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Astronomical Data Analysis Software and Systems VI ASP Conference Series, Vol. 125, 1997 Gareth Hunt and H. E. Payne, eds.

Image Reconstruction with Few Strip-Integrated Pro jections: Enhancements by Application of Versions of the CLEAN Algorithm
M. I. Agafonov Radiophysical Research Institute (NIRFI), 25 B. Pecherskaya st., Nizhny Novgorod, 603600, Russia, E-mail: agfn@nirfi.nnov.su Abstract. Iterative algorithms with non-linear constraints are very attractive in image reconstruction with only a few strip-integrated pro jections. We present research into various versions of the iterative CLEAN algorithm for the solution of this problem. We suggest a method to determine the area of p ermissible solutions in complicated cases for two CLEAN algorithms. This procedure was named 2-CLEAN Determination of Solution Area (2-CLEAN DSA).

1.

Introduction

Two dimensional image reconstruction from 1-D pro jections is often hamp ered by the small numb er of available pro jections, by an irregular distribution of p osition angles, and by p osition angles that span a range smaller than ab out 100 . These limitations are typical of b oth lunar occultations of celestial sources and observations with the fan b eam of a radio interferometer, and also apply to greatly foreshortened reconstructive tomography. 2. Deconvolution Problem

The problem requires the solution of the equation G = H F (+noise) , (1)

where F (x, y ) is the ob ject brightness distribution, H (x, y ) is the fan (dirty) beam, and G(x, y ) is the dirty (summary) image. The classical case (Bracewell & Riddle 1967) needs a numb er of pro jections N D/, where is the desired angular resolution, and D is the diameter of the ob ject. The incomplete sampling of H (u, v ) requires the extrap olation of the solution of F (x, y ) using non-linear processing methods. 2.1. The Iterative Algorithms
k +1

The general scheme of an iterative algorithm is F = r CF k + (G - HCF k ) , (2)

where is the loop gain (0 < < 2/max H (u, v )), C = C1 C2 ...Cn are the limitations, and r is the stabilizer. 202

© Copyright 1997 Astronomical Society of the Pacific. All rights reserved.


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The simple standard CLEAN (Hogb om 1974) is the b est known realization Ё of iteration schemes in radio astronomy. But the algorithm has defects (strip es and ridges) in areas of extended emission. CLEAN was used for the image reconstruction of the Crab Nebula from four lunar occultation profiles (Maloney & Gottesman 1979; Agafonov et al. 1986). However, more complete information is needed for an extended ob ject. Trim Contour CLEAN (TC-CLEAN) (Steer et al. 1984) gave hop e for the improvement of image quality with extended features, but it needed a study in different practical cases (Cornwell 1988). 2.2. Numerical Mo deling

The process of solution convergence by (ERROR of initial and control 1-D profiles) minimization with variation of parameters and TC (Trim Contour level) was investigated (Agafonov & Podvo jskaya 1989; Agafonov & Podvo jskaya 1990) for b oth algorithms using of the following procedure: · 2-D ob ject model 1-D profiles Dirty image · CLEAN () or TC-CLEAN (, T C ) using the Dirty b eam · Control test from clean maps: Calculation of (ERROR of control and initial 1-D profiles) · Correction of or , T C to min The process of parameter ( or , T C ) optimization to min was shown very well graphically (Agafonov & Podvo jskaya 1989). CLEAN The dirty map p eak is the target of each iteration. The choice of loop gain was not clear in the original scheme, but it significantly influences the solution. For example, by modeling the test ob ject (Crab Nebula map at 1.4 GHz) the optimum range of was found to b e ab out 0.05­0.10 (from the dep endence of () in this map). But the optimum value dep ends on the ob ject structure. The algorithm has high instability for distributed ob jects. Changes in the resulting maps as a function of with small changes of testifies to this instability. We attempted to increase stability by: (i) choosing solutions with min from the optimum interval and then averaging; and (ii) by sp ecial processing--such as a complex method like sp eckle-masking (but increasing the computational efforts). TC-CLEAN A smooth function is subtracted at each iteration. Trim contour (TC ) is used for the choice of comp onents p er iteration. TC must b e low, but ab ove the level corresp onding to the true ob ject dimensions. The algorithm has high stability, a simple choice of and TC , converges in few iterations, and is computationally efficient. 3. Discussion and Conclusions

A simple ob ject (consisting of the p eaks) may b e successfully restored by the standard CLEAN. The results obtained by b oth methods are practically identical for a simple ob ject consisting of individual comp onents, but TC-CLEAN is more computationally efficient. A difference b etween the solutions is observed


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for complicated ob jects with small comp onents in areas of extended emission. The standard CLEAN reconstruction has a "grooved" structure for such areas. For smoothed 1-D profiles with small "hillocks," the solution can b e obtained from the isolated individual comp onents (CLEAN), and also from the more smoothed comp onents (TC-CLEAN) by using the same min for the initial and control profiles. CLEAN increases the contrast of small comp onents, but the extended background decreases b ecause of the "grooves." If min (CLEAN) min (TC-CLEAN), the solutions will b e formally equivalent for b oth algo= rithms, and so we have two choices: (i) to prefer the result corresp onding to the physical p eculiarities of the ob ject in accordance with a priori information; or (ii) to assume the existence of a probable class of solutions b etween the "obtuse" (smooth) one from TC-CLEAN and the "sharp" one from CLEAN. CLEAN forms the solution from the sum of p eaks, and the result is the sharp est variant p ermissible within the established constraints. On the other hand, TC-CLEAN accumulates its result from the most extended comp onents that satisfy the constraints, producing the smoothest solution (Agafonov & Podvo jskaya 1990). We carried out the study of TC-CLEAN applied to image reconstruction with few pro jections. TC-CLEAN proved to b e an effective and stable solution. CLEAN emphasizes only individual features and needs a sp ecial treatment to obtain the solution stability (except the ob ject from the p eaks). We suggest determining the area of p ermissible solutions of complicated ob jects with the help of b oth algorithms. This procedure was named 2-CLEAN Determination of Solution Area (2-CLEAN DSA). It is also useful to study the p ossibility of reconstruction (the reality of the comp onents on the map) for any new case with p oor UV -filling by using a similar method used for our observational test ob ject. The 2-CLEAN DSA procedure can show, in complicated cases, a range of p ossible images from "obtuse" to "sharp" variants satisfying imp osed constraints and p oor a priori information.

4.

Maps from Real Lunar Occultation Observations

In certain cases, usually at lower frequencies, the angular resolution of synthesis radio telescop es is insufficient. However, observations of an ob ject during lunar occultations provide 1-D brightness profiles with high angular resolution. This is also useful in observations with optical instruments. The lunar limb is approximated by a plane screen, moving through different p osition angles. The first images of the Crab Nebula from four pro jections of lunar occultations was presented by using the standard CLEAN (Maloney & Gottesman 1979; Agafonov et al. 1986). But the maps had defects in extended areas due to the reconstruction algorithm. The application of TC-CLEAN was used in the reconstruction of the Crab Nebula map at 750 MHz with angular resolution 20в35 (Agafonov et al. 1990). The 1-D profiles were obtained by observations using the 70 meter dish (RT-70) in West Crimea. The method of 2-CLEAN DSA allowed us to determine that the area of p ermissible solutions lies formally b etween the "sharp" (CLEAN) and "smooth" (TC-CLEAN) variants. The two maps were generally similar. The standard CLEAN increased the contrast of small comp onents, while the TC-CLEAN map gave a b etter agreement with known a priori information ab out the Nebula, and was closer to the true brightness distribution of


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the Crab. The CLEAN variant of the map can b e used only for information ab out the location of the small comp onents. Acknowledgments. I am grateful to the National Radio Astronomy Observatory for the supp ort which made p ossible this presentation and my sp ecial gratitude to R. Simon and C. White for their endurance in the organization of this Conference. References Agafonov, M. I., Aslanyan, A. M., Gulyan, A. G., Ivanov, V. P., Martirosyan, R. M., Podvo jskaya, O. A., & Stankevich, K. S. 1986, Pis'ma v AZh, 12, 275 Agafonov, M. I., & Podvo jskaya, O. A. 1989, Izvestiya VUZ. Radiofizika, 32, 742 Agafonov, M. I., & Podvo jskaya, O. A. 1990, Izvestiya VUZ. Radiofizika, 33, 1185 Agafonov, M. I., Ivanov, V. P., & Podvo jskaya, O. A. 1990, AZh, 67, 549 Bracewell, R. N., & Riddle, A. C. 1967, ApJ, 150, 427 Cornwell, T. J. 1988, A&A, 202, 316 HЁ om, J. A. 1974, A&AS, 15, 417 ogb Maloney, F. P., & Gottesman, S. F. 1979, ApJ, 234, 485 Steer, D. G., Dewdney, P. E., & Ito, M. R. 1984, A&A, 137, 159