# Data-Driven Redundant Transform Based on Parseval Frames
## 2 Related Work  
Let \(\mathbf{H}\) be an \(N\)-dimensional discrete Hilbert space. A sequence \(\{\boldsymbol{\phi}_{i}\}_{i=1}^{M}\in\mathbf{H}\) is a frame if and only if there exist two positive numbers \(A\) and \(B\) such that [30]  
\[A\|\mathbf{x}\|_{2}^{2}\leq\sum_{i=1}^{M}|<\mathbf{x},\boldsymbol{\phi}_{i}>| ^{2}\leq\ B\|\mathbf{x}\|_{2}^{2}\quad\forall\mathbf{x}\in\mathbf{H}^{N}. \tag{2}\]  
\(A\) and \(B\) are called the bound of the frame and we call formula 2 the frame condition, as it is a termination of frame. Furthermore, \(\{\boldsymbol{\phi}_{i}\}_{i=1}^{M}\) is tight if \(A=B\) is possible [30]. In particular, \(\{\boldsymbol{\phi}_{i}\}_{i=1}^{M}\) is a Parseval frame if \(A=B=1\) is satisfied. There are two associated operators can be defined between the Hilbert space \(\mathbf{H}^{N}\) and a Square integrable Space \(\mathbf{l}_{2}^{M}(\cdot)\) once a frame is defined: One is the analysis operator, \(\mathbf{T}\), defined by  
\[(\mathbf{T}\mathbf{x})_{i}=<\mathbf{x},\boldsymbol{\phi}_{i}>,\quad\forall \mathbf{x}\in\mathbf{H}^{N}, \tag{3}\]  
and the other is its adjoint operator, \(\mathbf{T}^{*}\), which is called the synthesis operator:  
\[\mathbf{T}^{*}\mathbf{c}=\sum_{i=1}^{M}\mathbf{c}\boldsymbol{\phi}_{i}\quad \forall\mathbf{c}=(\mathbf{c}_{i})_{i\in I}\in\mathbf{l}_{2}^{M}(\mathbf{T}). \tag{4}\]  
Then, the frame operator can be defined by the following canonical expansion  
\[\mathbf{F}\mathbf{x}=\mathbf{T}^{*}\mathbf{T}\mathbf{x}=\sum_{i=1}^{M}< \mathbf{x},\boldsymbol{\phi}_{i}>\boldsymbol{\phi}_{i}. \tag{5}\]Let \(\mathbf{x}\in\mathbb{R}^{N}\) be an arbitrary vector in \(\mathbf{H}\). A reconstruction function is an expression with the following form  
\[\mathbf{x}=\sum_{i=1}^{M}<\mathbf{x},\boldsymbol{\psi}_{i}>\boldsymbol{\phi}_{i},\quad\forall\mathbf{x}\in\mathbf{H}, \tag{6}\]  
where the sequence \(\{\boldsymbol{\psi}_{i}\}_{i=1}^{M}\in\mathbf{H}\) is called the dual frame of \(\{\boldsymbol{\phi}_{i}\}_{i=1}^{M}\). Obviously, \(\{\boldsymbol{\psi}_{i}\}_{i=1}^{M}\) is not unique, unless \(\{\boldsymbol{\phi}_{i}\}_{i=1}^{M}\) is an orthogonal basis. In fact, for an arbitrary given frame \(\{\boldsymbol{\phi}_{i}\}_{i=1}^{M}\), there is a series of dual frames corresponding to it. The non-uniqueness of the dual frame allows us to achieve a better expression of the signal by optimizing the dual frame.  
The frame \(\boldsymbol{\Phi}\) and its dual frame \(\boldsymbol{\
abla}\) can be stacked 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. The matrices can be regard as sparse representation dictionaries, transform operators and so on. A frame \(\boldsymbol{\Phi}\) with the bounds \(A\) and \(B\) means that the maximum and minimum singular values of it are equal to \(A\) and \(B\) respectively. What' more, the singular values of tight frame are all equal, particularly, the singular values of Parseval frame are all equal to \(1\). Thus, when the frame \(\boldsymbol{\Phi}\) is applied as sparse representation dictionary or transform operator, its condition number are determined by \(\frac{B}{A}\). In this way, the model will never provide degenerate dictionary or transform. In fact, frames are matrices with special structure.