Algorithm 4. Sensor signal reconstruction.

1 Input: received data $X$, measurement matrix $\mathsf{\Phi }$, the number of atom is $n_{atom}$

2 Output: reconstruct data $\stackrel{\wedge }{{\displaystyle \mathrm{x}}}$

3 generate sparse basis $\mathsf{\Psi }$ using Algorithm 1

4 collect data $Y$ in the network using Algorithm 3

5 $A\leftarrow \mathsf{\Psi }\ast \mathsf{\Phi }$

6 initialization residual error $r_0=y$, ${\mathsf{\Lambda }}_0=\varnothing$, $A_0=\varnothing$, $it=1$

7 compute $\langle r_{i-1},a_j^{\prime }\rangle$, select the largest $la$ values from $\langle r_{i-1},a_j^{\prime }\rangle$; these values correspond to $A$’s column indexes $j$, constructing set $J_0$

8 set ${\mathsf{\Lambda }}_{it}={\mathsf{\Lambda }}_{it-1}\cup J_0$, $A_{it}=A_{it-1}\cup a_j$ (for all $j\in J_0$)

9 ${\stackrel{\wedge }{{\displaystyle x}}}_{it}={(A_{it}^T{A}_{it})}^{-1}{A}_{it}^{T}y$

10 update $r_{it}=y-A_{it}{\stackrel{\wedge }{{\displaystyle x}}}_{it}$

11 $it=it+1$, if $it\le K$ go back to step 7, or proceed to step 12

12 reconstruct $\stackrel{\wedge }{{\displaystyle \mathrm{x}}}$, which is the generation value of the last iteration ${\stackrel{\wedge }{{\displaystyle x}}}_{it}$.