Determination of the PSF

To optimally use the MCS algorithm, it is necessary to know the PSFS(~x) with a high accuracy. Indeed, it is the most important step to ensure that MCS will be successful.

For that purpose, an algorithm was developed, on the same basis as MCS, to determine the PSF on images consisting of possibly blended point sources (Magain et al., 2007).

This method, often called PSFsimult to ease the conversation, works well, even in very crowded fields, when no point source is sufficiently isolated to derive an accurate PSF from standard techniques. However, it assumes that the portion of the image used to determine the PSF only contains point sources.

For a single one-dimensional image containing one point source with intensity a, centered in c and with the usual notations, the function to minimize is:

K^S =

where wij is a length scale for the smoothing term. Practically, we choose wij equal to rij because, if a deconvolved frame cannot contain any frequency higher thanR(~x), then neither should the PSFS(~x) used for its construction. The parameters to modify

during the iterations of the algorithm are the following: the center of the point source, its intensity and all the pixels si from the partial PSF frame.

What if we have at our disposal a frame containing several possibly blended point sources ? Let us consider a single observed frameD (~x). It can be expressed as follows:

D (~x) =

F (~x) ×+S (~x)

+ N (~x) , (5.27)

S(~x) being the partial PSF. The usual problem is to recover F (~x) knowing D (~x) and S (~x). If we consider a part of the frame which contains no numerical background but M point sources of intensity ak and centered in ~ck, the deconvolved imageF(~x) can be expressed as follows:

The aim of the PSF determination program PSFsimult is to obtain S(~x) while D(~x) is known and F(~x) is given by Eq. 5.28. If, indeed, the shape of F(~x) is known, the intensities and positions of the point sources are considered as free parameters and must be determined at the same time as the partial PSFS(~x). If the image contains N pixels, the function to minimize is the following:

K =

where g is a gaussian function. The width of g and the value of λ are adjusted so that χ²≈ N, as N is approximately equal to the number of degrees of freedom.

In the case of an image composed of blended sources, a bump from a neighbouring point source can be interpreted by the program as another source or as a bump in the wings of the PSF or even as a mixture of both. We thus have to prevent the algorithm from finding an acceptable solution (forS (~x)) in terms of χ², but presenting bumps in the wings from the neighbouring stars. To avoid such local minima, we proceed in two steps. First the PSF is approximated by an analytical function such as a Moffat profile, which is defined as follows:

M(x, y) = a1 + b1(x− c^x)²+ b2(y− c^y)²+ b3(x− c^x)(y− c^y)^−β

. (5.30) This profile is centered on (cx, cy) with an intensity a. The parameters b1, b2 and b3

specify the ellipticity and orientation of the Moffat function and β controls the width of the wings. By definition, there will be no bumbs in such a PSF. But this is not sufficient to get an accurate instrumental profile: as a second step, a numerical component is added to this analytical one. And to avoid the bumps, we proceed gradually in fitting first the central regions of the PSF on the different point sources and then in gradually enlarging the modified area. In doing so, the algorithm first fits reasonably the intensity of the sources in the central regions of the PSF and does not add spurious bumps in the wings. Let us note that the smoothing term in Eq. 5.29 is only applied to the numerical part of the PSF and not to the analytical one.

5.3 The MCS algorithm and its advantages 63

In this work we also treat HST/NICMOS-2 images, HST standing for Hubble Space Telescopeand NICMOS for Near Infrared Camera and Multi-Object Spectrometer. The Tiny Tim software package (Krist & Hook, 2004) allows the generation of a numerical estimation of HST PSFs for each instrument, filter and observing configuration. More-over, it depends on the spectral type of the observed object. Most of the astrophysicists are usually satisfied with that PSF to get rid of the instrumental shape. But the actual PSF always departs significantly from this approximated version, especially in the core.

And to reach the best results, we need a better estimation. That is why we apply PSFsimultto the HST frames. However, instead of departing from an analytic function as a Moffat profile, we start from a Tiny Tim PSF. As the Tiny Tim software computes the total PSF T (~x), we first have to deconvolve it by the final resolution R(~x). In practice we notice that doing this significantly improves the results. We then run the PSF determination algorithm, in order to improve the deconvolved Tiny Tim PSF in adjusting it to the point sources of the frame.

Moreover, as the MCS algorithm allows the user to oversample the deconvolved frame compared to the original data⁶, the instrumental profile itself has to be oversampled by the desired factor. That is why we take advantage of the presence of several point sources on one frame: they are located on different positions on the CCD and so centered differently on the pixel grid. They can be deconvolved simultaneously, each one of them constraining the numerical component of the unique oversampled output PSF. That gives a PSF with a higher accuracy than one adjusted on a single source.

Chapter 7 presents a method which extends the one of Magain et al. (2007) to images containing point sources lying on a diffuse background. It is based on an iterative scheme, in which both the PSF and diffuse background of the frame are improved step by step.

6This is useful for the deconvolution of images of an object for which the position on the detector is shifted from frame to frame. Let us remind that it is called dithering.

Part II