics, 1 (2014), 391-421.
doi: 10.3934/jcd.2014.1.391. |
[23] |
E. Wan, Sigma-point filters: An overview with applications to integrated navigation and vision assisted control, in 2006 IEEE Nonlinear Statistical Signal Processing Workshop, (2006), 201–202.
doi: 10.1109/NSSPW.2006.4378854. |
[24] |
K. Willcox and J. Peraire,
Balanced model reduction via the proper orthogonal decomposition, AIAA Journal, 40 (2002), 2323-2330.
doi: 10.2514/6.2001-2611. |
[25] |
M. O. Williams, C. W. Rowley and I. G. Kevrekidis, A kernel-based method for data-driven Koopman spectral analysis, J. Comput. Dyn., 2 (2015), 247–265. arXiv: 1411.2260.
doi: 10.3934/jcd.2015005. |
[26] |
M. O. Williams, I. G. Kevrekidis and C. W. Rowley,
A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25 (2015), 1307-1346.
doi: 10.1007/s00332-015-9258-5. |
[27] |
M. J. Zimoń, J. M. Reese and D. R. Emerson,
A novel coupling of noise reduction algorithms for particle flow simulations, J. Comput. Phys., 321 (2016), 169-190.
doi: 10.1016/j.jcp.2016.05.049. |








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