We made computer simulations to compare the quality of images from
direct demodulation with that from cross-correlation deconvolution and
other methods for the same simulated RMC data. The following
normalized cross-correlation function was used in our calculations
To estimate the background, we calculated
then found the bin with maximum s(i) and subtracted the contribution of the apparent discrete source at point i from the data. The background data was obtained by repeating the above procedure until the maximum value of s(i) was less than a given threshold. The background b can be found by solving Eq. (10) iteratively. In the iterations, a continual constraint was used, i.e. the difference between two neighbouring bins should not be greater than a certain value.
In the simulations, we used an RMC telescope structured as in Fig. 1 (click here) with strip width a=1.0 cm, the separation of the two grids H=95 cm, the total active area of the detector 1000 cm2 and FOV. To avoid "ghost'' peaks in the image, we set up a 0.5 cm offset between the grids in the two planes. The background is assumed to be 0.09 counts cm-2 s-1 and the observation time is 24 hours. We first supposed that only a single point source exists in the FOV at the position (see Fig. 3 (click here)a) with a flux of ph cm-2 s-1. The rotating period was divided into 400 units (m=400). A Monte Carlo sample of observed data of the RMC telescope was produced. The cross-correlation map and CLEAN cross-correlation map from the simulated data are shown in Figs. 3 (click here)b and c respectively. For the same simulated data, we did image analysis by the direct demodulation method and got the result shown in Fig. 3 (click here)d. From the direct demodulation map, we find the position of the point source at and flux ph cm-2 s-1, which is consistent with the assumed values. The errors of the position and counts are the standard deviations of values of the quantities measured for images from twenty Monte Carlo samples.
Figure 3: Images from the simulated data of a single point source observed by a RMC system with height H=95 cm, strip width a=1.0 cm and FOV=. a) Object. b) Cross-correlation map. c) CLEAN cross-correlation map. d) Image from direct demodulation
It is clear from comparing Fig. 3 (click here)d with Fig. 3 (click here)b that the direct demodulation technique can significantly improve the angular resolution. The FWHM of the point-source image in Fig. 3 (click here)d from DDM is , much smaller than that in Fig. 3 (click here)b from CCM, .
The cross-correlation maps, Fig. 3 (click here)b and c, show severe sidelobes around the image of the point source. The image derived by using the direct demodulation technique with the same simulated data, Fig. 3 (click here)d, has a much clearer background.
Figure 4: Discriminating two nearby sources. a) Object scene. b) Cross-correlation map. c) CLEAN cross-correlation map. d) Image from direct demodulation. e) Image from Richardson-Lucy iterations
For further comparing the angular resolution ability of the two methods, we set up two point sources placed at and with fluxes ph cm-2 s-1 and ph cm-2 s-1, respectively. The results of image analyses by CCM, CLEAN CCM and DDM are shown in Figs. 4 (click here)b, c and d, respectively. From Fig. 4 (click here)d, one can see that the direct demodulation technique can clearly discriminate the two point sources with a separation smaller than the intrinsic resolution of the telescope, while the cross-correlation and the CLEAN cross-correlation cannot. Figure 4 (click here)d gives two point sources sited at and with fluxes ph cm-2 s-1 and ph cm-2 s-1, respectively. We also tried the Richardson-Lucy algorithm (Richardson 1972; Lucy 1974) and got an image shown in Fig. 4 (click here)e, in which many false sources appear and from which the fluxes of the two sources are ph cm-2 s-1 and ph cm-2 s-1, much less than the assumed values.
It is difficult to give correct source locations and intensities in the case of multi-source imaging with a single RMC and the cross-correlation deconvolution. Simultaneous imaging for both point and extended sources is also difficult by a single RMC. Figure 5 (click here)a shows an object scene with two point sources and an extended source. The fluxes of the two point sources are assumed to be and ph cm-2 s-1, respectively, and the total flux of the extended source ph cm-2 s-1. The background, the observational duration and the RMC system configuration are assumed to be the same as that in subsection 3.1. Figure 5 (click here)b shows the cross-correlation map from the simulated data and Fig. 5 (click here)c the CLEAN cross-correlation map, from which only the strong point source can be identified. For the same simulated data, we made a direct demodulation reconstruction and got the result shown in Fig. 5 (click here)d. Besides the strong point source, an image of the weak point source and structure of the extended source can also be seen from Fig. 5d. The fluxes estimated from Fig. 5 (click here)d are and ph cm-2 s-1 for the two point sources and ph cm-2 s-1 for the extended source, respectively.
Figure 5: Imaging for point and extended sources by a single RMC with strip width a=1.0 cm, height H=95 cm and FOV=. a) The object scene. b) Cross-correlation deconvolution. c) CLEAN cross-correlation. d) Direct demodulation reconstruction