Axial imaging necessitates loss of lateral shift invariance: proof with the Wigner analysis

We present a proof of the conjecture reported in Stein and Barbastathis ["Axial imaging necessitates loss of lateral shift invariance," Appl. Opt.41, 6055-6061 (2002)] that axial imaging necessitates loss of lateral shift invariance. We apply the Wigner distribution function (WDF) to the axial imaging process and compare laterally shift variant and invariant imaging systems. Two conditions for axial imaging are established: (i) objects with spatially variant local spatial frequency content to create sufficient change in the WDF with defocus and (ii) properly designed shift variant imaging kernels to estimate the slope of the sheared WDF. The lateral shift variance is a necessary condition for axial imaging. We use two examples from Stein and Barbastathis to show how axial imaging is interpreted in Wigner space.