*7.1. Convergence Analysis*

The convergence of the presented dictionary pair learning algorithm is evaluated in Figure 1. Here, we train a pair of dictionaries **Φ** and **Ψ** of size 100 × 200 from 65,000 patches, which are of size 10 × 10 randomly sampled from six natural images. We apply the frame reconstruction function **<sup>Φ</sup>**S*λ*(**Ψ***T***x**) to reconstruct the patches. The convergence of the presented dictionary pair learning algorithm is evaluated in Figure 1. The dictionary pair is illustrated in Figure 2. They exhibit that our dictionary pair learning method is able to capture the feature of the image along with the convergence property.

**Figure 1.** Convergence analysis. The *X*-labels are the iteration number. The *Y*-labels are the is the objective function of System (20) (left) and the restoration result (measured by 'PSNR') (right). It is shown that our dictionary pair learning algorithm is a convergence one.

**Figure 2.** The exemplified dictionary pair (**Φ**, **Ψ**) in our stable sparse model with non-tight frame (SSM-NTF) model training by natural images.

#### *7.2. Image Denoising*

In this subsection, we evaluate the performance of our proposed SSM-NTF model on image denoising. Benefitting from the concept of non-tight frame, the proposed SSM-NTF model contains a pair of dictionaries: The frame and its dual frame. As a result, our proposed SSM-NTF model contains an analysis system and a synthesis system. The analysis-like system is denoted as

$$\mathbf{y} = \mathcal{S}\_{\lambda}(\mathbf{Y}^T \mathbf{x})\_{\prime} \quad \|\mathbf{y}\| \le s \tag{43}$$

which analyzes the signals in **Ψ** domain. The synthesis system is denoted as

$$\mathbf{x} = \Phi \mathbf{y}, \quad \|\mathbf{y}\| \le \mathbf{s} \tag{44}$$

which reconstructs the analyzed signals. The two systems share the same sparse coefficients **Y**.

Therefore, we compare our proposed SSM-NTF with synthesis and analysis models, respectively. It is well known that the synthesis sparse model has advantage in dealing with the natural image while the analysis sparse model is mostly used to address the piecewise constant image. Therefore, we respectively, perform the denoising experiments on natural images and piecewise images comparing with the most related approaches.
