## 1 Introduction
Study [1] presents a data-driven method that the dictionary atoms associated with the tight frame are generated by filters. These approaches achieve much better image processing performance than previous methods, and meanwhile the tight frame condition which requires the frame almost-orthogonality will limit the flexibility in sparse representation. Study [31] derives stable recovery result for \(l_{1}\)-analysis minimization in redundant, possibly non-tight frames. Inspired by this result and the relationship between RIP and frame, we aim to establish a stable sparse model with RIP based on non-tight frame. We call a sequence \(\{\mathbf{\phi}_{i}\}_{i=1}^{M}\in\mathbf{H}\) a frame if and only if there exist two positive numbers \(A\) and \(B\) such that  
\[A\|\mathbf{x}\|_{2}^{2}\leq\sum_{i=1}^{M}|<\mathbf{x},\mathbf{\phi}_{i}>|^{2}\leq \ B\|\mathbf{x}\|_{2}^{2}\ \ \ \forall\mathbf{x}\in\mathbf{H}^{N} \tag{2}\]  
Here, \(A\) and \(B\) are called the bounds of the frame. We find that every submatrix \(\mathbf{\Phi}_{k}\) satisfied RIP is a non-tight frame with \((1-\delta_{k})\) and \((1+\delta_{k})\) as its frame bounds with a given \(k\). Obviously, there is an essential connection between the non-tight frame and the RIP. In this paper we focus on a stable sparse model and more specifically on the development of an algorithm that would learn a pair of non-tight frame based dictionaries from a set of signal examples. We propose a stable sparse model via applying the non-tight frame condition to approximate the RIP. This model shares the favorite overcomplete structures with the common sparse models, and meanwhile it contains RIP and closed-form sparse coefficient expression which ensure stable recovery. Recognizing that the optimal framebounds are essentially the maximum and minimum singular values of the frame, RIP is actually enforced on the dictionary pair (the frame and its dual frame) by constraining the singular values of them. We also formulate a dictionary pair learning model via applying the second-order truncated Taylor series to approximate the inverse frame operator. Then we present an efficient algorithm to learn the dictionary pair via a two-phase iterative approach. To summarize, this paper makes the following contributions:  
1. We propose a stable sparse model along with a dictionary pair learning model. Non-tight frame condition is utilized to develop a relaxation of RIP to guarantee stable recovery of sparse representation. Moreover, the sparse coefficients are also modeled, which leads to a more stable recovery especially for seriously noisy image. 2. It is nearly impossible to solve the dictionary pair learning model in a straightforward way since the inverse frame operator is involved. We provide an effective way to modify the model via applying a second-order truncated Taylor series to approximate the inverse frame operator, and provide an efficient algorithm for the modified one. 3. We present the stability analysis of the proposed model and demonstrate it on natural and synthetic image denoising, super resolution and image inpainting. The denoising results show that the proposed approach outperforms synthesis models such as the KSVD and the data-driven tight frame based methods for natural image case in terms of average PSNR. Moreover, it also gains comparable performance to the Analysis KSVD for a piecewise-constant (PWC) image in terms of average PSNR. The meaningful structures in the trained dictionary pairs for natural images and a PWC image are observed. The super resolution results show that the SSM-NTF produces better performance than the Bicubic interpolation method and the method in [32]. The inpainting results show that our model is able to eliminate text of fonts completely. This paper is organized as follows: Section 2 reviews the related work on frame, synthesis sparse model and analysis sparse model. Section 3 presents our stable sparse model with non-tight frame SSM-NTF along with a dictionary pair learning model.