ATM - Analytic Continuation

[Richard Schwartz <richas@idt.net>]

An analytic function is one with bounded derivatives in a region that
contains a circle and its interior.   The function can be extrapolated
so that the result is also analytic, and that extrapolation is unique!

So image enhancement is done by doing a fourier transform on the focal
plane image (including phase) to get the wavefront across the telescope
aperture.   Then that function is extrapolated to a much larger aperture
by analytic continuation.  Finally, a new sharper image is synthesized
by a fourier transform of the extrapolated wavefront.

Of course, in complex variables courses, you are told exactly what the
analytic function is by a math formula.  In data collection, however, it
is subject to corruption by noise and measurement error, so you have to
estimate it.  And you better believe that I last studied complex
variables a LONG LONG time ago.  It was the only math class in which I
got a grade of B.  (But the teacher was German, young, very pretty and
wore short skirts, thank you!)

I apologize for use of the word "aliasing" to describe the artifacts.
That word refers to sampled data systems where the input has frequencies
more than half the sampling rate.

. . . Richard