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.
© 2009 Optical Society of America
Original Manuscript: May 15, 2009
Manuscript Accepted: August 24, 2009
Published: October 19, 2009
Se Baek Oh and George Barbastathis, "Axial imaging necessitates loss of lateral shift invariance: proof with the Wigner analysis," Appl. Opt. 48, 5881-5888 (2009)