# Data-Driven Redundant Transform Based on Parseval Frames
## 4 Image Denoising
### Image Denoising
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\sigma\) & **Image** & **Barbara** & **Boat** & **Couple** & **Hill** & **Lena** & **Man** & **Average** \\ \hline \multirow{8}{*}{20} & K-SVD [14] & 31.01 & 30.50 & 30.15 & 30.27 & 32.51 & 30.26 & 30.78 \\ \cline{2-9}  & T.KSVD [3] & 30.02 & 29.30 & 29.25 & 29.21 & 31.45 & 29.01 & 29.71 \\ \cline{2-9}  & DTF [33] & 31.07 & 30.35 & 30.20 & 30.31 & 32.56 & 30.07 & 30.76 \\ \cline{2-9}  & SSM-NTF & 31.01 & 30.47 & 30.24 & 30.34 & 32.50 & 30.23 & 30.80 \\ \cline{2-9}  & BM3D [35] & 32.01 & 31.02 & 30.88 & 30.85 & 33.19 & 30.83 & 31.47 \\ \cline{2-9}  & WNNM [36] & 32.31 & 31.09 & 30.92 & 30.94 & 33.18 & 30.84 & 31.55 \\ \cline{2-9}  & K-SVD [14] & 28.75 & 28.60 & 28.07 & 28.51 & 30.59 & 28.43 & 28.83 \\ \cline{2-9}  & T.KSVD [3] & 27.78 & 27.86 & 27.46 & 27.23 & 29.25 & 27.13 & 27.79 \\ \cline{2-9}  & DTF [33] & 29.07 & 28.48 & 28.22 & 28.64 & 30.60 & 28.26 & 28.88 \\ \cline{2-9}  & SSM-NTF & 29.00 & 28.63 & 28.24 & 28.66 & 30.73 & 28.49 & 28.96 \\ \cline{2-9}  & BM3D [35] & 30.12 & 29.22 & 28.95 & 29.23 & 31.40 & 29.04 & 29.66 \\ \cline{2-9}  & WNNM [36] & 30.32 & 29.30 & 29.02 & 29.33 & 31.50 & 29.10 & 29.76 \\ \cline{2-9}  & K-SVD [14] & 27.03 & 27.23 & 26.54 & 27.23 & 29.13 & 27.17 & 27.39 \\ \cline{2-9}  & T.KSVD [3] & 26.35 & 26.45 & 25.98 & 26.45 & 28.20 & 26.14 & 26.60 \\ \cline{2-9}  & DTF [33] & 27.58 & 27.20 & 26.87 & 27.49 & 29.25 & 26.99 & 27.56 \\ \cline{2-9}  & SSM-NTF & 27.50 & 27.39 & 27.00 & 27.56 & 29.35 & 27.34 & 27.69 \\ \cline{2-9}  & BM3D [35] & 28.68 & 27.92 & 27.58 & 28.08 & 30.11 & 27.83 & 28.37 \\ \cline{2-9}  & WNNM [36] & 28.85 & 27.99 & 27.64 & 28.18 & 30.25 & 27.90 & 28.47 \\ \cline{2-9}  & K-SVD [14] & 25.71 & 26.05 & 25.42 & 26.29 & 27.92 & 26.18 & 26.26 \\ \cline{2-9}  & T.KSVD [3] & 25.10 & 25.56 & 25.03 & 25.89 & 27.01 & 25.40 & 25.67 \\ \cline{2-9}  & DTF [33] & 26.45 & 26.15 & 25.84 & 26.63 & 28.15 & 26.09 & 26.55 \\ \cline{2-9}  & SSM-NTF & 26.43 & 26.32 & 25.99 & 26.79 & 28.40 & 26.40 & 26.72 \\ \cline{2-9}  & BM3D [35] & 27.48 & 26.89 & 26.49 & 27.20 & 29.06 & 26.94 & 27.34 \\ \cline{2-9}  & WNNM [36] & 27.70 & 26.97 & 26.60 & 27.35 & 29.23 & 27.01 & 27.48 \\ \cline{2-9}  & K-SVD [14] & 24.45 & 25.18 & 24.57 & 25.69 & 27.01 & 25.40 & 25.38 \\ \cline{2-9}  & T.KSVD [3] & 24.50 & 24.88 & 24.36 & 25.40 & 26.60 & 24.78 & 25.09 \\ \cline{2-9}  & DTF [33] & 25.64 & 25.33 & 25.04 & 25.91 & 27.22 & 25.38 & 25.75 \\ \cline{2-9}  & SSM-NTF & 25.50 & 25.45 & 25.14 & 26.03 & 27.33 & 25.67 & 25.85 \\ \cline{2-9}  & BM3D [35] & 26.36 & 26.02 & 25.61 & 26.44 & 28.14 & 26.18 & 26.46 \\ \cline{2-9}  & WNNM [36] & 26.59 & 26.12 & 25.74 & 26.60 & 28.33 & 26.26 & 26.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Average PSNR results of different noise levels on six images. learning method but not focus on image denoising. Our model is plain without applying any extra prior, besides the basic sparsity characteristics of the signals. The experimental results demonstrate that our proposed models can achieve better performance than traditional sparse models in image denoising. However, the methods BM3D and WNNM are based on image nonlocal self-similarity (NSS). The NSS prior refers to the fact that for a given local patch in a natural image, one can find many similar patches to it across the image. Intuitively, by stacking nonlocal similar patch vectors into a matrix, this matrix should be a low-rank matrix and have sparse singular values. The exploitation of NSS has been used to significantly boost image denoising performance. We have not involved this prior into our model.