Help for MEM

PURPOSE

MEM performs a non-linear restoration (deconvolution) using the Maximum 
Entropy Method.  Deconvolution is the process of undoing convolution:  
given a degraded image that results from an image being convolved with a 
known point spread function (PSF) and having noise added, find the original 
image.  There are, in general, many solutions to the deconvolution problem;  
of these MEM selects the solution having the greatest entropy:

	H = Sum over all pixels ( -b*log(b/m) )

where b is the pixel brightness and m is the default image.



EXECUTION

Normally use:
MEM  (DEGRADED.IMG,PSF.IMG)  MEM.IMG  RMSERROR=? 

Using more parameters:
MEM  (DEGRADED.IMG,PSF.IMG,DEFAULT.IMG)  MEM.IMG  RMSERROR=?   				MAXITER=?  MAXCRIT=?


The images can be in any non-complex format;  the output image has
the same format as the degraded image.  Be sure that the maximum in the
degraded image is quite a bit less than the maximum pixel value, so
that the output image won't have to be clipped.  The degraded image should
be nearly all positive (because the output maximum entropy image is totally
positive), and the background level should be close to  zero.  A sub-image 
of the degraded image can be used by specifying the SIZE or (SL,SS,NL,NS) 
parameters.  The image size must be less than or equal to 4096 in both 
dimensions.


The point spread function (PSF) is the image that the degraded image 
has been convolved with.  In some cases it will be known exactly; in
others it must be estimated.  If the degraded image contains point sources
(stars) then one of these can be used for the PSF.  Use COPY to cutout the 
point source.  A lot of noise on the point source image will cause problems 
since it will not then be the true PSF.  In this case a model image can be 
made using SPOT or F2 to approximate the PSF.  If there are no point sources 
then an edge can sometimes be used to estimate the PSF.  If the degradation 
is known (or hypothesized) to be a linear smear at constant velocity then 
the PSF is a line;  look at the FFT of the degraded image in order figure out
the length and direction of the line.  MEM normalizes the volume of the PSF 
to unity so its height is unimportant.  The center of the PSF should be near 
the center of the PSF image.

Usually the default image can be left out.  In some cases better results
may be obtained by using a default image that represent apriori information
about what the image should look like.  For example, the maximum entropy
method may work better on planets if an image containing a disk of the 
correct size and position is used as the default.


The RMSERROR parameter gives the amount of noise in the data.  This can be 
estimated from the standard deviation of a blank part of the image (use
standard deviation from HIST).  Specifying too large a number will lead to 
an overly smooth image.  Too small a number will prevent convergence.  If 
the PSF is inexact or the problem too difficult the desired RMS error maybe
unobtainable.  In these cases use a higher value for RMS error.  If you can't
estimate RMSERROR then run MEM with some low value, and then use the RMS error 
that is obtained when MEM starts to slow down.

The MAXITER parameter is the maximum number of iterations the program will do.
If the solution is reached sooner MEM will stop before doing that number of
iterations.  Typically around 10 iterations are required.  More difficult 
problems and higher signal-to-noise cases will take more iterations.  Easy
problems can often be done in about 5 iterations.  There is little to be 
gained from doing more than about 30 to 40 iterations.

The MAXCRIT parameter specifies the maximum allowed solution criterion.  The
solution criterion is a measure of how close to the maximum entropy solution
the image is for the particular RMS error it is currently at.  Values near
zero indicate that it is at a solution; values near one indicate problems.
The default value for MAXCRIT can nearly always be used.

The MEM program can be restarted using a previous image by specifying the
'RESTART keyword.  The previous (and output) image is specified in the
output image slot.  The parameters and/or images can be changed when restarting
so that adjustments can be made along the way.


OPERATION

The MEM program starts with the default image and changes the image each
iteration to better fit the data.  If no image is specified the default 
image is a constant equal to the average pixel value in the input image.
The algorithm maximizes the entropy with the constraint of the data by
using the method of Lagrange multipliers.  The algorithm uses a gradient 
search method where the gradient direction is modified by an approximation
to the second derivative.  At each iteration the image should be at the 
maximum entropy solution for the particular RMS error that is has.
The program stops (converges) if the RMS error is within ten percent of
the desired error and the solution criterion is less than ten percent
greater than the MAXCRIT parameter.  This algorithm is a modification
of the one described in the first reference (Cornwell and Evans).

MEM performs convolutions using either FFT's or directly, depending on the
size of the PSF.  If the PSF is larger than 16 in either direction then the
FFT's are used.  The size of the FFT is the power of two that is larger than 
the input image size plus the size of the PSF image.  Thus it saves CPU time 
if this slightly less than a power of two rather than slightly more.  If the
convolutions are being performed directly then the PSF should be as small as
possible in order to save CPU time.  MEM creates a bunch of scratch file 
images of real format with the file name MEMn , where n is a digit from 1 to 7.
If the FFT's are being used seven scratch files are created, otherwise five 
are used.  The two FFT scratch files are the size of the FFT, and other five 
are the size of the output image.  If there is a crash or if you stop the 
program, the latest MEM image is in the scratch file MEM7 .  Most of the CPU
time is spent in the two convolutions that are performed for each iteration.  


THEORY

The entropy measure has its background in statistical and information
theoretical arguments, but these are not widely accepted.  Pragmatically,
the entropy can be thought of as a non-linear measure of distance from
the default image, which measures the lack of dispersion in pixel values.
MEM finds the image that has the maximum entropy while still fitting the
data to within a specified RMS error.  The RMS error is the root mean
square of the difference between the degraded input image and the MEM image
convolved with the PSF.  This formulation assumes that the noise has 
a gaussian distribution and is added in the image domain after the convolution.

The purpose of maximizing entropy is to try to recover some of information 
lost in the degradation process.  The information the entropy puts in
is not necessarily correct for all applications.  The method tends to
produce images with enhanced gaussian peaks on a flat zero background level.
Maximum entropy works better if the image fits this picture, and it can do
poorly if the image is a lot different.  For example, it will put oscillations 
in high flat plateaus.  Another important property is that the resolution 
varies depending on the signal to noise ratio, with greater resolution on 
brighter features.  Maximum entropy forces the image to be positive and 
slightly biased.  These properties are a result of maximizing the entropy 
with the constraint of the data, and are not caused by the particular 
algorithm that performs the maximization.  For more information on the 
properties of maximum entropy images and the technique in general see 
the second reference.

The maximum entropy method restoration can be used for resolution enhancement
and for deblurring.  The resolution cannot be increased very much, however,
without introducing artifacts, because some information is permanently lost.
The method performs better on deblurring problems.  Often maximum entropy
will perform much better than linear methods (such as RESTORW), but sometimes
the linear methods can do better.  In general, maximum entropy will excel
(compared to linear methods) in cases of low signal-to-noise.



REFERENCES

    Cornwell, T. J. and Evans, K. F., 1985.  "A New Simple Maximum Entropy
Deconvolution Algorithm",  Astronomy and Astrophysics, 143:77-83.

    Narayan, R. and Nityananda, R., 1986.  "Maximum Entropy Image Restoration
in Astronomy",  Annual Reviews of Astronomy and Astrophysics.





Cognizant Programmer:  L.W.Kamp

Original Programmer:  K. F. Evans	January 1986

PARAMETERS: