ATM An efficint algorithm for robotic Foucault testing

[Dale Eason <atmpob@yahoo.com>]

Concerning maskless robotic Foucault testing of
mirrors.

One of the problems with the robotic Foucault Analysis
is that the time it takes is proportional to the
number of zones you want to measure.  Of course that
is true for the manual version as well.  I have
discovered a much faster algorithm that is not
proportional to the number of zones and thus you can
have as many zones as there are pixels across the
mirror.  It is proportional to the number of
measurements and only needs two (more are better).  I
will explain it here.

In one method (I call the zone null search) you pick a
zone to measure and the program adjust the knife back
and forth until both the left and right zones are
equal at a specific intensity. To do this the program
hunts for the right position and may have to move the
knife several times.  There is a second method the we
call the flip and diff on the list but it too requires
at least one measurement per zone.    I have noticed
that during the null hunt the difference of the left
and right zone values follow a linear relationship to
the longitudinal knife position.  This has given rise
to a new algorithm I call  Linear Null Approximation
as described next.

Set the knife to a longitudinal position.  Adjust the
lateral intensity until the average of all zones
matches your measurement intensity.  If all the zones
have values inside the max and low thresholds Record
the longitudinal position and all left and right zone
differences.

Move to a second longitudinal position and do the same
again.  You now have enough information to computer
where the null will be for each zone using a line
equation for each zone.  You can also take more
measurements and then do a least squares fit to help
eliminate measurement error and noise.

I have done this with about 30 knife longitudinal
positions for each zone and have verified that  the
linear condition holds.  I have a excel spread sheet
of the data that I will send any interested parties. 
I have also use the standard algorithm to compute the
same nulls.  They agree with the Linear Null
Approximation within .002 inch.  That is as good or
better than a manual tester usually gets.

In conclusion only a few longitudinal knife positions
are required to test all the zones.  Now for the fun
part.  You can have as many zones as there are pixels!
 There are enhancements to this.  You could average
several neighboring pixels to reduce noise and other
errors.  But the bottom line is that you can have more
reading than any of our analysis programs can handle. 
There are some conditions that must be met to assure
the linear relationship.  The most import I have found
so far is the left and right zone readings must be
within the threshold of the camera pixel values ( > 0 
and < max).  That is the value must not be clipped. 
Otherwise the relationship is not linear but is
Sigmoidal.

Another advantage of Linear Null Approximation is that
it is much easier to use with only a still camera and
manual stage since only a few images need to be taken.
 The hard part is making all the images at the same
average intensity.  I’m not sure how different average
intensities affect the results.  I will experiment
with that next.

The part I am haveing trouble with is converting the
results into a mirror profile.  Can one of you
Foucault analysis authors or anyone help with that?


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