## 3 Results
### Dynamic Denoising and Reconstruction  
As the second test, the unforced Duffing equation taken from [7] was used to generate the test dataset. The governing differential equation is  
\[\ddot{x}=-\delta\dot{x}-x(\gamma+\alpha x^{2}) \tag{22}\]  
Figure 3: Five sample snapshots of the fully-sampled dataset reconstruction for the double-gyre dataset. The top row shows the reference snapshots titled with the snapshot numbers. The second row shows the random samples taken. The third and the fourth rows show csDMD and DMDct reconstructions, respectively. Figure 2: The real parts of the mode shapes of the first five DMD modes corresponding to the eigenvalues with non-negative imaginary parts for the double-gyre dataset. The top row shows the mode shapes obtained by applying exact DMD on the fully-sampled dataset. The second and third rows show the aligned csDMD and DMDct reconstructions, respectively. The corresponding eigenvalues are represented above the columns. The Pearson correlation coefficients between the aligned reconstructed mode shapes and those of exact DMD are shown as well. where \(\delta=0.5\), \(\gamma=-1\), and \(\alpha=1\). The equation was solved over the region \(x,\dot{x}\in[-2,2]\), which was discretized as a \(41\times 41\) mesh. For each node of the mesh, the corresponding values of \(x\) and \(\dot{x}\) were taken as the initial conditions and the ODE was solved for 5 s during which the snapshots were taken every 0.1 s resulting in a total of 51 snapshots. Even though the numerical solution resulted in both \(x\) and \(\dot{x}\) values, only \(x\) values were taken and used as the test dataset. Figure 4 shows six sample snapshots of the reference dataset. Two cases are presented here for comparison. The first case does not have a gap, whereas the second case has a rectangular gap. Both cases were evaluated with noise-free and noisy samples. In all cases, 20% of the available data of each snapshot were taken as the measurement samples and were used for reconstruction. For each case, twenty different random sampling masks were tested. For each mask, the sampling locations remained the same over all snapshots. The noisy cases were to study the effect of measurement noise and to see how well the two methods could denoise the data. To make the noisy dataset, random Gaussian noise with the standard deviation of 0.25 was added to the reference dataset. The PVNR metric was calculated only for the noisy reconstructions. The same set of basis vectors was used by both methods. For the noise-free samples, the maximum number of basis vectors were used (\(s_{x}=s_{y}=41\)), whereas, for the noisy samples, a reduced set of basis vectors was incorporated (\(s_{x}=s_{y}=20\)), hence dropping the high-frequency components from reconstruction. The eigenvalues derived by csDMD were used for DMDct reconstruction as well. The number of DMD modes to use was found through the method of singular value hard thresholding (SVHT) [22]. According to SVHT, the number of DMD modes for the noise-free and noisy samples was taken as 25 and 5, respectively. Figure 4b shows the amplitudes of the dynamic mode shapes of the reference Duffing dataset. In the figures depicting the snapshots, the first (#0), the middle (#25), and the last (#50) snapshots of the first sampling mask are presented for comparison. The csDMD method aims at reconstructing the mode shapes and not the fully-sampled dataset. Since no fully-sampled snapshot is available, it is not possible to reconstruct the whole dataset solely based on Exact DMD framework by simply marching forward/backward in time using Equation (2). One possible workaround is to find the optimal amplitudes of DMD modes by minimizing the RMS of reconstruction error as proposed in [6] which leads to  
\[\mathbf{b}_{opt}=\left((\mathbf{\Phi}_{s}^{*}\mathbf{\Phi}_{s})\otimes\left( \overline{\mathbf{V}\mathbf{V}^{*}}\right)\right)^{-1}\overline{\text{diag}( \mathbf{V}\mathbf{Z}^{*}\mathbf{\Phi}_{s})} \tag{23}\]  
Figure 4: The reference Duffing dataset. (**a**) Six sample snapshots are shown.