## 4 Dictionary Pair Learning Algorithm
### Sparse coding phase
## 7 Experimental Results

We demonstrate the effectiveness of our SSM-NTF model by first discussing the convergence of our dictionary pair learning algorithm and then evaluating the performance on natural and piecewise constant image denoising, super resolution and image inpainting. ### Convergence Analysis

The convergence of the presented dictionary pair learning algorithm is evaluated in Figure 1. Here, we train a pair of dictionaries \(\mathbf{\Phi}\) and \(\mathbf{\
abla}\) of size \(100\times 200\) from 65,000 patches, which are of size 10 \(\times\) 10 randomly sampled from six natural images. We apply the frame reconstruction function \(\mathbf{\Phi}\mathcal{S}_{\lambda}(\mathbf{\
abla}^{T}\mathbf{x})\) 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. ### 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

\[\mathbf{y}=\mathcal{S}_{\lambda}(\mathbf{\
abla}^{T}\mathbf{x}),\ \ \|\mathbf{y}\|\leq s \tag{43}\]

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 (\(\mathbf{\Phi}\), \(\mathbf{\
abla}\)) in our stable sparse model with non-tight frame (SSM-NTF) model training by natural images. which analyzes the signals in \(\mathbf{\Psi}\) domain. The synthesis system is denoted as

\[\mathbf{x}=\mathbf{\Phi}\mathbf{y},\ \ \|\mathbf{y}\|\leq s \tag{44}\]

which reconstructs the analyzed signals. The two systems share the same sparse coefficients \(\mathbf{Y}\). 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. #### 7.2.1 Natural Image Denoising

We now turn to present experimental results on six classical natural images named 'Barbara', 'Boat', 'Couple', 'Hill', 'Lena' and 'Man' which are shown in [1], to evaluate the performance of the training algorithm. The denoising problem which has been widely studied in sparse representation is used as the target application. We add Gaussian white noise to these images at different noise levels \(\sigma=20,30,40,50,60,70,80,90,100\). Then we use the learned dictionary pair to denoise the natural images, with overlap of 1 pixel between adjacent patches of size \(10\times 10\). The patch denoising stage is followed by weighted averaging the overlapping patch recoveries to obtain the final clean image. The parameters in our scheme are \(\gamma_{1}=1.1\) and \(\gamma_{3}=1.2(L/M)^{2}\)where \(L\) and \(M\) are the sample and dictionary size, respectively. We have stated in Section 3.2 that we usually set \(A\) to a positive number around but smaller than 1. In fact, we set \(A\) from 0.6 to 1 by a step of 0.03 to test the denoising performance to determine the specific value of it. Then, with fixed \(A\), we set \(B\) from 1 to 4 with a step length of 0.3 to run experiments on every noise level to determine the values of \(B\).