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
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