**2. Related Work**

Let **H** be an *N*-dimensional discrete Hilbert space. A sequence {*φ<sup>i</sup>* }*M <sup>i</sup>*=<sup>1</sup> ∈ **H** is a frame if and only if there exist two positive numbers *A* and *B* such that [30]

$$\mathcal{A} \|\mathbf{x}\|\_{2}^{2} \le \sum\_{i=1}^{M} |<\mathbf{x}, \boldsymbol{\Phi}\_{i} >|^{2} \le \|\mathbf{B}\|\mathbf{x}\|\_{2}^{2} \quad \forall \mathbf{x} \in \mathbf{H}^{N}.\tag{2}$$

*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, {*φ<sup>i</sup>* }*M <sup>i</sup>*=<sup>1</sup> is tight if *A* = *B* is possible [30]. In particular, {*φ<sup>i</sup>* }*M <sup>i</sup>*=<sup>1</sup> is a Parseval frame if *A* = *B* = 1 is satisfied. There are two associated operators can be defined between the Hilbert space **H***<sup>N</sup>* and a Square integrable Space **l***<sup>M</sup>* <sup>2</sup> (·) once a frame is defined: One is the analysis operator, **T**, defined by

$$(\mathbf{T}\mathbf{x})\_i = <\mathbf{x}, \boldsymbol{\Phi}\_j >\_{\prime} \quad \forall \mathbf{x} \in \mathbf{H}^N,\tag{3}$$

and the other is its adjoint operator, **T**∗, which is called the synthesis operator:

$$\mathbf{T}^\*\mathbf{c} = \sum\_{i=1}^{M} \mathbf{c}\boldsymbol{\Phi}\_i \quad \forall \mathbf{c} = (\mathbf{c}\_i)\_{i \in f} \in \mathbf{l}\_2^M(\mathbf{T}).\tag{4}$$

Then, the frame operator can be defined by the following canonical expansion

$$\mathbf{Fx} = \mathbf{T}^\* \mathbf{T} \mathbf{x} = \sum\_{i=1}^{M} < \mathbf{x}\_i \boldsymbol{\Phi}\_i > \boldsymbol{\Phi}\_i. \tag{5}$$

Let **<sup>x</sup>** <sup>∈</sup> <sup>R</sup>*<sup>N</sup>* be an arbitrary vector in **<sup>H</sup>**. A reconstruction function is an expression with the following form

$$\mathbf{x} = \sum\_{i=1}^{M} < \mathbf{x}\_{\prime} \boldsymbol{\Psi}\_{i} > \boldsymbol{\Phi}\_{i\prime} \quad \forall \mathbf{x} \in \mathbf{H}\_{\prime} \tag{6}$$

where the sequence {*ψ<sup>i</sup>* }*M <sup>i</sup>*=<sup>1</sup> ∈ **H** is called the dual frame of {*φ<sup>i</sup>* }*M <sup>i</sup>*=1. Obviously, {*ψ<sup>i</sup>* }*M <sup>i</sup>*=<sup>1</sup> is not unique, unless {*φ<sup>i</sup>* }*M <sup>i</sup>*=<sup>1</sup> is an orthogonal basis. In fact, for an arbitrary given frame {*φ<sup>i</sup>* }*M <sup>i</sup>*=1, 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 **Φ** and its dual frame **Ψ** can be stacked as the matrices **Φ** = [*φ*1, *φ*2, ... , *φM*] and **Ψ** = [*ψ*1, *ψ*2, ... , *ψN*], respectively. The matrices can be regard as sparse representation dictionaries, transform operators and so on. A frame **Φ** 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 **Φ** is applied as sparse representation dictionary or transform operator, its condition number are determined by *<sup>B</sup> <sup>A</sup>* . In this way, the model will never provide degenerate dictionary or transform. In fact, frames are matrices with special structure.
