# Data-Driven Redundant Transform Based on Parseval Frames
## 3 Data-Driven Redundant Transform Model Based on Tight Frame
### Transform Learning for the Drtbf Model
#### 3.2.1 Sparse Coding Phase  
This subsection presents the sparse coding method for the proposed DRTBF model, in which the sparse coefficients of \(\mathbf{Y}\) are obtained by OMP, and the threshold values \(\boldsymbol{\lambda}\) are obtained by a designed elementwise method.  
The **Y** Subproblem  
The pursuit of **Y** is equivalent to solving the following problem with fixed \(\mathbf{\Phi}\), \(\mathbf{\Psi}\), and \(\mathbf{\lambda}\):  
\[\hat{\textbf{Y}}=\operatorname*{arg\,min}_{\textbf{Y}}\|\textbf{X}-\mathbf{\Phi} \textbf{Y}\|_{F}^{2}+\eta_{1}\|\textbf{Y}-\mathcal{S}_{\mathbf{\lambda}}(\mathbf{\Psi} ^{T}\textbf{x})\|_{F}^{2}+\eta_{2}\|\textbf{Y}\|_{0}, \tag{14}\]  
which can be easily solved by OMP [14; 34], as (14) can be easily converted to the classical synthesis sparse expression \(\min\|\textbf{Z}-\textbf{D}\textbf{Y}\|_{F}^{2}\) such that \(\|\textbf{Y}\|_{0}\), where \(\textbf{Z}=[\textbf{X}\sqrt{\eta_{1}}\mathcal{S}_{\mathbf{\lambda}}(\mathbf{\Psi}^{T} \textbf{x})]\) and \(\textbf{D}=[\mathbf{\Phi}\ \sqrt{\eta_{1}}\textbf{I}]\).  
The \(\mathbf{\lambda}\) Subproblem  
With fixed \(\mathbf{\Phi}\), \(\mathbf{\Psi}\), and **Y**, finding \(\mathbf{\lambda}\) is equivalent to solving the following problem  
\[\hat{\mathbf{\lambda}}=\operatorname*{arg\,min}_{\mathbf{\lambda}}\|\textbf{Y}- \mathcal{S}_{\mathbf{\lambda}}(\mathbf{\Psi}^{T}\textbf{x})\|_{F}^{2}, \tag{15}\]  
which can be decomposed into \(M\) individual optimization problems \(\operatorname*{arg\,min}_{\lambda_{i}}\|\mathbf{\overline{y}}_{i}-\mathcal{S}_{ \lambda_{i}}(\mathbf{\psi}_{i}^{T}\textbf{X})\|_{2}^{2}\), \(i=1,\ldots,M\). By denoting \(\mathcal{J}_{i}:=\operatorname*{supp}(\mathcal{S}_{\lambda_{i}}(\mathbf{\psi}_{i} ^{T}\textbf{X}))\) to be the set of indices of non-zero elements of \(\mathcal{S}_{\lambda_{i}}(\mathbf{\psi}_{i}^{T}\textbf{X})\), we have  
\[\mathcal{S}_{\lambda_{i}}(\mathbf{\Psi}^{T}\textbf{x}_{j}) =\mathbf{\Psi}^{T}\textbf{x}_{j},\ \forall j\in\mathcal{J}_{i}\] \[\mathcal{S}_{\lambda_{i}}(\mathbf{\Psi}^{T}\textbf{x}_{j}) =0,\ \forall j\in\{1,\ldots,L\}\setminus\mathcal{J}_{i}.\]  
As the cardinality of \(\mathcal{J}_{i}\) depends on \(\lambda_{i}\), we transform (15) to another optimization problem:  
\[\hat{\lambda}_{i}=\operatorname*{arg\,min}_{\lambda_{i}}\ \underbrace{\sum_{j\in\{1,\ldots,L\} \setminus\mathcal{J}_{i}}y_{ij}^{2}}_{f(\lambda_{i})}+\underbrace{\sum_{j\in \mathcal{J}_{i}}(y_{ij}-\mathbf{\psi}_{i}^{T}\textbf{x}_{j})^{2}}_{g(\lambda_{i})}, \tag{16}\]  
where \(y_{ij}\) denotes the \((i,j)\)th entry of **Y** and \(\textbf{x}_{i}\) denotes the \(i\)th column of **X**. Denote \(l(\lambda_{i})\) as  
\[l(\lambda_{i})=\sum_{j\in 1,2,\cdots,L\setminus\mathcal{J}}(y_{ij}-\mathbf{\psi}_{i} ^{T}\textbf{x}_{j})^{2} \tag{17}\]  
We observe that the function \(f(\lambda_{i})\) is a monotonically increasing function and that \(g(\lambda_{i})\) is monotonically decreasing. We take \(\mathbf{\psi}_{i}^{T}\textbf{x}_{i},i=1,2,\ldots,L\) as candidates and compute all the values of \(f(\lambda_{i})+g(\lambda_{i})\). Then, the optimal \(\lambda_{i}\) should lie in an interval determined by \(\mathbf{\psi}_{i}^{T}\textbf{x}_{k}\) and \(\mathbf{\psi}_{i}^{T}\textbf{x}_{l}\), which correspond to the smallest and the second smallest values of \(f(\lambda_{i})+g(\lambda_{i})\), respectively. Then, any suitable value for \(\lambda_{i}\) can be selected. The algorithm for the threshold is summarized as Algorithm 1.