How to average to find 'circular' symmetry in a SQUARE matrix?

[Robert A. Macy]

This might be slightly OT, but I posted WITH NO ANSWERS to
sci.math.  Maybe there are appropriate octave functions,
I've overlooked...

Assume there is a square matrix 301 by 301 of x and y
coordinates containing data that has rotational symmetry in
the center about 151,151. 

What is the BEST way to find that circular symmetry, the
average, while still remaining in x & y system? 

I have used quadrant averaging and mirror averaging, but is
there another way, better way to find circular symmetry? 

Is there some method that takes better advantage of the
fact the structure has circular symmetry? 

Or, is the quadrant/mirror averaging mathematically the
best that can be done? 

  - Robert -