Algorithm 3 Sensory Data Reconstruction

Input:$K$, $D_k(t_0)$($k=1,\cdots ,K$), $L$, $N_k$, $\{k_1,\cdots ,k_{N_k}\}$($k=1,\cdots ,K$), $t_0$, $T$, $T$ Output:${\widehat{D}}_1(t_0),\cdots ,{\widehat{D}}_K(t_0)$ for each$k\in \{1,\cdots ,K\}$do /* Constructssampling indicator matrix$S_k(t_0)$ */ for each$l\in \{0,\cdots ,L-1\}$do for each$i\in \{k_1,\cdots ,k_{N_k}\}$do if$d_i(t_0-lT)$ in $D_k(t_0)$ been sampled $s_i(t_0-lT)=1$ else $s_i(t_0-lT)=0$ end if end for end for $S_k(t_0)={\left(s_i(t_0-lT)\right)}_{L\times N_k}$ /* Derive space constraint matrix$H(t_0)$ */ for each$i\in \{1,\cdots ,N\}$do for each$j\in \{1,\cdots ,N\}$do if$n_i$ and $n_j$ are neighbors at $t_0$ ${h^{\prime }}_{i,j}(t_0)=1$ else ${h^{\prime }}_{i,j}(t_0)=0$ end if end for end for $H^{\prime }(t_0)={({h^{\prime }}_{i,j}(t_0))}_{N\times N}$ for each$i\in \{1,\cdots ,N\}$do $temp={\displaystyle {\sum }_{j=1}^N{h^{\prime }}_{i,j}(t_0)}$ for each$j\in \{1,\cdots ,N\}$do if$temp=0$ $h_{i,j}(t_0)=0$ else if$i=j$ $h_{i,j}(t_0)=1$ else $h_{i,j}(t_0)=-{h^{\prime }}_{i,j}(t_0)/temp$ end for end for $H(t_0)={(h_{i,j}(t_0))}_{N\times N}$ $\{L,R^{\ast }\}=ArgMin\left({\Vert S_k(t_0)\cdot (L{R}^{\ast })-D_k(t_0)\Vert }_F^2+\lambda \left({\Vert L\Vert }_F^2+{\Vert R^{\ast }\Vert }_F^2\right)+{\Vert H(t_0)L{R}^{\ast }\Vert }_F^2+{\Vert L{R}^{\ast }T\Vert }_F^2\right)$ ${\widehat{D}}_k(t_0)=L{R}^{\ast }$ end for return${\widehat{D}}_1(t_0),\cdots ,{\widehat{D}}_K(t_0)$