comes from the fact that $\bsW_n\in\mD$ for all $n\in\N$. This completes the proof.

Experimental results {#Experiments}
====================

In this section, we present the experimental results on the phantom image and the real MR image used in [@G.Ongie2016], to compare the proposed DDTF based CS-MRI model \[ProposedCSMRIModel\] with several existing methods. Note that the major focus of this paper is to reconstruct a piecewise constant MR image from a given undersampled k-space data. For this purpose, we choose to compare with the total variation (TV) model [@M.Lustig2007] $$\begin{aligned}
\label{TVModel}
\min_{\bu}~\f{1}{2}\left\|\bmP_{\MM}\bmsF\bu-\bsf\right\|_2^2+\gamma\left\|\Na\bu\right\|_1,\end{aligned}$$ and the Haar framelet (Haar) model (e.g. [@J.F.Cai2012]) $$\begin{aligned}
\label{HaarModel}
\min_{\bu}~\f{1}{2}\left\|\bmP_{\MM}\bmsF\bu-\bsf\right\|_2^2+\left\|\gga\cdot\bsW\bu\right\|_1,\end{aligned}$$ where $\gga=\left\{\gamma 2^{-l}:l=0,\cdots,L-1\right\}$. Both \[TVModel,HaarModel\] are solved by the split Bregman method [@J.F.Cai2009/10; @T.Goldstein2009]. We also compare with the following Schatten $p$-norm minimization model $$\begin{aligned}
\label{Schatten}
\min_{\bsv\in\mC}~\left\|\bmP_{\MM}\bsv-\bsf\right\|_2^2+\gamma\left\|\bmH\left(\La\bsv\right)\right\|_p^p,\end{aligned}$$ with $p=0$, $0.5$, and $1$, solved by the generic iterative reweighted annihilating filters (GIRAF) method [@G.Ongie2017] based on the split Bregman algorithm, which are referred to as “GIRAF$0$”, “GIRAF$0.5$”, and “GIRAF$1$”, respectively. All experiments are implemented on MATLAB $\mathrm{R}2014\mathrm{a}$ running on a laptop with $64\mathrm{GB}$ RAM and Intel(R) Core(TM) CPU $\mathrm{i}7$-$8750\mathrm{H}$ at $2.20\mathrm{GHz}$ with $6$ cores.

      Dataset                 Model                $K$       $r$     $\mu$    $\gamma$    $\beta_1$   $\beta_2$   $\beta_3$  
  --------------- ----------------------------- --------- --------- -------- ----------- ----------- ----------- 