# Data-Driven Redundant Transform Based on Parseval Frames
## 3 Data-Driven Redundant Transform Model Based on Tight Frame
### Data-Driven Redundant Transform  
In this subsection, we first propose a threshold-based reconstruction function, with the assumption that the signal is sparse in the dual frame domain. Then, we present the data-driven redundant transform based on Parseval frames model. Let \(\{\boldsymbol{\phi}_{i}\}_{i=1}^{M}\) be a frame and \(\{\boldsymbol{\psi}_{i}\}_{i=1}^{M}\) be its dual frame. For convenience, we stack them as the matrices \(\boldsymbol{\Phi}=[\boldsymbol{\phi}_{1},\boldsymbol{\phi}_{2},\ldots, \boldsymbol{\phi}_{M}]\) and \(\boldsymbol{\
abla}=[\boldsymbol{\psi}_{1},\boldsymbol{\psi}_{2},\ldots, \boldsymbol{\psi}_{N}]\), respectively. Let \(\mathbf{x}=\hat{\mathbf{x}}+\mathbf{e}\) be a signal vector, where \(\hat{\mathbf{x}}\) is the original noiseless signal and \(\mathbf{e}\) is a zero-mean white Gaussian noise. The frame reconstruction function (6) can be formulated as \(\mathbf{x}=\boldsymbol{\Phi}\boldsymbol{\
abla}^{T}\mathbf{x}=\boldsymbol{ \Phi}\boldsymbol{\
abla}^{T}(\hat{\mathbf{x}}+\mathbf{e})\). By assuming the sparse prior of signals over the \(\boldsymbol{\
abla}\) domain, we apply a columnwise hard thresholding operator \(\mathcal{S}_{\lambda}(\cdot)\) (which shall be defined in the next subsection) on \(\boldsymbol{\
abla}^{T}(\hat{\mathbf{x}}+\mathbf{e})\), such that  
\[\hat{\mathbf{x}}=\boldsymbol{\Phi}\mathcal{S}_{\lambda}(\boldsymbol{\
abla}^{ T}\mathbf{x}), \tag{7}\]  
where \(\lambda\) is a vector with elements \(\lambda_{i}\) corresponding to \(\boldsymbol{\psi}_{i}\), \(i=1,2,\ldots,M\). Apparently, \(\mathcal{S}_{\lambda}(\boldsymbol{\
abla}^{T}\mathbf{x})\) is the sparse coefficients of \(\mathbf{x}\) under \(\boldsymbol{\
abla}\) in the sense of an analysis model, while it also serves as the sparse coefficients under \(\boldsymbol{\Phi}\) in the sense of a synthesis model. In other words, Equation (7) admits that the synthesis and analysis models share almost the same sparse coefficients. As we all know, the standard orthogonal basis, which is a significant tool in signal representation and transformation, is a special kind of frame with frame bounds \(A=B=1\). In fact, the standard orthogonal basis is a special case of a Parseval frame. In order to exceed the so-called _perfect reconstruction property_ of the standard orthogonal basis in signal representation and transform, we refer to the Parseval frame. Therefore, we propose the data-driven redundant transform based on Parseval frame (DRTPF), as follows\[\mathbf{y} \leftarrow\mathcal{S}_{\lambda}(\mathbf{\Psi}^{T}\mathbf{x}), \tag{8}\] \[\widehat{\mathbf{x}} \leftarrow\mathbf{\Phi}\mathbf{y},\] (9) s.t. \[\mathbf{\Psi}\mathbf{\Phi}=\mathbf{I},\] (10) \[|\mathbf{y}\|_{0}\leq s,\] \[\sum_{i=1}^{M}| <\mathbf{x},\boldsymbol{\phi}_{i}>|^{2}=\|\mathbf{x}\|_{2}^{2}, \tag{11}\]  
where (8) is the forward transform and (9) is the backward transform. The relationship between \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\) is formulated as (10), which implies the relationship between the frame and its dual frame. The forward transform operator \(\mathbf{\Psi}\) is also a Parseval frame, as it is a dual frame of \(\mathbf{\Phi}\). Thus, the projection of the signal \(\mathbf{x}\) over the \(\mathbf{\Psi}\) domain can be formulated as  
\[\sum_{i=1}^{M}|<\mathbf{x},\boldsymbol{\psi}_{i}>|^{2}=\|\mathbf{x}\|_{2}^{2}. \tag{12}\]  
Equation (12) indicates that the transform coefficients of the proposed DRTBF are bounded by the original signal \(\mathbf{x}\). This constraint leads to a more robust result than traditional sparse models. To convert DRTPF into an optimization problem, (11) can be written as the more compact expression \(\mathbf{\Phi}\mathbf{\Phi}^{T}=\mathbf{I}\), which characterizes \(\mathbf{\Phi}\) in a way that is unrelated to the data. This property indicates that the rows of the frame \(\mathbf{\Phi}\) are orthogonal, thus satisfying the so-called _perfect reconstruction property_ which ensures that a given signal can be perfectly represented by its canonical expansion (in a manner similar to orthogonal bases).