Algorithm 1 L1-minimization

Input: $A=\Phi \Psi ,y,k$  Inizialize:  $\left\{\begin{array}{c}P=[p_1\dots p_N],P=I-A^{†}A,\hfill \\ x_0=A^{†}y\hfill \\ t=0\hfill \end{array}\right.$  Output: k-sparse coefficient vector x  while $t<N_{iter}$ do   $s_t=sgn\left(x_t\right),w_t=(s_t+1)/2$  // reduce $s_t$ to binary vector $w_t$   if $t>0$ then    $v_t=w_t\oplus w_{t-1}$    ${\Omega }_t=\{j/v_t\left(j\right)=1\}$  // define the set of indices ${\Omega }_t$ corresponding to a change from $s_{t-1}$ to $s_t$    $q_t=q_{t-1}+2{\sum }_{j\in {\Omega }_t}{p}_j{s}_t\left(j\right)$   else    $q_t=P{s}_0$    $\mu =\frac{{ϵ}_{max}N}{{\parallel q_0\parallel }_2}$   end if   $x_{t+1}=x_t-\mu \frac{{\parallel x_t\parallel }_1}{N}{q}_t$   $t=t+1$  end while