Tikhonov reconstruction from two-plane scattering data ($J_{\alpha }(V)=\Vert A_{+}V-d_{+}{\Vert }^2+\Vert A_{-}V-d_{-}{\Vert }^2+\alpha \Vert V{\Vert }^2$)

The two Born approximation measurement operators $A_{\pm }$ map a supported scattering potential to fields on planes above and below it. The displayed Tikhonov regularization functional balances their residuals against a positive quadratic penalty. With an $L^2$ penalty, the unconstrained minimizer is $(A^{\ast }A+\alpha I{)}^{-1}{A}^{\ast }d$, where $A=(A_{+},A_{-})$. Regularization stabilizes estimation but cannot remove the nonuniqueness of Born non-scattering contrasts at one incident direction without extra information.