# Data-Driven Redundant Transform Based on Parseval Frames
## 4 Image Denoising
### Sparsification of Nature Images  
A classic sparsifying transform learning model [21] is formulated as  
\[\min_{\mathbf{W},\mathbf{Y}}\|\mathbf{W}\mathbf{X}-\mathbf{Y}\|_{F}^ {2}-\lambda\log\det\mathbf{W}+\mu\|\mathbf{W}\|_{F}^{2}\] \[\text{s.t. }\|\mathbf{Y}_{i}\|_{0}\leq s\lor i, \tag{28}\]  
where \(\mathbf{X}\) is the training data, \(\mathbf{Y}\) are the sparse coefficients, and \(\mathbf{W}\) is the learnt transform. The quality of the learnt transforms in the experiment [21] was judged based on their condition number and sparsification error. Similar to the experimental setting in [21], we also evaluated the effectiveness of the transforms learnt from our DRTPF by their condition number and sparsification error. The \(\mathbf{I}_{2}\)-norm condition number of the transform operator \(\mathbf{\Phi}\) is denoted as the ratio of the maximum singular value to the minimum singular value of \(\mathbf{\Phi}\); that is,  
\[\mathcal{K}_{\mathbf{\Phi}}=\frac{\delta_{max}(\mathbf{\Phi})}{\delta_{min}( \mathbf{\Phi})}. \tag{29}\]  
In our case, the condition number \(\mathcal{K}_{\mathbf{\Phi}}=1\), as the maximum and minimum singular values (which are determined by the optimal frame bounds) must be equal to \(1\). Similarly, we can obtain that \(\mathcal{K}_{\mathbf{Y}}=1\). It is the best case when the transform operators have condition number equal to \(1\). The sparsification error of the model (28) is defined as  
\[SE=\|\mathbf{W}\mathbf{X}-\mathbf{Y}\|_{F}^{2}. \tag{30}\]  
Similarly, we define the'sparsification error' of the proposed DRTPF, to measure the energy loss due to sparse representation, which is formulated as  
\[\widetilde{SE}=\|\mathbf{Y}-\mathcal{S}_{\lambda}(\mathbf{\Psi}^{T}\mathbf{X} )\|_{F}^{2}. \tag{31}\]  
Figure 3: Reconstruction of _barbara_ using DRTPF (**left**) and K-SVD [14] (**right**). Top: \(\sigma=30\); bottom: \(\sigma=50\). The'sparsification error' indicates the compact ability of the transform \(\mathbf{\Psi}\) with reasonable ignorance of the thresholding operator \(\mathcal{S}_{\lambda}(\cdot)\). To demonstrate that our model and algorithms are insensitive to the initialized transforms, we applied the proposed sparse coding and transform operator pair learning algorithms to train a pair of transforms. The training data are patches of size \(10\times 10\) extracted from the image 'Babara' which is shown in Figure 4. The trained transform pair are of size \(100\times 200\). We extracted the patches with non-overlap and removed the DC values of every sample. 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. The matrices used for initialization were the 1D DCT matrix, the matrix with random columns sampled from the training data, and the redundant identity matrix. As the transform for DRTPF is redundant, the redundant identity matrix here is formed as \(\left[\mathbf{I}\ \mathbf{I}\right]\) where \(\mathbf{I}\) is the identity matrix of size \(100\times 100\). The convergence curve of the objective function and the'sparsification error' are shown in Figure 5. From the left sub-figure of Figure 5 we know that our proposed algorithm for DRTPF is converged, and all the initializations converge to the same result after about 20 iterations which demonstrate that our proposed DRTPF and the corresponding algorithm are insensitive to different initializations. The right sub-figure of Figure 5 shows the'sparsification error' of the three initialized methods, the 2D DCT transform of and the KLT transform. The 2D DCT is formed by the Kronecker product of two 1D DCT transform, i.e., \(\mathbf{D}=\mathbf{D}_{0}\otimes\mathbf{D}_{0}\), where \(\mathbf{D}_{0}\) is the 1D DCT transform of size \(8\times 8\) and \(\otimes\) denotes the Kronecker product. The KLT transform \(\mathbf{K}\) of size \(64\times 64\) is obtained by principle component analysis (PCA) method. The'sparsification error' of 2D DCT and KLT are calculated via the model in [21] at iteration zero.