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Astronomical Data Analysis Software and Systems IV
ASP Conference Series, Vol. 77, 1995
R. A. Shaw, H. E. Payne, and J. J. E. Hayes, eds.
MEM Task for Image Restoration in IRAF
N. Wu
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore,
MD 21218
Abstract. This presentation is devoted to the MEM (Maximum En­
tropy Method) task, version D, for image restoration in IRAF. Described
in detail are its enhanced functions of Poisson noise only case handling
and subpixelization technique; improved algorithms for maximization in­
cluding the preconditioned conjugate method, accurate Newton method,
the accurate one­dimensional search, and the model updating technique.
The task's limitations and possible development are also discussed.
1. Introduction
The IRAF mem0 package, version B, was released to the public in October 1992,
and was upgraded to version C in December 1993. Version C resides in the
anonymous ftp site at NOAO (iraf.noao.edu, directory ¸ /contrib/). Now, the
most important MEM task of the mem0 package has been upgraded to version
D; the other four tasks remain unchanged. This task is included in the sub­
package stsdas.analysis.restore with the name mem. Its executable code on
SUN/UNIX has also been put into the ftp site at NOAO with the task name
dmem. The two relevant files are dmem.notes and dmem.tar.Z.
The task mem, version D, features the following sophisticated functions and
techniques:
1. Poisson noise only case handling.
2. Subpixelization technique.
3. Model updating technique.
4. The preconditioned conjugate method and accurate Newton method.
5. The accurate one­dimensional search in maximization.
They are described in detail in the following section.
2. Algorithm and Programming
2.1. The Case of Poisson Noise Only
CCD images from the Wide Field and Planetary Camera (WF/PC) of Hubble
Space Telescope (HST ) have both readout noise, of Gaussian type, and signal
noise, of Poisson type. In this case the total noise variance is calculated using the
parameters noise, in electrons, for the former, and adu (analogue­to­digital unit
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2
conversion constant or gain), in electrons/DN, for the latter. Then this variance
is used in the Gaussian likelihood function or, equivalently, in the expression of
ü 2 .
On the other hand, images from the photon counting detectors of the Faint
Object Camera (FOC) of HST contain only signal noise of Poisson type. Not
only should the parameter noise be set to zero, but, more importantly, the
Poisson likelihood function must be used in calculation. Failure to do so will
lead to wrong and unacceptable results.
Earlier versions of the task mem could not handle correctly the case of Poisson
noise only. In version D, setting the parameter poisson to yes selects the correct
expression for the likelihood function, automatically sets the parameter noise to
zero, and takes other actions especially designed for the case of Poisson noise
only.
2.2. Subpixelization Technique
It has been shown that the subpixelization (subsampling) technique may improve
resolution in restored images, or at least result in more pleasant appearances for
objects (Weir & Djorgovski 1990).
Subpixelization means to restore an image on a grid finer than that on
which the input data image is defined. In this process one ``normal size'' pixel
of the input image is ``split'' to several ``subpixels'' of the restored image. This
mechanism must be built into the program, but cannot be accomplished simply
by replicating each pixel of the input image before restoration.
Subpixelization is activated by setting the parameter nsub to a value greater
than one. In this case all input images, including the point spread function (PSF)
but excluding the data image, must be subpixelized by the user by a factor of
nsub in each dimension. The required core memory and computational time are
approximately proportional to nsub 2 .
2.3. Model Updating Technique
This technique was described in detail in Wu (1994). The MEM program uses
a double iteration scheme: the values of the Lagrange multipliers ff and fi,
respectively for the data constraint and the total power constraint, are revised
in the outer iteration, while the inner iteration is for finding the ME solution
for the particular ff and fi of each outer iteration.
The basic idea behind the model updating technique is to use the ME
image converged for particular ff and fi in each outer iteration as the model to
start the next iteration. In this way in maximization, the approximate solution
to the linear equations used in the preconditioned conjugate method is more
accurate, or the accurate solution to the linear equations required in the accurate
Newton method is easier to be found. Therefore, the total number of iterations
is considerably reduced and much computational time is saved. The restored
image also has improved photometric linearity.
2.4. Preconditioned Conjugate and Accurate Newton Methods
In previous versions of the task mem, only the zeroth­order approximate Newton
method of maximization was available. Hence the name of the package: mem0.
Zeroth­order approximation means that in the solution of a large set of linear

3
equations (or equivalently the inversion of a large matrix) non­diagonal elements
are ignored, under the assumption that the diagonal ones dominate. In this way,
solving the equations becomes a simple operation. However, this simple method
may result in very slow convergence.
Now, in version D, much more sophisticated methods are used to calculate
the change in the iteration, i.e., to determine the search direction in maximiza­
tion. The first method is the preconditioned conjugate method (as opposed to
the ``standard'' conjugate method commonly used), or the conjugate method
based on the approximate Newton method described in the above. More specif­
ically, in each outer iteration for particular ff and fi, the approximate Newton
method is used to calculate the search direction for the first inner iteration.
Thereafter, in each inner iteration, a direction is calculated using the approx­
imate Newton method. The component orthogonal or conjugate to the search
direction used in the previous inner iteration is calculated, and used as the new
search direction for the maximum point. By careful programming, the core mem­
ory requirement and the number of FFTs are the same as in the approximate
Newton method.
The second method is the accurate Newton method (as opposed to the
approximate Newton method). Here the accurate solution to a set of linear
equations of large size is calculated by iteration, each of which requires two
convolutions, i.e., four FFTs.
The accurate Newton method is the most efficient in the sense that the
fewest total number of (inner) iterations is needed because the search direction
is determined accurately. However, many FFTs may well be required in each
(inner) iteration to solve the linear equations. In contrast, the preconditioned
conjugate method requires a greater total number of (inner) iterations, but no
extra FFTs in each (inner) iteration are needed to calculate the conjugate di­
rection.
By default, in the case of Poisson noise only (poisson=yes) the accurate
Newton method is used, otherwise (poisson=no) the preconditioned conjugate
method is used. This is the best choice. In the case of Poisson noise only, the
Hessian of the objective function, which is a measure of the curvature of the
image space, changes rapidly from (inner) iteration to iteration. Consequently,
the preconditioned conjugate method, as a method taking advantage of memory
in the iteration, is not effective in determining the search direction.
2.5. Accurate One­Dimensional Search in Maximization
After determining the search direction, an optimal step (length) in this direction
should be calculated. In earlier versions of the task mem, quadratic extrapolation
and cubic interpolation are used for this purpose. They are both approximate
methods for calculating the optimal step. Now, in version D, the accurate one­
dimensional (1­D) search is available for interpolation. Specifically, the approx­
imate maximum point found by cubic interpolation is used as the initial guess
to start a search for the accurate maximum point, using the Newton method in
a single variable. In such a way, the maximum point is found with little extra
effort but much higher accuracy.

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The (approximate) quadratic extrapolation remains. In most cases, es­
pecially at the late stage of iteration, interpolation but not extrapolation is
desirable in the 1­D search.
2.6. Other Revisions
Apart from employing the methods described in the previous subsections to
enhance the task's function and to speed up convergence, having a good user
interface is also important. Every effort has been made to create a user friendly
interface. The number of positional parameters is kept to a minimum. The
default values of hidden parameters are carefully chosen. Automatic schemes
to adjust some variables and to deal with some predicted ill­conditioned cases
are built in the program. Options and parameters used only for testing are
hidden from the user. The diagnostic messages are informative and grouped
logically, and can be output at three different levels of verboseness to meet the
user's needs. The most detailed level (message=3) is primarily for debugging
purposes. The help file is well written and should be read before the first attempt
to run the task.
3. Concluding Remarks
The current version of the task mem has its limitations: Like other MEM pro­
grams, it can only handle the case of space­invariant PSF. It cannot be used
for multi­channel and multi­data set restoration like the package MEM/MemSys5
(Weir 1991). The algorithm used in the accurate Newton method to find the ac­
curate solution of a large set of linear equations should be improved, e.g., using
the preconditioned conjugate method. Finally, criteria for convergence should
be investigated, especially in the case of Poisson noise only, and more reasonable
ones should be adopted. For this version of the task, the user's judgment is very
important in obtaining satisfactory results.
References
Weir, N. & Djorgovski, S. 1990, in The Restoration of HST Images and Spec­
tra, ed. R. L. White & R. J. Allen (Baltimore, Space Telescope Science
Institute), p. 31
Weir, N. 1991, in Proceedings of the 3rd ESO/ST­ECF Data Analysis Workshop,
ed. P. J. GrosbÜl & R. H. Warmels (Garching, ST­ECF), p. 115
Wu, N. 1994, in The Restoration of HST Images and Spectra II, ed. R. J.
Hanisch & R. L. White (Baltimore, Space Telescope Science Institute),
p. 58