Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.adass.org/adass/proceedings/adass96/reprints/cheny.pdf Дата изменения: Wed Jan 14 23:16:22 1998 Дата индексирования: Tue Oct 2 10:27:11 2012 Кодировка: Поисковые слова: crab nebula

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

Imaging by an Optimizing Method
Y. Chen, T. P. Li, and M. Wu High Energy Astrophysics Lab, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, PRC, E-mail: cheny@astrosv1.ihep.ac.cn Abstract. The imaging problem can b e describ ed as an optimizing problem in mathematics. Thus optimizing theory and algorithms can b e used to solve it. In this pap er we present an optimizing method, in which we take the imaging problem as an optimizing problem with linear constraints. We choose the ob jective function carefully. Both the mathematical exp ectation and the variance of the observed data are considered. Upp er and lower limit source and background intensities can b e conveniently considered. We adopt an algorithm very similar to the affine scaling approach in convex programming. Computer simulations of rotating modulation collimator imaging show that the quality of images from this method is b etter than that from the traditional cross-correlation method. Both p oint and extended sources can b e imaged in the same field of view. We also apply the algorithm to ROSAT PSPC p ointed observation data of the Crab nebula. The image quality is improved significantly. The extended structure of the Crab nebula can clearly b e seen.

1.

Introduction

The imaging problem may b e describ ed as inferring the sky brightness distribution from observations and prior knowledge (Cornwell 1992). In this pap er, we will introduce an optimizing method and adopt it to the imaging problem. Then we will apply it to simulations of a rotating modulation collimator (RMC), and ROSAT PSPC (the Position Sensitive Prop ortional Counter) observations. 2. Imaging Problem d = Pf + n (1)

The imaging problem is: where d is the observational data, P is the p oint spread function, f is the unknown sky, and n is the noise. Usually, there are some constraints for f and n in this linear system of equations. The optimizing problem is: min. F (x) subj ect to Ax = b x0 178 (2) (3) (4)

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


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where F (x) is an ob jective function. In astronomy, the noise nk usually follows a Poisson or Gaussian distribution. Thus it has a certain exp ected value and variance. The sky intensity f may have a upp er limit up and a lower limit low. Then we can make an ob jective function
k k m

F (f, n) = (
i

n2 /di - k)2 + a( i
i

ni )2 - b
i

(ln (fi - lowi )+ln (upi - fi )) (5)

where a and b are coefficients, k is the numb er of bins of observational data, and m is the numb er of sky bins. The constraint condition is
m

pji fi + nj = dj (j = 1, ..., k)
i

(6)

Both fi and nj are unknown. This problem is similar to the convex programming problem. 3. Affine Scaling Algorithm

The affine scaling (AS) algorithm is one of the simplest and most efficient of interior p oint method algorithms (Dikin 1967). For the optimizing problem (2) (4) the AS algorithm in detail is: 1. Try to find an initial solution. 2. Calculate Hk = [
2 - f (xk )+ Xk 2 ]- k 1

(7) (8) (9)

[AHk AT ] k = AH sk =

f (xk )
k

f (xk ) - AT

3. Check whether the stopping criteria is satisfied. 4. Find a transition direction. dk = -Hk s x
k

(10)

5. Calculate the step length k . Search for that k which minimizes the ob jective function. 6. Move to a new solution. xk
+1

xk + k dk x

(11)

7. Let k k + 1 and go to Step 2. We develop ed an algorithm based on the affine scaling algorithm (Goldfarb 1991; Fang 1993) for problem (5) (6).


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Chen, Li, and Wu

Figure 1. a) The assumed sources. b) Result of the AS algorithm. c) Result of cross-correlation. d) Result of CLEAN cross-correlation. 4. 4.1. Application Rotating Mo dulation Collimator

We have simulated a rotating modulation collimator. The configuration of the RMC is shown in Table 1. The background is assumed to b e 0.09 ph cm-2 s-1 . The fluxes of the two p oint sources are assumed to b e 1.0 в 10-2 and 6.7 в 10-3 ph cm-2 s-1 , and the total flux of the extended source 3.5 в 10-2 ph cm-2 s-1 (Figure 1a). The observing time is assumed to b e one day. Figure 1b shows the Table 1. distance b etween strips (cm) 1.0 The Configuration of the RMC distance b etween grid planes (cm) 34 FOV (o ) 6в6 total active area (cm2 ) 1000

result of the AS algorithm, while Figure 1c and d result from the cross-correlation method and the CLEAN cross-correlation, resp ectively. Both the p oint sources and the extended source can b e seen in Figure 1b. The angular resolution and image quality in Figure 1b are much b etter than in the cross-correlation images. 4.2. ROSAT PSPC Image of Crab

We used this algorithm to reconstruct ROSAT PSPC data. The results are shown in Figure 2. Figure 2a is the original image observed by PSPC. The


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Figure 2. a) The original ROSAT PSPC Crab nebula image. b) The image obtained from the AS algorithm. The FOV is 100в100 arcsec. c) A ROSAT HRI image of the Crab nebula. The FOV is 100в96 arcsec. result of the AS algorithm is shown in Figure 2b, where the extended structure is clearly seen. This extended structure can also b e seen by ROSAT HRI (the High Resolution Imager) (Figure 2c). 5. Discussions and Conclusions

The calculation time for the AS algorithm dep ends on the initial solution. We can use the solution, from some other algorithm such as Richardson-Lucy iteration or cross-correlation, as the initial solution in order to reduce the calculation time. In this pap er we develop ed an AS algorithm and applied it to the imaging problem. The results show that this algorithm can b e used in the imaging problem and usually results in a b etter image than that obtained from a traditional method such as cross-correlation. This algorithm can b e also used in the data reconstruction of an imaging instrument (e.g., ROSAT PSPC). Acknowledgments. analysis. References Cornwell, T. J. 1992, in ASP Conf. Ser., Vol. 25, Astronomical Data Analysis Software and Systems I, ed. D. M. Worrall, C. Biemesderfer, & J. Barnes (San Francisco: ASP), 163 Goldfarb, D., & Liu, S. 1991, Mathematical Programming, 49, 325 Fang, S.-C., & Puthenpura, S. 1993, in Linear Optimization and Extensions (Englewood Cliffs, NJ: Prentice Hall) We thank X. J. Sun for the help on the ROSAT data