## 1 Introduction
[PERSON] also verified the effectiveness of GAP and suggested the Constrained Least Squares (CLS) algorithm ([PERSON], 1998) using the constraint term of norm \(\ell_{2}\). In comparison with GAP and Orthogonal Matching Pursuit (OMP) ([PERSON], 1998), CLS can better balance reconstruction quality and speed. However, GAP has no obvious difference to CLS in terms of calculating efficiency, though it has much better peak signal to noise ratio (PSNR) performance. For OMP with similar PSNR evaluation, GAP is more efficient in calculation. Therefore, GAP still excels. [PERSON] et al. (2010, 2011) proposed a Gaussian Mixture Model (GMM)-based inversion method that can reconstruct video frames with greater fidelity than other methods for certain synthetic and real data through the on-line or off-line dictionary learning. An online-updated GMM algorithm was robust in different scenes and motions, especially in simple movements. For fast and complicated movements, the offline-learned GMM algorithm with related training videos can yield promising results. In their comparison, however, the GAP algorithm had not been optimized for certain scenes and adopted only the default values for the weighting coefficient. Hence, it is necessary to choose the algorithm in accordance with the natural scenes. Additionally, the GMM algorithm is faster than GAP in parallel calculation, but slower in serial calculation. The GAP algorithm is an undeniable success for CACTI. The GAP with transform regularization and GAP with total variation (TV) regularization algorithms are the state-of-the-art algorithms used in CACTI. The basic principles of these different algorithms are shown in Figure 1. The basic principle of GAP with transform regularization uses wavelet transform regularization in the spatial domain and DCT transform regularization in the temporal domain simultaneously ([PERSON] et al., 2015), which is calculated only for the data from one observation, as shown in Figure 1b. However, the GAP algorithm cannot easily find a good transformation to be imposed on different data, and the selection of groups and weights affects the results significantly ([PERSON] et al., 2015). Moreover, this transformation will consume more computational time. Therefore, [PERSON] proposed the TV regularization method to optimize GAP ([PERSON] et al., 2016). By replacing the constraint term of weighting-\(\ell_{2,1}\) norm with the TV regularization term, this method can help realize the approximate performance of GAP, while simplifying it. However, the TV regularization was only verified against one kind of video image in simulated comparison, and only evaluated with fixed compression ratio. It has yet to be further proved with extensive experiments on compressive sensing of video with different sparsity and motion. Furthermore, the TV regularization is only imposed on the 2D image, which may result in information loss or over-smoothness of reconstructed images, as shown in Figure 1a. While avoiding the image distortion, CACTI can considerably increase the temporal resolution. Nevertheless, the system requires the high-resolution mask or the spatial light modulator as an active coding element, which in turn reduces the luminous flux to the system, and thus renders the imaging results more sensitive to noise. Present methods conduct the video compression mainly in idealized conditions without sufficient consideration of the effect of sparsity and noises of natural scenes on the reconstruction algorithms. This paper proposes a robust reconstruction approach to undermine the noise effect, and to stably adapt to a high compressive ratio. To utilize inter-frame redundant information and sparsity in multi-transform domains, a maximum a posteriori probability and Markov random field (MAP-MRF) model for CACTI is proposed. The proposed approach adopts the coordinate descent method to perform joint estimation on model parameters including spatiotemporal super-resolution images, motion vector flow, and temporal super-resolution image sequences.