## 2 Related Work  
In this section, we briefly review the related work on frame, synthesis sparse model and analysis sparse model. Frame: A frame \(\mathbf{\Phi}\)is called a tight frame if the frame bounds are equal in the Equation (2) [32]. There are two associated operators can be defined between the Hilbert space \(\mathbf{H}^{N}\) and Square Integrable Space \(\mathbf{l}_{2}^{M}(\cdot)\) once a frame is defined. One is the analysis operator \(\mathbf{T}\) defined by  
\[(\mathbf{T}\mathbf{x})_{i}=<\mathbf{x},\mathbf{\phi}_{i}>,\quad\forall\mathbf{x} \in\mathbf{H}^{N} \tag{3}\]  
and the other is its adjoint operator \(\mathbf{T}^{*}\) which is called the synthesis operator  
\[\mathbf{T}^{*}\mathbf{c}=\sum_{i=1}^{M}\mathbf{c}\mathbf{\phi}_{i}\quad\forall \mathbf{c}=(\mathbf{c}_{i})_{i\in I}\in\mathbf{l}_{2}^{M}(\mathbf{T}) \tag{4}\]  
then, the frame operator can be defined as the following canonical expansion  
\[\mathbf{F}\mathbf{x}=\mathbf{T}^{*}\mathbf{T}\mathbf{x}=\sum_{i=1}^{M}< \mathbf{x},\mathbf{\phi}_{i}>\mathbf{\phi}_{i} \tag{5}\]In Euclidean space, a given frame \(\Phi\) can be represent in manner of matrix with its columns of it as the frame elements. Then one of its adjoint operator can be represented as \(\mathbf{\psi}_{i}=\mathbf{F}^{-1}\mathbf{\phi}_{i}\)[32]. Let\(\mathbf{x}\in\mathbb{R}^{N}\) be an arbitrary vector, a reconstruction function can be expressed as the following form  
\[\mathbf{x}=\sum_{i=1}^{M}<\mathbf{x},\mathbf{\psi}_{i}>\mathbf{\phi}_{i} \tag{6}\]  
Synthesis sparse model: The conventional synthesis sparse model represents a vector \(\mathbf{x}\) by the linear combination of a few atoms from a large dictionary \(\mathbf{\Phi}\), denoted as \(\mathbf{x}=\mathbf{\Phi}\mathbf{y},\|\mathbf{y}\|_{0}\leq L\), where \(L\) is the sparsity of \(\mathbf{y}\). The computational techniques for approximating sparse coefficient \(\mathbf{y}\) under a given dictionary \(\mathbf{\Phi}\) and \(\mathbf{x}\) includes greedy pursuit (e.g., OMP [9]) and convex relaxation optimization, such as Lasso [33] and FISTA [8]. In order to improve the performance of sparse representation, some modified models such as the nonlocal sparse model [16], the frame based sparse model [21], and the MD sparse model [18] are also investigated. Analysis sparse model: The analysis sparse model is defined as: \(\mathbf{y}=\mathbf{\Omega}\mathbf{x},\|\mathbf{y}\|_{0}=p-l\) where \(\mathbf{\Omega}\in\mathcal{R}^{p\times d}\) is a linear operator (also called as a dictionary), and \(l\) denotes the co-sparsity of the signal \(\mathbf{x}\). The analysis representative vector \(\mathbf{y}\) is sparse with \(l\) zeros. The zeros in \(\mathbf{y}\) denote the low-dimensional subspace to which the signal \(\mathbf{x}\) belongs. The analysis sparse coding [14] and dictionary learning [34] approach are also been proposed. However, all these models ignore the stability recovery property which provides stable reconstruction of the signals in presence of noise. Dictionary learning methods: The dictionaries include analytical dictionaries, such as DCT, DWT, curvelets and contourlets and learned dictionaries. Some dictionary learning method are proposed, such as the classical KSVD [9] algorithm, the efficient sparse coding which convert the original dictionary learning problem to two least squares problem by applying the Lagrange dual [3], the non-local sparse model [16] which learns a set of PCA sub-dictionaries by cluster the samples into K clusters using image nonlocal self-similarity prior and its improved version which using the \(l_{q}\)-norm to instead the \(l_{2}\)-norm in order to handle different image contents. With the realization of stability, some mutual-coherence based methods are proposed. In [20] a sparsity-based orthogonal dictionary learning method is proposed to minimize the mutual-coherence of the dictionary. The authors of [21] propose an in coherent dictionary learning scheme by integrating a low rank gram matrix of the dictionary into the dictionary learning model.