## 1 Introduction  
Sparse representation of signals in dictionary domains has been widely studied and has provided promising performance in numerous signal processing tasks such as image denoising [1; 2; 3; 4; 5], super resolution [6; 7; 8], inpainting [9; 10] and compression [11; 12]. It is well known that images are represented by a linear combination of certain atoms of a dictionary. Overcomplete sparse representation is the overcomplete system with a sparse constraint. Common overcomplete systems differ from the traditional bases, such as DCT, DFT and Wavelet, because they offer a wider range of generating elements; potentially, this wider range allows more flexibility and effectiveness in signal sparse representation. However, it is a severely under-constrained illposed problem to find the underlying overcomplete representation due to the redundancy of the systems. When the underlying representation is sparse and the overcomplete systems have stable properties, the ill-posedness will disappear [13]. Sparse models are generally classified into two categories: Synthesis sparse models and analysis sparse model [14]. The commonly referred to sparse models are synthesis sparse models. The analysis ones characterize the signal by multiplying it with an analysis overcomplete dictionary, leading to a sparse outcome. A variety of effective sparse models have been investigated and established such as the classical synthesis sparse model [9; 15], the classical analysis sparse model [14], the nonlocal sparse model [16; 17] and the 2D sparse model [18]. Unfortunately, these models ignore the stability recovery property which claims that once a sufficient sparse solution is found, all alternative solutions necessarily reside very close to it [9]. Recently, the stable recovery of sparse representation has drawn attention in signal processing theory. Generally speaking, stable recovery can be guaranteed by two properties: Sufficient sparsity and a favorable structure of the dictionary [19]. [PERSON] defines the concept of mutual incoherence of the dictionary and applies it to prove some possibility of stable recovery [19]. The authors of [20] propose sparsity-based orthogonal dictionary learning method to minimize the mutual incoherence. The authors of [21] propose an incoherent dictionary learning scheme by integrating a low rank gram matrix of the dictionary into the dictionary learning model. A more powerful stable recovery guarantee developed by [PERSON] and [PERSON], termed Restricted Isometry Property (RIP), makes consequent analysis easy [22]. A matrix \(\mathbf{\Phi}\) is said to satisfy the RIP of order \(k\) if there exists a constant \(\delta_{k}\in(0,1)\) such that  
\[(1-\delta_{k})\|\mathbf{y}\|_{2}^{2}\leq\|\mathbf{\Phi}\mathbf{y}\|_{2}^{2}\leq(1 +\delta_{k})\|\mathbf{y}\|_{2}^{2} \tag{1}\]  
holds for all \(k\)-sparse vectors \(\mathbf{y}\). \(\delta_{k}\) is defined as the smallest constant which satisfies the above inequalities and is called the restricted isometry constant of \(\mathbf{\Phi}\). Most RIP research substantially investigates applying RIP as a stablility analysis instrument [17; 23; 24] or finding optimal RIP constant [25; 26] which are all theoretical analyses rather than practical applications. According to the research of [21], the intrinsic property of a dictionary has a direct influence on its performance. All familiar algorithms are staggeringly unstable with a coherent or degenerate dictionary [19]. Recognizing the gap between theoretical analyses and practical applications of RIP, this paper aims to build a stable sparse model satisfying the RIP. Recently, the frame as a stable overcomplete system has drawn some attention in signal processing as the given signal can be represented by its canonical expansion in a manner similar to conventional bases under the frame. Some data-driven approaches are proposed in [1; 27; 28; 29; 30]. The authors of [27; 29; 30] utilize redundant tight frame in compressed sensing and [28] applies tight frame to few-view image reconstruction.