## 2 Coded Aperture Compressive Temporal Imaging (CACTI)
### Preliminary Estimation of Reconstruction Image Sequence  
In accordance with the forward model mentioned above, the \(N_{f}\) frames of video can be reconstructed from one observation of coded images \(y\). The reconstructive algorithm requires Equation (6) to reach the results. For most algorithms, data fidelity needs to be enforced by using the quadratic term and the regularization term. The total variation-based algorithms have demonstrated high performance on various CS problems, which can improve the fidelity and robustness of solvers [21; 22]. The problem in Equation (6) can be represented as:  
\[\min_{x}\lVert\mathrm{TV}(x)\rVert\leq C,\ \ \ \mathrm{subject}\ \ \ \ \ \mathrm{to}\ \ \ \boldsymbol{\Phi}x=y \tag{15}\]  
where \(C\) is the radius of \(\ell_{1}-\) ball based on TV of the signal, and \(\mathrm{TV}(x)\underset{i,j}{\sum}\sqrt{\left(\
abla_{x}x\right)_{i,j}^{2}+ \left(\
abla_{y}x\right)_{i,j}^{2}}\). The alternating projection algorithm is adopted to solve the above equation:  
\[\left(x^{(t)},\theta^{(t)}\right)=\underset{x,\theta}{\mathrm{argmin}}\frac{1 }{2}\lVert x-\theta\rVert_{2}+\lambda\lVert\mathrm{TV}(\theta)\rVert,\ \ \ \mathrm{subject}\ \ \ \mathrm{to}\ \ \ \boldsymbol{\Phi}x=y \tag{16}\]  
Figure 3: The framework of the MAP-MRF-based approach. The initial estimation of video sequence which is observed by multiple snapshot, and the low resolution (LR) image patch are candidates for MRF model. Then motion vectors field estimation by SIFT Flow, and which are used to calculate weighted coefficients of the 3D weight NS and multi-regularization. Then LR sequence are reconstructed by proposed multi-regularization algorithm, the LR image patch are candidates for MRF model. Finally, the high resolution (HR) image is constructed by Belief Propagation algorithm. where \(\lambda\) is the regulation coefficient and can be set as 0.07-0.15; \(t\) is the number of iteration times. Equation (16) can be solved by alternating the iteration algorithm with updated \(\theta\) and \(x\). For a given \(\theta\), the update of \(x\) is simplified to:  
\[x^{(t+1)}:=\arg\!\min_{x}\frac{1}{2}\|y-\mathbf{\Phi}x\|_{2}^{2}+\frac{\eta}{2}\|x- \theta^{(t)}\|_{2}^{2} \tag{17}\]  
It can be converted into the Euclidean Projection problem directly or resolved with Alternating Direction Method of Multipliers (ADMM) algorithm [23]. The two methods are equivalent to each other as proven in literature [13]. For simplicity, the parameter can be set to values as 1. \[x^{(t+1)}=\theta^{(t)}+\mathbf{\Phi}^{T}\Big{(}\mathbf{\Phi}\mathbf{\Phi}^{T}\Big{)}^{-1} \Big{(}y-\mathbf{\Phi}\theta^{(t)}\Big{)} \tag{18}\]  
Given \(x\), the sub-problem about updating \(\theta\) can be regarded and solved as the denoising or deblurring problem [24]. Taking the noises into account, the sub-problem can be rewritten as:  
\[\theta^{(t+1)}:=\arg\!\min_{x}\|x^{(t+1)}-\theta\|_{2}^{2}+\beta\|b-\text{TV} (\theta)-t\|_{2}^{2}+\lambda\|\text{TV}(\theta)\| \tag{19}\]  
The problem can be solved with the Accelerated Linearized Bregman algorithm, which accelerates the classical gradient descent method in that it reduces the iteration complexity significantly without increasing the per-iteration computational effort [25]. \(\beta\) is the weight and can be set as 0.15 for most applications. This algorithm can be applied in three easy steps:  
\[\theta^{(t+1)}:=\arg\!\min_{\theta}\|x^{(t+1)}-\theta\|_{2}^{2}+\beta\|b- \text{TV}(\theta)-t\|_{2}^{2} \tag{20}\]  
\[b^{(t+1)}:=\arg\!\min_{b}\beta\|b-\text{TV}(\theta)-t\|_{2}^{2}+\lambda\| \text{TV}(\theta)\| \tag{21}\]  
\[t^{t+1}:=t^{t}+\Big{(}\theta^{t+1}-b^{t+1}\Big{)} \tag{22}\]  
Equation (22) is explicit and convenient for evaluation. Equation (20) is a differentiable optimization problem, which can be solved by Gauss-Seidel algorithm [26].