# Data-Driven Redundant Transform Based on Parseval Frames
## 4 Image Denoising
### Image Denoising  
In this subsection, we evaluate the performance of our DRTPF model using six natural images of size \(512\times 512\), which are shown in Figure 4. We added Gaussian white noise to these images at different noise levels (\(\sigma=20,30,40,50,60\)). We set the parameters \(\eta_{1}=1.1\) and \(\eta_{3}=1e+7\), and \(\eta_{2}\) was replaced by the \(\ell_{0}\) thresholding \(0.6\sigma\), as before. We compared DRTPF with the three most related methods of sparse representation: K-SVD [14], the overcomplete transform (T.KSVD) [3], the learning-based frame (DTF [33]), the BM3D [35] and WNNM [36]. The BM3D and WNNM are nonlocal-based methods with the parameters setting as in corresponding paper. We note that DTF works on filters, instead of image patches. In the experiment, our DRTPF method and K-SVD were the same as in Section 5.1. All methods were trained iteratively (25 times). The DTF method was initialized by 64 3-level Harr wavelet filters of size \(16\times 16\). The operator size of the T.KSVD method was \(128\times 64\) and the patch size it worked on was \(8\times 8\) overlapping mean-subtracted patches. The hard thresholding was \(s=30\). Table 1 shows the comparison results, in terms of average PSNR. As shown in Table 1, our DRTPF method and the DTF method outperformed K-SVD and T.KSVD on most images, i.e., our proposed DRTPF outperforms K-SVD for 0.47 dB and outperforms T.KSVD for 0.76 dB at noise level \(\sigma=60\). This result implies that methods using frames are more robust against noise. Furthermore, the higher the noise level, the better the results of DRTPF method and the DTF method than K-SVD and T.KSVD. We can also see that our DRTPF method outperformed DTF on most of the images, especially when the noise level was very high. In fact, in our model, the sparse coefficients are calculated accurately by the inner product of the signals and the frame \(\mathbf{\
abla}\), and are limited to a certain range. Theoretically,  
Figure 5: Convergence Curve and Sparsification Error. **Left**: The \(X\)-label is the iteration number. **Right**: The \(Y\)-label is the objective function and the sparsification error, respectively. It can be seen that our DRTPF learning algorithm is convergent and insensitive to initialization. Figure 6: The average Sparsity and Recovery PSNR. **Left**: The \(X\)-label is the iteration number and the \(Y\)-label is the average sparsity. **Right**: The \(X\)-label is the average sparsity and the \(Y\)-label is the average Recovery PSNR. it should perform better than the compared method. Figure 7 shows two exemplified visual results on the images 'Boat' and 'Man' at noise level \(\sigma=40\). The PSNR of the K-SVD, T.KSVD, DTF, and the proposed DRTPF are 27.17 dB, 26.14 dB, 26.99 dB, 27.34 dB for 'Man' and 27.23 dB, 26.45 dB, 27.20 dB and 27.39 dB for 'Boat'. Our proposed DRTPF and the DTF method provide more features and higher PSNR values of the two images than K-SVD and T.KSVD. Though the DTF provides higher PSNR values than K-SVD and T.KSVD, and better visual performance, the results of this method suffer from deformation and margin smoothing as it based on filter. The proposed DRTPF shows much clearer and better visual results than the other competing methods without any deformation. All the six methods can be classified to two categories (1) without any extra constraint, e.g., nonlocal similarity, and (2) with additional prior like nonlocal similarity. Our proposed DRTPF belongs to category (1). We would like to point out that our goal was to establish a redundant transform