## 3 The Proposed SSM-NTF
### Stable Sparse Model with Non-Tight Frame  
In this section, we derive our stable sparse model with non-tight frame where the non-tight frame condition serves as an approximation to the RIP. According to [35], a \(k\)-th RIP constant can be express as  
\[\delta_{k}(\mathbf{\Phi})=\frac{\Gamma_{k}(\mathbf{\Phi})-1}{\Gamma_{k}(\mathbf{\Phi})+1} \tag{7}\]  
where  
\[\Gamma_{k}(\mathbf{\Phi})=\frac{\theta_{max}^{k}}{\theta_{min}^{k}} \tag{8}\]\[\theta^{k}_{max}=\max_{\|\mathbf{y}\|_{0}=k}\frac{\|\mathbf{\Phi}\mathbf{y}\|_{2}^{2}} {\|\mathbf{y}\|_{2}^{2}},\theta^{k}_{min}=\min_{\|\mathbf{y}\|_{0}=k}\frac{\| \mathbf{\Phi}\mathbf{y}\|_{2}^{2}}{\|\mathbf{y}\|_{2}^{2}} \tag{9}\]  
The Equation (7) provides a new perspective in integrating the RIP to sparse model via applying \(\theta^{k}_{max}\) and \(\theta^{k}_{min}\) instead of the RIP constant \(\delta_{k}(\mathbf{\Phi})\). The difficulty in building a stable sparse model decreases. However, the sparsity \(k\) varies with the noise level, and also, in a feasible numerical calculation method, it is impossible to sweep through all the samples satisfying \(\|\mathbf{x}\|_{0}=k\) to pursue an unknown dictionary \(\mathbf{\Phi}\). Let \(\mathbf{x}\) be a signal vector, the frame reconstruction function can be formulated as \(\mathbf{x}=\mathbf{\Phi}\mathbf{\Psi}^{T}\mathbf{x}\) where \(\mathbf{\Psi}\) is a dual frame of \(\mathbf{\Phi}\). Adding a reasonable sparsity prior to the signal \(\mathbf{x}\) over \(\mathbf{\Psi}\) domain, we can derive  
\[\frac{\|\mathbf{\Phi}(\mathbf{\Psi}^{T}\mathbf{x})\|_{2}^{2}}{\|\mathbf{\Psi}^{T}\mathbf{ x}\|_{2}^{2}}=\frac{\|\mathbf{x}\|_{2}^{2}}{\|\mathbf{\Psi}^{T}\mathbf{x}\|_{2}^{2}}= \frac{1}{\frac{\|\mathbf{\Psi}^{T}\mathbf{x}\|_{2}^{2}}{\|\mathbf{x}\|_{2}^{2}}} \tag{10}\]  
Denoting the optimal frame bounds of \(\mathbf{\Phi}\) as \(A\) and \(B\), the frame condition of \(\mathbf{\Psi}\) can be formulated as \(\frac{1}{8}\leq\frac{\|\mathbf{\Psi}^{T}\mathbf{x}\|_{2}^{2}}{\|\mathbf{x}\|_{2}^{2 }}\leq\frac{1}{A}\). Then a pair of bounds for Equation (10) can be obtained as \(A\leq\frac{\|\mathbf{\Phi}(\mathbf{\Psi}^{T}\mathbf{x})\|_{2}^{2}}{\|\mathbf{\Psi}^{T} \mathbf{x}\|_{2}^{2}}\leq B\). A formula similar to Equation (9) is derived as \(B=\max_{\mathbf{x}\in\mathcal{J}}\frac{\|\mathbf{\Phi}(\mathbf{\Psi}^{T}\mathbf{x})\|_ {2}^{2}}{\|\mathbf{\Psi}^{T}\mathbf{x}\|_{2}^{2}}\), \(A=\min_{\mathbf{x}\in\mathcal{J}}\frac{\|\mathbf{\Phi}(\mathbf{\Psi}^{T}\mathbf{x})\|_ {2}^{2}}{\|\mathbf{\Psi}^{T}\mathbf{x}\|_{2}^{2}}\) where \(\mathcal{J}\) is the data set. Imitating Equation (7), we can obtain a RIP-like constant expression  
\[\hat{\delta}(\mathbf{\Phi})=\frac{\hat{\Gamma}(\mathbf{\Phi})-1}{\hat{\Gamma}(\mathbf{ \Phi})+1} \tag{11}\]  
where \(\hat{\Gamma}(\mathbf{\Phi})=\frac{B}{A}\). Obviously, \(\hat{\delta}(\mathbf{\Phi})\) can be regarded as an approximation of the RIP constant which benefits the computation due to the ignorance on sparsity degree. In a word, the RIP constraint can be satisfied by constraining the frame bounds. Thus, a stable overcomplete system with a sparsity prior can be established. Now we discuss the characteristic of the frame bounds \(A\) and \(B\). The Frame Condition (2) has a more compact form \(\sqrt{A}\leq\eta_{\mathbf{\Phi}}\leq\sqrt{B}\) where \(\eta_{\mathbf{\Phi}}\) denotes any singular value of \(\mathbf{\Phi}\). More specifically, \(\sqrt{A}=\eta_{min}\), \(\sqrt{B}=\eta_{max}\) where \(\eta_{max}\) and \(\eta_{min}\) denote the maximum and minimum singular values of \(\mathbf{\Phi}\), respectively. Then, we can obtain \(\eta_{max}\geq\theta^{k}_{max}\), \(\eta_{min}\leq\theta^{k}_{min}\). It is easy to know that \(\hat{\delta}(\mathbf{\Phi})\geq\delta_{k}(\mathbf{\Phi})\). Obviously, \(\hat{\delta}(\mathbf{\Phi})\) is a reasonable relaxation of \(\delta_{k}(\mathbf{\Phi})\) as \(\hat{\delta}(\mathbf{\Phi})\) is slightly exceed \(\delta_{k}(\mathbf{\Phi})\) but resides very close to it as long as the data is not seriously degraded.