4 Simulation study

We have tested the acceleration provided by OS-EM using some synthetic images generated by S. Correia and A. Richichi from Osservatorio Astrofisico di Arcetri. A complete description of these images is contained in Correia & Richichi (2000).

The first is a $128\times 128$ image of the spiral galaxy NGC 1288. We have computed 4 sets of simulated observations, corresponding to 4,6,8 and 10 equispaced values of the parallactic angle between $0^\circ$ and $180^\circ$ and to an integration time of 1000 s per parallactic angle. For these computations we have used ideal diffraction-limited PSF's, without taking into account the rotation of the baseline during the integration time nor the residual effect of the adaptive optics correction. The FWHM of these PSF's in the direction of the baseline was about 4 pixels. More realistic PSF models will be considered in future work. Finally each image has been corrupted by Poisson and read-out noise.

We have applied both LR/EM and OS-EM to each set of observations. The quality of the restoration provided by the k-th iteration has been measured by computing the relative RMS error defined as follows

$$\varrho^{(k)}={{\Vert\vec{f}^{(k)}-\vec{f}\Vert}\over{\Vert\vec{f}\Vert}}$$ (11)

where $\vec{f}$ is the original image of the galaxy and the norm is the Euclidean one.

Figure 1: Plot of the relative restoration error $\varrho ^{(k)}$ as a function of the number of iteration k, both in the case of LR/EM (full-line) and in the case of OS-EM (dotted line). The case considered is that of 8 equispaced observations of the NGC 1288 galaxy

In Fig. 1 we plot the behaviour of $\varrho ^{(k)}$ as a function of k for the two methods (full line for LR/EM, dotted line for OS-EM) when p=8. In both cases we find a minimum of the restoration error. The number of iterations corresponding to the minimum will be denoted by $k_{{\rm opt}}$. It is evident that $k_{{\rm opt}}$(OS-EM) is considerably smaller than $k_{{\rm opt}}$(LR/EM). The best restoration provided by OS-EM, which practically coincides with that provided by LR/EM, is shown in Fig. 2. All the results obtained in this example are summarized in Table 1 where $\varrho_{{\rm min}}$ denotes the minimum value of $\varrho ^{(k)}$. Several interesting features of OS-EM result from this table.

 

 

Table 1: Values of the minimum restoration error and of the corresponding number of iterations for various values of p both in the case of LR/EM and in the case of OS-EM. The example is the galaxy NGC 1288

p	$\varrho_{{\rm min}}$	$k_{{\rm opt}}$	$\varrho_{{\rm min}}$	$k_{{\rm opt}}$
	(LR/EM)	(LR/EM)	(OS-EM)	(OS-EM)
4	4.4%	314	4.4%	79
6	4.0%	380	4.0%	64
8	3.9%	393	3.9%	50
10	3.5%	452	3.5%	46

Figure 2: a) Original image of the galaxy NGC 1288. b) One of the simulated LBT psf (parallactic angle = $67.5^\circ$). c) One of the simulated LBT images (parallactic angle = $0^\circ$). d) Restoration by means of 8 images, using the OS-EM method

First we observe that OS-EM provides the same restoration error as LR/EM. This restoration error is rather small, thanks to the high SNR assumed for the LBT images (the peak SNR is 80). As it must be, it decreases for increasing number of observations. Secondly the ratio between $k_{{\rm opt}}$(LR/EM) and $k_{{\rm opt}}$(OS-EM) is approximately p. As a result, while $k_{{\rm opt}}$(LR/EM) increases for increasing p, $k_{{\rm opt}}$(OS-EM) decreases. Since the computational cost of each iteration is roughly proportional to p, this result means that the computational cost of LR/EM rapidly increases for increasing p, while the cost of OS-EM increases very slowly.

The second example refers to binary stars of different relative magnitude. In the synthetic object each star is located in one pixel, with a separation between stars of about 14 pixels, which is about 3.5 times the FWHM of the PSF's in the direction of the baseline.

Figure 3: Plot of the relative magnitude $\Delta m_{\rm r}$ as a function of the number of iterations both in the case of the LR/EM-method (full line) and in the case of the OS-EM method (dotted line). All the computations are performed using 8 different baselines. a) the case of binary stars with $\Delta m_{\rm r}=2.5$. b) the case of binary stars with $\Delta m_{\rm r}=1$

Moreover two cases are considered: 1) the magnitudes of the two stars are 27.5 and 30 and the average peak SNR's in the image are 11.3 for the main star and 1.1 for the companion; 2) the magnitudes are 29 and 30 while the corresponding average peak SNR's are 5.5 and 2.2 respectively (Correia & Richichi 1999). Also for these examples we have computed sets consisting of 4,6,8 and 10 equispaced observations.

These examples confirm that OS-EM provides an acceleration of LR/EM by a factor of p. However the convergence is much slower than in the example of the galaxy; a number of iterations of the order of thousand is required by LR/EM and, of course, a number of iterations of the order of hundreds by OS-EM. Our simulations confirm a well known feature of LR/EM: extended objects are recovered with a number of iterations smaller than that required for point-like objects.

As concerns the RMS restoration errors, they are higher than in the case of the galaxy, as a consequence of the smaller SNR: about 10% for example 1) and about 25-30% for example 2). These RMS's are essentially unchanged if they are computed not on the complete image but on two square regions of $5\times 5$ pixels centered on the positions of the two stars.

These simulations have also been used for comparing the astrometric and photometric performances of the two methods. Astrometric position is fully retrieved by both methods also in the case of the faint companion. Analogously both methods provide essentially the same photometric results. Indeed we performed aperture photometry on the restored images using $5\times 5$ pixels regions centered on the brightest pixel of the reconstructed stars. In Fig. 3 we plot, as a function of the number of iterations, the relative magnitude for the restoration of the binary with $\Delta m_{\rm r}=2.5$ (Fig. 3a) and of that with $\Delta m_{\rm r}=1$ (Fig. 3b), both in the case of LR/EM (full-line) and in the case of OS-EM (dotted-line). The computations are performed using 8 images. The figures show that both methods provide essentially the same results after about 100 iterations. A more detailed discussion of photometry and of its dependence on the number of iterations and noise can be found in Correia & Richichi (1999).