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.
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Figure 1. The results of the unforced Duffing equation dataset. On the left, the noise-free, noisy, and reconstructed datasets are shown. The numbers at the top of the columns show the corresponding snapshot numbers. The two top rows show the noise-free (reference) and noisy datasets, respectively. The reconstructions of TV and DMD-KF-W methods are shown as indicated. The numbers inside the parenthesis are the overall RMSE values, where the lower is better. On the right, per-snapshot (solid lines) and overall (dotted) RMSE values of the noisy and the reconstructed datasets are presented
Figure 2. The results of second test case (wake behind a cylinder at Re = 100). On the left, the reference, noisy, and reconstructed datasets are shown. The images show the snapshot number 21. The bottom row shows the reconstruction errors with the overall RMSE values, where the lower is better. On the right, per-snapshot (solid lines) and overall (dotted) RMSE values of the noisy and the reconstructed datasets are presented
Figure 3. The effects of the tuning parameters of TV and DFW on the reconstruction error for the second test dataset. For TV (on left), the effect of the weighting parameter of the regularization term is shown where $ \lambda = 0.07 $ resulted in the best reconstruction. For DFW (on right), the effects of the smallest wavelet level size and the number of cycles are shown. In this case, 5 cycles with the smallest wavelet level size of 10 resulted in the best reconstruction
Figure 4. The effects of tuning parameters of TV and DFW on the reconstruction error for the synthetic 4D-Flow MRI dataset. For TV (on left), the effect of the weighting parameter of the regularization term is shown where $ \lambda = 0.03 $ resulted in the best reconstruction. For DFW (on right), the effects of the smallest wavelet level size and the number of cycles are shown. In this case, 4 cycles with the smallest wavelet level size of 12 resulted in the best reconstruction
Figure 5. The results for the synthetic 4D-Flow MRI dataset. The synthetic data resembles a cardiac cycle. (a) The top row shows sample slices where at each point, the norm of the velocity is shown. The bottom row shows the norm of difference between the datasets and the noise-free reference. The bottom row figures are titled with the corresponding total RMSE values. All areas outside the geometry mask are zeroed before RMSE values are calculated. (b) Per-snapshot (solid lines) and total (dotted) RMSE values of the noisy and the reconstructed datasets
Figure 8. Evaluating the effectiveness of wavelet denoising step of the proposed method in improving the RMSE values for the synthetic test cases. The horizontal axis shows the test case number. The vertical axis shows the ratio of the RMSE of reconstruction to the RMSE of the noisy input dataset in percent. A lower ratio corresponds to better denoising
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The results of the unforced Duffing equation dataset. On the left, the noise-free, noisy, and reconstructed datasets are shown. The numbers at the top of the columns show the corresponding snapshot numbers. The two top rows show the noise-free (reference) and noisy datasets, respectively. The reconstructions of TV and DMD-KF-W methods are shown as indicated. The numbers inside the parenthesis are the overall RMSE values, where the lower is better. On the right, per-snapshot (solid lines) and overall (dotted) RMSE values of the noisy and the reconstructed datasets are presented
The results of second test case (wake behind a cylinder at Re = 100). On the left, the reference, noisy, and reconstructed datasets are shown. The images show the snapshot number 21. The bottom row shows the reconstruction errors with the overall RMSE values, where the lower is better. On the right, per-snapshot (solid lines) and overall (dotted) RMSE values of the noisy and the reconstructed datasets are presented
The effects of the tuning parameters of TV and DFW on the reconstruction error for the second test dataset. For TV (on left), the effect of the weighting parameter of the regularization term is shown where
The effects of tuning parameters of TV and DFW on the reconstruction error for the synthetic 4D-Flow MRI dataset. For TV (on left), the effect of the weighting parameter of the regularization term is shown where
The results for the synthetic 4D-Flow MRI dataset. The synthetic data resembles a cardiac cycle. (a) The top row shows sample slices where at each point, the norm of the velocity is shown. The bottom row shows the norm of difference between the datasets and the noise-free reference. The bottom row figures are titled with the corresponding total RMSE values. All areas outside the geometry mask are zeroed before RMSE values are calculated. (b) Per-snapshot (solid lines) and total (dotted) RMSE values of the noisy and the reconstructed datasets
The results for the two in-vivo 4D-Flow MRI datasets. For both datasets, the binary mask is used to exclude all points outside the vessel. The spatial resolutions of datasets (a) and (b) are
The effect of changing the type of Daubechies wavelet used for denoising on the RMSE value
Evaluating the effectiveness of wavelet denoising step of the proposed method in improving the RMSE values for the synthetic test cases. The horizontal axis shows the test case number. The vertical axis shows the ratio of the RMSE of reconstruction to the RMSE of the noisy input dataset in percent. A lower ratio corresponds to better denoising