invariant wavelet frame. For $\bq\in\ell_1(\Z)$, define a convolution operator $\bsS_{\bq}:\ell_2(\Z)\to\ell_2(\Z)$ by $$\begin{aligned}
\label{DiscreteConv}
(\bsS_{\bq}\bu)[n]=(\bq\ast\bu)[n]=\sum_{k\in\Z}\bq[n-k]\bu[k]~~~~~\text{for}~~~\bu\in\ell_2(\Z).\end{aligned}$$ Given a set of finitely supported filters $\{\bq_1,\cdots,\bq_m\}$, define the analysis operator $\bsW$ and the synthesis operator $\bsW^T$ by $$\begin{aligned}
\bsW&=\left[\bsS_{\bq_1[-\cdot]}^T,\bsS_{\bq_2[-\cdot]}^T,\cdots,\bsS_{\bq_m[-\cdot]}^T\right]^T,\label{AnalConv}\\
\bsW^T&=\left[\bsS_{\bq_1},\bsS_{\bq_2},\cdots,\bsS_{\bq_m}\right],\label{SyntConv}\end{aligned}$$ respectively. Then, the direct computation can show that the rows of $\bsW$ form a tight frame on $\ell_2(\Z)$ (i.e. $\bsW^T\bsW=\bsI$) if and only if the filters $\{\bq_1,\cdots,\bq_m\}$ satisfy the following [*unitary extension principle*]{} (UEP) condition [@B.Han2011]: $$\begin{aligned}
\label{SomeUEP}
\sum_{l=1}^m\sum_{k\in\Z}\bq_l[n+k]\bq_l[k]=\dde_n=\left\{\begin{array}{cl}
1~&\text{if}~n=0,\vspace{0.4em}\\
0~&\text{if}~n\neq0.
\end{array}\right.\end{aligned}$$ Moreover, once the one dimensional filters generate a wavelet tight frame on $\ell_2(\Z)$, the higher dimensional wavelet tight frame could be obtained via the tensor product of the one dimensional filters.

In the literature, wavelet frames are widely used for the sparse approximation of an image. This is due to the multiscale structure of the wavelet frame systems, short supports of the framelet functions with varied vanishing moments, and the presence of both low pass and high pass filters in the wavelet frame filter banks, which are desirable in sparsely approximating piecewise smooth functions [@B.Dong2017]. For better sparse approximation of an image, the authors in [@J.F.Cai2014] proposed a *data driven tight frame* approach. Specifically, given an image $\bu$, a tight frame system 