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 *σ* = 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 × 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 *γ*<sup>1</sup> = 1.1 and *γ*<sup>3</sup> = 1.2(*L*/*M*)2where *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*. The values of frame bounds *A* and *B* are shown in Table 1. For example, when the noise level *σ* = 40, *A* and *B* are set to be 0.8 and 1.8, respectively.

**Table 1.** The valuesof *A* and *B*.


Table 2 shows the comparison results in terms of PSNR. There are three related image denoising methods involved, including the classical dictionary learning algorithm KSVD [9], the data-driven tight frame based denoising method [1] and the incoherent dictionary learning based method [21]. The patch size of KSVD [9] and the method in [21] are 8 × 8 with stripe 1 and the dictionaries are of size 64 × 256 at their optimal state according to the previous work. We point out that [1] works on filters of size 16 × 16 instead of image patches and initialized by 64 3-level Harr wavelet filters in size 16 × 16. All the three compared methods can achieve their best performance with 50 iterations.

Table 2 shows that the incoherent dictionary learning method [21] outperforms the KSVD [9] in average as the mutual incoherent of dictionary can provide stable recovery. That [1] outperforms [21] implies that the tight frame is a more stable system. Then our stable sparse model based method outperforms [1] in average suggests that applying non-tight frame to approximate RIP can provide even better and more stable reconstruction results. Figure 3 shows two exemplified visual results on images 'Man' and 'Couple' at noise levels *σ* = 50 and *σ* = 40, respectively. The proposed method shows much clearer and better visual results than the other competing methods.

**Figure 3.** Visual comparison of reconstruction results by different methods on 'Man' (*σ* = 50) and 'Couple' (*σ* = 40). From left to right: original image, noise image, KSVD [9], method of [1], method of [21] and our proposed method.

**Table 2.** PSNR (dB) for nature image denoising results.



**Table 2.** *Cont.*

#### 7.2.2. Piecewise Constant Image Denoising

In this subsection, we demonstrate the analytical property of our SSM-NTF model using a synthetic image. The denoising problem which has been widely studied in sparse representation is used as the target application. We start with a piecewise constant of size 256 × 256 contaminated by Gaussian white noise with noise level *σ* = 5 and extract all possible 5 × 5 image pathes. For the denoising we apply the dictionary pair learning algorithm with the parameters *γ*<sup>1</sup> = 1.5, *γ*<sup>3</sup> = 1.2 *L*/*M*, *A* = 0.8 and *B* = 1.8 in parallel with patch denosing with the synthesis KSVD [9] and the analysis KSVD [14]. We apply 100 iterations of the our dictionary learning method on this training set, and learning dictionary pair of size 25 × 50. The experimental set of the synthesis KSVD [9] and the analysis KSVD [14] are at their optimal state according to the previous work.

The learned dictionary pair **Φ** which exhibits much like the synthesis dictionary and **Ψ** which exhibits a high resemblance to the analysis dictionary are illustrated in Figure 4. The resulting PSNRs of the denoised images are 45.32 dB for Analysis KSVD, 43.60 dB for Synthesis KSVD, and 45.17 dB for our proposed algorithm. The figure shows that our dictionary pair learning method is able to capture the features of the piecewise constant image. Figure 5 shows the absolute difference images for each of the three methods. Note that these images are displayed in the dynamic range [0, 20]to make the differences more pronounced. Our proposed approach leads to a much better denoising result than the synthesis KSVD and is comparable with the analysis KSVD.

**Figure 4.** The exemplified dictionaries training by the piecewise constant. From left to right: Synthesis KSVD (25 × 100), our proposed dictionary pair (25 × 50) and analysis KSVD (25 × 50).

**Figure 5.** Visual quality comparison of denoising results for piecewise constant image. Images of the absolute errors are displayed in the dynamic range [0,20] (from left to right): Original image, noise image, analysis KSVD [14], synthesis KSVD [9], our proposed method.
