## 2 Coded Aperture Compressive Temporal Imaging (CACTI)
### Bayesian Model for CACTI Super-Resolution (SR) Reconstruction  
The Bayesian MAP [20] is adopted to carry out the joint estimation on high resolution (HR) image \(I\), spatiotemporal transformation function \(M_{i}\) and super temporal resolution image sequence \(f_{i}\) as:  
\[\left\{I^{*},\{M_{i}\}^{*},\{f_{i}\}^{*}\right\}=\underset{I,\{M_{i}\},\{f_{ i}\}}{\arg\max}\,p(I,\{M_{i}\},\{f_{i}\}|\{y_{i}\}) \tag{9}\]  
Figure 2: The illustration of the coding mechanisms within the coded aperture compressive temporal imaging (CACTI) system. (**a**) The system structure of CACTI; (**b**) the measuring principle of CACTI. where the posterior is the product of prior and likelihood:  
\[p(I,\{M_{i}\},\{f_{i}\}|\{y_{i}\})\propto p(I)\prod_{i}p(M_{i})\prod_{i}p(f_{i}) p(y_{0}|I,f_{0})\prod_{i\
eq 0}p(y_{i}|I,M_{i},f_{i}) \tag{10}\]  
Corresponding to motion vector matrix \(M_{i}\), the motion vector is assumed to be separated into the horizontal component \(u_{i}\) and the vertical component \(v_{i}\). With the gradient sparsity of motion vector, super temporal resolution image sequence, and the HR image as the image prior, the distribution of probability is:  
\[p(I)=\frac{1}{Z_{I}(\eta)}\exp\{-\eta\|\
abla I\|\} \tag{11}\]  
\[p(f_{i})=\frac{1}{Z_{K}(\xi)}\exp\{-\xi(\|\
abla f_{i}\|)\} \tag{12}\]  
\[p(M_{i})=\frac{1}{Z_{\omega}(\lambda)}\exp\{-\lambda(\|\
abla u_{i}\|+\| \
abla v_{i}\|)\} \tag{13}\]  
where \(\
abla\) is the gradient operator; \(Z_{I}(\eta),Z_{\omega}(\lambda),Z_{K}(\xi)\) are the normalization constants respectively, depending on \(\eta,\xi,\lambda\) only. With the known \(p(\{f_{i}\})\) and \(p(M_{i})\) of the current frame, the prior probability distribution of the observed image can be presented as:  
\[p(y_{i}|I,f_{i},T_{i})=\frac{1}{Z_{f}(\xi)}\exp\{-\xi\|y-\mathbf{DKM}_{i} \mathbf{F}I\|\} \tag{14}\]  
where matrices \(\mathbf{D}\) and \(\mathbf{K}\) correspond to down-sampling and filtering with blur kernel \(K\); \(\mathbf{M}_{i}\) is the warping matrix corresponding to motion vector; matrices \(\mathbf{F}\) correspond to the Hadamard (element-wise) product between the coding matrix and the frame in CACTI; \(Z_{f}(\xi)\) are the normalization constants respectively depending on \(\xi\) only. After the probability distributions are obtained from both prior and likelihood, the method of coordinate descent is applicable for the solution. The CACTI performance is directly related to the sparsity of objective scene, the velocity and manner of relative movement between the passive coding element and the pixel, the size of the coded element, as well as the velocity of the relative movement between the objective and the sensor. However, if these conditions are definite, the imaging results will be determined by the performance and robustness of the reconstruction algorithms. Reconstruction Approach Based on Maximum a Posteriori Probability and Markov Random Field (MAP-MRF)  
Based on the Bayesian model for CACTI, the model parameters are estimated by the coordinate descent method. The model parameters include a super temporal resolution image sequence, motion vector flow and high-resolution images. First, a robust MAP-MRF-based reconstruction approach is proposed, which renders the noises into total variation (TV) minimization problem for compressive sensing and solves it by accelerated linearized Bregman algorithm. This algorithm can perform preliminary estimation of the reconstruction image sequence, as the initial low resolution (LR) sequence shown in Figure 3. The estimation of motion vector for the weighted 3D neighbor system is estimated by scale-invariant feature transform (SIFT) Flow as shown in Figure 3. The proposed reconstruction algorithm utilizes the multi-regularization and the motion vector among different frames to reconstruct the frame. Furthermore, a super-resolution reconstruction approach using the Belief Propagation method is proposed to enhance the clarity and fidelity of the reconstruction results.