5 Application to stellar field post-processing

One of the first applications of wide FOV imaging in astronomy is the observation of large stellar fields. In this case, we have a strong a priori information on the object: it is a sum of Dirac functions [Gunsay & Jeffs1995,Fusco et al.1999b]:

$$o(\text{\boldmath$\alpha$ })=\sum_{k=1}^{n}\gamma_k\delta(\text{\boldmath$\alpha$ }-\text{\boldmath$\alpha$ }_k)$$ (17)

where $\text{\boldmath$\alpha$ }_k$ and $\gamma_k$ are the position vector and the intensity of the $k^{\rm th}$ star respectively. They are the object parameters we are seeking. We consider that n, the number of stars, is a known parameter. Note that, using appropriate prior statistics (Bernouilli Gaussian or Poisson-Gaussian for example), it is possible to generalize the method and to incorporate n as an unknown of the problem.

5.1 The deconvolution method

To solve this inverse problem, i.e., estimate the unknowns ( $\text{\boldmath$\alpha$ }_k, \gamma_k$), we use the approach presented in [Fusco et al.1999b] with a isoplanatic PSF, but modified to include the anisoplanatic effects. Note that this appraoch can be seen as a "PSF fitting'' in the Fourier domain.

The criterion to be minimized with respect to $\alpha_k$ and $\gamma_k$ can be written in the Fourier domain as:

$$\begin{align} &J(\gamma_k,\text{\boldmath$\alpha$ }_k){=}\nonumber \\ &{\Big\Ve... ...ldmath$\alpha$ }_k}({\bf f})-\tilde{i}({\bf f})\Big\Vert^2 \nonumber \end{align}$$ (18)

where $\text{OTF}_{\text{\boldmath$\alpha$ }_k}({\bf f})$ is the OTF for the $k^{\rm th}$ star and ${\bf f}$ the spatial frequency and $\tilde{.}$ denotes a Fourier transform. $\text{OTF}_{\text{\boldmath$\alpha$ }_k}({\bf f})$ is slowly variable with $\alpha_k$. Then a crude estimation of $\alpha_k$ using a first deconvolution with a constant PSF in the whole field of view allows the calculation of the associated ATF's. In a first approximation, these ATF estimates are assumed to be the true ones and are not adjusted in the deconvolution process. The criterion to be minimized is then:

$$\begin{align} J(\gamma_k,\text{\boldmath$\alpha$ }_k)&= \Big\Vert\sum_{k=1}^{... ...mber & \text{ATF}_{\alpha_k}({\bf f})-\tilde{i}({\bf f})\Big\Vert^2. \end{align}$$ (19)

Note that a more complete, but more complex approach should consider $\text{ATF}_{\alpha_k}({\bf f})$ as unknow of the problem.

This minimization is done using a conjugate gradient method. The object reparametrization allows an accurate precision on the parameters (sub-pixel precision on the star positions). Nevertheless, the criterion is not convex and a good first initialization for the star positions is suitable to avoid problems related to local minima. This initialization is made using a pixel by pixel deconvolution (Wiener filter or Lucy Richardson Algorithm for example).

Let us consider an object which is a field of 21 stars with the same magnitude. The separation between each stars is one arcsecond (see Fig. 1). In order to focus on the limitation induced by the anisoplanatic problem and the gain brought by the ATF estimation, we first consider a noise free image, but the noise influence will be discussed later in the paper.

5.2 Noise free image

We compare, on a noise free image (see Fig. 1), the gain brought by the ATF introduction in the image processing. ATF's for each star are computed using an estimation of $\alpha_k$ given by the first pixel by pixel deconvolution (Wiener filter). $\text{OTF}_0$ is simply given by the fourier transform of the on-axis PSF. We plot in Fig. 7 the error, given in percent, on the stars magnitude estimate using "isoplanatic'' deconvolution process (deconvolution using criterion defined in Eq. (18) but assuming that the PSF is constant in the whole FOV) and our modified approach using the ATF estimation for each star position Eq. (19). Figure 7 shows the gain brought by the use of the ATF in the a posteriori processing. The error, which increases as a function of angle, in the "isoplanatic'' deconvolution case, up to 34% for $\alpha=20\hbox{$^{\prime\prime}$ }$, is close to be constant and only about 1% when we use the ATF estimation. We believe that this residual error is due to the non-stationarity assumption made in the ATF estimation (see Sect. 3.3).

Figure 7: Error in percent on the magnitude estimation for each star in the case of a noise free image.[$\Diamond$] estimation using the same PSF ( $\alpha =0$) for the whole FOV, [*] estimation using the ATF associated to each star

5.3 Influence of the photon noise

Of course, in a realistic case, the image is not only degraded by the turbulence effect but also affected by photon and detector noise. Let us assume that in the case of AO long exposures, the dominant noise is the photon noise. It leads to an error on the parameter estimation.

In the case of a "isoplanatic'' deconvolution (PSF constant in the whole FOV), for a given flux (that is for a given noise level), a limit angle can be defined below which the anisoplanatic error is lower than the noise error. The lower the flux, the greater the limit angle (see Fig. 8). For example, in the case of 10³ photons the anisoplanatic error becomes greater than the noise error for $\alpha \ge 7\hbox{$^{\prime\prime}$ }$. This limit angle can be seen as a definition of an isoplanatic angle for our deconvolution method.

Figure 8: Error in percent on the magnitude estimation using the same OTF ( $\alpha =0$) in the whole FOV, in the case of a noise free image, and in the case of photon noise with 106, 105, 104 and 103 photons in the whole image respectively. The error is only averaged on 10 noise outcomes for each flux case

Now, let us use the ATF correction of the OTF in the post-processing. In that case, the noise error is dominant for the whole FOV, as soon as it is of the order or greater than 1% which corresponds to 10⁵ photons in our case (see Fig. 9).

Figure 9: Error in percent on the magnitude estimation using the ATF correction of the OTF in the post processing in the case of a noise free image, and in the case of photon noise with 106, 105 and 103 photons in the whole image respectively. The error is only averaged on 10 noise outcomes for each flux case

Even if the ATF under-estimation due to the phase non-stationarity error is the dominant noise for a flux greater than 10⁵ photons, the gain brought by the use of the ATF is still very important with respect to a "isoplanatic'' deconvolution (see Figs. 8 and 9). And, at low flux, the error in the FOV is only limited by the photon noise.

5.4 Experimental results

Let us now apply the wide FOV deconvolution method (Eq. 19) on the two experimental images of $\xi$ Cephee and $\Theta$ Orionis presented in Sect. 4. In these two cases, because of the large angular separation between the two components, an accurate estimation of the magnitude difference and of the separation can be obtained using the sole images and computing a center of gravity and an integral of the flux for each image. Since we have a good estimation of the parameters of interest by this aperture photometry method, it is interesting to validate our deconvolution approach on this data and to evaluate the gain brought by the ATF estimation.

The results of the estimation of the separations and the magnitude differences are summarized in Table 3 for each processing case:

- a crude aperture photometry for the magnitude estimation plus a center of gravity estimation for the seperation estimation;

- a deconvolution with a sole PSF (on axis PSF, that is the brighter star image) for the whole image ("isoplanatic'' deconvolution);

- and the use of the ATF model in the deconvolution process.

 

 

Table 3: Angular separation and magnitude difference estimation for $\xi$Cephee and $\Theta$ Orionis using a center of gravity calculation, a "isoplanatic'' deconvolution (a sole PSF for the whole image) and a deconvolution with the anisoplanatic OTF

$\xi$ Cephee	Center of	"Isoplanatic''	ATF estimation
	gravity	deconvolution	deconvolution
Angular
separation	$7.95\hbox{$^{\prime\prime}$ }$	$7.94\hbox{$^{\prime\prime}$ }$	$7.95\hbox{$^{\prime\prime}$ }$
Magnitude
difference	1.89	2.01	1.87
$\Theta$ Orionis	Center of	"Isoplanatic''	ATF estimation
	gravity	deconvolution	deconvolution
Angular
separation	13.30	13.45	13.41
Magnitude
difference	1.48	1.96	1.50

These results clearly show the gain brought by the ATF correction of the OTF on the estimation of star parameters: mainly on the magnitude estimation. The "isoplanatic'' deconvolution process, which is not efficient in the case of large FOV, because of the PSF variation, is strongly improved by the introduction of the theoretical degradation of this PSF as a function of angle.