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Astronomical Data Analysis Software and Systems VII
ASP Conference Series, Vol. 145, 1998
R. Albrecht, R. N. Hook and H. A. Bushouse, e
Ö Copyright 1998 Astronomical Society of the Pacific. All rights reserved.
ds.
Determination of the Permissible Solutions Area by Image
Reconstruction from a few Projections: Method 2­CLEAN
DSA
Michail I. Agafonov
Radiophysical Research Institute (NIRFI), 25 B.Pecherskaya st., Nizhny
Novgorod, 603600, Russia, E­mail: agfn@nirfi.nnov.su
Abstract. We have proposed the 2­CLEAN DSA (Determination of
Solution Area) method for the estimation of the area of possible images
(from the ``obtuse'' (smooth) to the ``sharp'' variants) in complicated cases
with constraints and poor a priori information. The area of permissible
solutions can be determined with the help of two CLEAN algorithms:
standard CLEAN and Trim Contour CLEAN (TC­CLEAN). The pro­
cedure has high e#ciency and simple criteria by errors minimization of
initial and control 1­D profiles. We present here a description of some
valuable features of the reconstruction technique.
1. Introduction
Iterative algorithms with non­linear constraints are very attractive in image re­
construction with only a few strip­integrated projections (Vasilenko & Taratorin
1986). The process of convergence to solutions for the various realizations of iter­
ative schemes using the di#erent versions of the CLEAN algorithm have already
been investigated for this problem (Agafonov & Podvojskaya 1989). Two di­
mensional image reconstruction from 1­D projections is often hampered by the
small number of available projections, by an irregular distribution of position
angles, and by positions angles that span a range smaller than about 80 degrees.
These limitation are typical of both lunar occultations of celestial sources and
observations with the fan beam of radio telescope, and also apply to greatly
foreshortened reconstructive tomography. Our previous paper (Agafonov 1997)
also contains the basic description of the problem and the features of 2­CLEAN
DSA method. This paper contains some examples and useful diagrams and also
a valuable addition for the development of the reconstruction technique for this
problem.
2. Application
The problem requires the solution of the equation
G = H # F (+noise) , (1)
where F (x, y) is the object brightness distribution, H(x, y) is the fan (dirty)
beam, and G(x, y) is the dirty (summary) image. An example of the fan beam
58

Permissible Solutions Area by Image Reconstruction 59
Figure 1. Fan beam H(x, y) for four projections (typical example).
Figure 2. The same fan beam H(u, v) on the UV plane (sampling
of about 0.1 of the UV plane in the limit # l to the fixed radius of the
frequency).
for four projections is shown in Figure 1. The classical case (Bracewell & Riddle
1967) needs a number of projections N # #D/#, where # is the desired angular
resolution, and D is the diameter of the object. The incomplete sampling of
H(u, v) (see Figure 2) requires the extrapolation of the solution using non­linear
processing methods.
A two dimensional image reconstruction of a complicated object using a
poor fan beam function H(u, v) needs to carry out the following procedure:
1. Test experiment: evaluation of the possibility and quality of restoration
on the similar images versions; modeling for available projections (number
N , position angles, signal/noise ratio) and desirable frequency limit # l .
This point can include following steps:
. 2­D object model # 1­D profiles # Dirty image
. CLEAN (#) or TC­CLEAN (#, TC) using of Fan beam

60 Agafonov
. Control test from clean maps: Calculation of # (ERROR of con­
trol and initial 1­D profiles)
. Correction of # or #, TC to min #. This step can help to determine
the best algorithm parameters range.
The process of solutions convergence using CLEAN (Hogbom 1974) and
Trim Contour CLEAN (TC­CLEAN) (Steer et al. 1984) has been ana­
lyzed (Agafonov & Podvojskaya 1989; Agafonov & Podvojskaya 1990) by
optimizing the parameter # or #, TC (Trim Contour level) to min #. A
real example of such a process for both algorithms is shown graphically in
Figures 3 and 4.
2. Reconstruction from real observational 1­D profiles. This process
can also include the correction of # or #, TC (Trim Contour level) to min #.
Figure 3. The error of original and control profiles as a function of
the loop gain (using standard CLEAN).
Figure 4. The error of original and control profiles as function of the
loop gain loop using two di#erent Trim Contour levels of TC­CLEAN.

Permissible Solutions Area by Image Reconstruction 61
3. Conclusions
CLEAN forms the solution from the sum of peaks and the result is the sharpest
variant permissible within the established constraints. On the other hand TC­
CLEAN accumulates its result from the most extended components that satisfy
the constraints, producing the smoothest solution (Agafonov & Podvojskaya
1990).
A simple object (consisting of the peaks) may be successfully restored by the
standard CLEAN. The results obtained by both methods are practically identical
for a simple object consisting of individual components, but TC­CLEAN is more
computationally e#cient. For smoothed 1­D profiles with small ``hillocks'', the
solution can be obtained from the isolated individual components (CLEAN), and
also from the more smoothed components (TC­CLEAN). CLEAN increases the
contrast of small components, but the extended background decreases because
of ``grooves''. If min # (CLEAN) # = min # (TC­CLEAN) (see the example
shown in Figures 3 and 4), the solutions will be formally equivalent for both
algorithms, and so we have two choices: (i) to prefer the result corresponding to
the physical peculiarities of the object in accordance with a priori information;
or (ii) to assume the existence a probable class of solutions between the ``smooth''
one from TC­CLEAN and the ``sharp'' one from CLEAN.
The area of permissible solutions of complicated objects can be determined
with a help of both algorithms. The 2­CLEAN DSA procedure can show a
range of possible images from ``smooth'' to ``sharp'' variants satisfying imposed
constraints and poor a priori information.
The application of TC­CLEAN and CLEAN was presented as a reconstruc­
tion of the Crab Nebula map at 750 MHz (Agafonov et al. 1990). The method
of 2­CLEAN DSA allowed us to determine that the area of the permissible solu­
tions lies formally between the ``sharp'' (CLEAN) and ``smooth'' (TC­CLEAN)
variants. Two maps were generally similar. The standard CLEAN increased the
contrast of small components while the TC­CLEAN map gave a better agree­
ment with known a priori information.
Acknowledgments. I am grateful to the Space Telescope European Co­
ordinating Facility and European Southern Observatory for the support which
made possible this presentation and my special gratitude to Rudi Albrecht,
Richard Hook and Britt Sjoeberg for their attention and endurance.
References
Agafonov, M. I. 1997, in ASP Conf. Ser., Vol. 125, Astronomical Data Analysis
Software and Systems VI, ed. Gareth Hunt & H. E. Payne (San Francisco:
ASP), 202
Agafonov, M. I., & Podvojskaya, O. A. 1989, Izvestiya VUZ. Radiofizika, 32,
742
Agafonov, M. I., & Podvojskaya, O. A. 1990, Izvestiya VUZ. Radiofizika, 33,
1185
Agafonov, M. I., Ivanov, V. P., & Podvojskaya, O. A. 1990, AZh, 67, 549
Bracewell, R. N., & Riddle, A. C. 1967, ApJ, 150, 427

62 Agafonov
Hogbom, J. A. 1974, A&AS, 15, 417
Steer, D. G., Dewdney, P. E., & Ito, M. R. 1984, A&A, 137, 159
Vasilenko, G. I., & Taratorin A. M. 1986, Image Restoration (in Russian), Radio
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