## 2 Method
### Formulation of DMDct  
In DMD reconstruction, given as Equation (3), we know the matrix product \(\mathbf{\Phi}\mathbf{D}\) as the spatial component while the matrix \(\mathbf{V}\) represents the temporal evolution of the spatial component. Let us assume there is a set of basis vectors \(\mathbf{u}_{l}\) represented as an \(N\times s\) matrix \(\mathbf{U}=\left[\mathbf{u}_{1}\dots\mathbf{u}_{s}\right]\) based on which the matrix product \(\mathbf{\Phi}\mathbf{D}\) can be approximated as  
\[\mathbf{\Phi}\mathbf{D}\approx\mathbf{U}\mathbf{C} \tag{5}\]  
where \(\mathbf{C}\) is the \(s\times r\) matrix \[\rightarrow\mathbf{\hat{z}}_{k}=\mathbf{U}\sum_{j=1}^{r}\left(\alpha_{jk}\mathbf{a}_ {j}-\beta_{jk}\mathbf{b}_{j}\right) \tag{7}\]  
where \(\mathbf{a}_{j}\) and \(\mathbf{b}_{j}\) are the respective real and imaginary parts of the \(j\)-th column of \(\mathbf{C}\) and \(\lambda_{j}^{k}=\alpha_{jk}+i\beta_{jk}\). For the class of two dimensional problems addressed here, each snapshot \(\mathbf{S}_{k}\) is an \(n_{y}\times n_{x}\) matrix of real values for which Equation (7) may be rewritten as  
\[\mathbf{\hat{S}}_{k}=\mathbf{U}_{y}\left(\mathbf{U}_{x}\sum_{j=1}^{r}\left( \alpha_{jk}\mathbf{A}_{j}^{T}-\beta_{jk}\mathbf{B}_{j}^{T}\right)\right)^{T} \tag{8}\]  
where \(\mathbf{\hat{S}}_{k}\) is the real-valued reconstruction of \(k\)-th snapshot, \(\mathbf{U}_{y}\) is the \(n_{y}\times s_{y}\) matrix of the basis vectors along the columns of \(\mathbf{S}_{k}\), \(\mathbf{U}_{x}\) is the \(n_{x}\times s_{x}\) matrix of the basis vectors along the rows of \(\mathbf{S}_{k}\), and \(\mathbf{A}_{j}\) and \(\mathbf{B}_{j}\) are the \(s_{y}\times s_{x}\) matrices of unknown coefficients corresponding to the \(j\)-th dynamic mode. The columns of \(\mathbf{U}_{x}\) and \(\mathbf{U}_{y}\) are the basis vectors. For the special case of DCT basis vectors, Equation (8) may be rephrased as  
\[\mathbf{\hat{S}}_{k}=\mathcal{D}_{y}^{-1}\left(\mathcal{D}_{x}^{-1}\left( \sum_{j=1}^{r}\left(\alpha_{jk}\mathbf{A}_{j}^{T}-\beta_{jk}\mathbf{B}_{j}^{T} \right)\right)^{T}\right) \tag{9}\]  
where the operator \(\mathcal{D}_{x}\) and its inverse \(\mathcal{D}_{x}^{-1}\) are defined as  
\[\mathcal{D}_{x}\left(\mathbf{X}\right)\triangleq\mathbf{U}_{x}^{T}\mathbf{X},\ \mathcal{D}_{x}^{-1}\left(\mathbf{X}\right)\triangleq\mathbf{U}_{x}\mathbf{X} \tag{10}\]  
The forward and inverse operators \(\mathcal{D}_{x}\) and \(\mathcal{D}_{x}^{-1}\), respectively, apply the forward and inverse DCT transforms of length \(s_{x}\) to the columns of their arguments. The forward and inverse operators \(\mathcal{D}_{y}\) and \(\mathcal{D}_{y}^{-1}\) are defined similarly. Most numerical analysis packages provide forward and inverse DCT transforms as built-in functions hence eliminating the need to define the matrices \(\mathbf{U}_{x}\) and \(\mathbf{U}_{y}\) explicitly. Given the sampling mask \(M\), the reconstruction error of the \(k\)-th snapshot is defined as  
\[\mathbf{E}_{k}=\left[e_{ii^{\prime}}^{(k)}\right]_{n_{y}\times n_{x}},\ e_{ii^{ \prime}}^{(k)}=\left\{ \begin{array}{ll}\hat{s}_{ii^{\prime}}^{(k)}-\hat{s}_{ ii^{\prime}}^{(k)}&,(i,i^{\prime})\in M\\ 0&,(i,i^{\prime})\
otin M\end{array} \right. \tag{11}\]  
\[E_{k}\triangleq\left\|\mathbf{E}_{k}\right\|_{F}^{2}=\sum_{\forall(i,i^{ \prime})\in M}\left(\hat{s}_{ii^{\prime}}^{(k)}-\hat{s}_{ii^{\prime}k}^{(k)} \right)^{2} \tag{12}\]  
where \(\otimes\) is the element-wise product of two matrices and \(\hat{s}_{ii^{\prime}}^{(k)}\) and \(\hat{s}_{ii^{\prime}}^{(k)}\) are the respective \((i,i^{\prime})\) elements of the matrices \(\mathbf{\hat{S}}_{k}\) and \(\mathbf{S}_{k}\).