Algorithm 1.

Parameterized joint reconstruction method

Input: ${\mathbf{c}}_p^{(0)}$, ${\mathbf{p}}_0^{(0)}$,β

Output: $\widehat{{\mathbf{c}}_{\mathbf{p}}}$, $\widehat{p_0}$

1: k ← 0 {k is the algorithm iteration number.}

2: while stopping criterion is not satisfied do

3:  Calculate ${\mathbf{G}}_p^{(k)}$ and ${\mathbf{G}}_c^{(k)}$

4:  Choose ${\alpha }_k^p$ and ${\alpha }_k^c$ via Algorithm 2

5:  ${\mathbf{p}}_0^{(k+1)}\leftarrow {\mathrm{prox}}_{{\alpha }_k^p\beta R}\left({\mathbf{p}}_0^{(k)}-{\alpha }_k^p{\mathbf{G}}_p^{(k)}\right)$

6:  ${\mathbf{c}}_p^{(k+1)}\leftarrow {\mathbf{c}}_p^{(k)}-{\alpha }_k^c\mathbf{B}{\mathbf{G}}_c^{(k)}$

7:  k ← k + 1

8: end while

9: $\widehat{{\mathbf{c}}_p}\leftarrow {\mathbf{c}}_p^{(k)}$

10: $\widehat{{\mathbf{p}}_0}\leftarrow {\mathbf{p}}_0^{(k)}$