Hi HETGS Spatial-Spectral folks, 

  In doing a little reading after out meeting on 3/8/01 I found
two short-ish descriptions of "inverse problems" and maximum
entropy that felt illuminating to me...  They are sections 18.4
and 18.7 of Numerical Recipes (2nd Fortran edition) and
an article by T.J. Cornwell in the first ADASS conference series.
I've xeroxed these and will pass/mail them around shortly. 
The Cornwell one is nice given where we are and he stresses
"simulate and test" among other good things.

  I learned that just minimizing chi-squared has multiple crazy
solutions but another constraint is added to the process, then
the problem is stable and unique (!?!).  This additional constraint
is called "regularization", maximum entropy is just one of a bunch
of regularizing functions that can be used.  So, if chi-squared is
minimized subject to also being at a maximum of entropy then the
problem is better behaved.

  Note that Chi-squared is a measure between the forward folded
model and the measured data; whereas the entropy is a measure applied to
the pre-folded model.

  There are several "entropy" measures kicked around, all the sum
over all model "pixels" of a function:
i) -f ln(f) where f = I/Iave  with I the value of the model intensity
ii) ln(I)
iii) sqrt(I)   

  An exciting modification for HETGS work is that if there is a
known a priori variation of the image intensities other than the
"flat" starting image, then an entropy can be defined with 
respect to this "default" image:

ia) -I ln(I/I_o) where I_o is the "default map".

It seems to me that if we have a "model" that is a data cube
of X,Y images at different wavelengths, then the zeroth-order
image in a PHA band is a good "default map" for the image planes.
For line-dominated sources we might consider a few planes for
each lines, e.g., each plane is a given velocity offset from
the rest line wavelength and the velocity spacing can be adjusted
to correspond to the spectrometer resolution (e.g. we'd have more
velocity planes for O VIII at 19A than for Si Ly alpha at 6.2 A, 
more for HEG and MEG, etc.

  I've made a simple figure of the various components of an
HETGS MEM (or whatever regularization is used) scheme and put
it and some other goodies (including a quick look at the IDL
MEM routine that Michael Wise poined me to) on a (public) web page:

http://space.mit.edu/HETG/MEM

  A lot of the math in the articles is pretty messy (talking about
metrics and conjugate gradients) but that's why we have John Davis!

  OK, I've got other stuff to work on for a while - let me/us
know if you make any progress or have thoughts,

  -Dan