## The fitting procedure

The fitting technique treats the PSF as a template which may
be scaled and translated. A sub-image centered on the star of
interest is extracted and approximated with the model

where *s*_{0} is the fixed contribution of known
stars outside the sub-image support, *N*_{stars} is
the number of point sources within the sub-image, *
x*_{n}, *y*_{n}, *f*_{n} are
the position and flux of the* n-th* source, *p(x,y)* is
the PSF and *b*_{0}, *b*_{1}, *
b*_{2} are the coefficients of a slanting plane
representing the local background.

The optimization is performed with an iterative Newton-Gauss
technique, which requires the computation of the model
derivatives. For this purpose the Fourier shift theorem is
applied to the PSF:

where *FT* is the discrete Fourier transform, *N* is
the sub-image size and *u*, *v* are spatial
frequencies. The derivative with respect to *x*_{n}
is

and requires the interpolation of the PSF array to compute *
p(x-x*_{n}, y-y_{n} ).A similar technique has
been described by J.-P. Véran and F. Rigaut (*Proc.SPIE
3353,426,1998*).

Many algorithms may be applied to perform the required PSF
Shift. An example is the cubic convolution interpolation
implemented in the IDL function *Interpolate*, which
approaches very close the optimal "*sinc* "
interpolation with well-sampled data. We are also investigating
the accuracy achievable on marginally under-sampled (e.g. HST)
images.

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