Algorithm 1: Lagrange-Newton Method for EIT/UT Dual-Modality Image Reconstruction

Step 1: Initialization.   Calculate the initial value of x using linear back-projection (LBP) algorithm [13]: $x_0=J^{T}y$,   Given the value of the iteration termination parameter: $\epsilon \ge 0,\epsilon =1\times {10}^{-6}$,   Given the value of the intermediate parameter in iteration: $\kappa \in \left(0,1\right),\kappa =0.5,\eta =1$. Step 2: Termination Condition Judgment.   Calculate $G\left(x_k,{\lambda }_k\right)$ by Equation (27), and judge:   If $G\left(x_k,{\lambda }_k\right)\le \epsilon$, stop iteration and return the $x_k$ as the optimal estimation value,   Otherwise, calculate ${\left(\delta x\right)}_k$ and $\left(\delta \lambda \right){}_k$ by the Equation (24). Step 3: Compute Step Length.   Calculate $G\left(x_k+\eta {\left(\delta x\right)}_k,{\lambda }_k+\eta {\left(\delta \lambda \right)}_k\right)$ and $G\left(x_k,{\lambda }_k\right)$ by Equation (27), and judge:   If $G\left(x_k+\eta {\left(\delta x\right)}_k,{\lambda }_k+\eta {\left(\delta \lambda \right)}_k\right)\le \left(1-\kappa \eta \right)G\left(x_k,{\lambda }_k\right)$, go to Step 4,   Otherwise, set $\eta =\eta /4$, continue to the Step 3. Step 4: Iteration Update.   Update the conductivity difference x: $x_{k+1}=x_k+\eta {\left(\delta x\right)}_k$,   Update the Lagrange multiplier λ: ${\lambda }_{k+1}={\lambda }_k+\eta {\left(\delta \lambda \right)}_k$,   Set k=k+1 and go to Step 2.