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DOTA / ONERA, BP 72, 92322 Châtillon cedex, France.

LESIA / Observatoire de Paris-Meudon, 92195 Meudon, France

DOTA / ONERA, BP 72, 92322 Châtillon cedex, France.

We present a new, Maximum Likelihood (ML) based,
method for the estimation of the shift between two images. It notably outperforms the classical cross-correlation method especially in the case of low photon levels. Moreover, it is arbitrarily subpixel, without any resampling of the image, through the maximisation of a criterion.
The method was tested with simulations and was applied to the case of infrared astronomical imaging where the signal is usually very weak. We have also extended our method to the joint estimation of the shifts in a sequence of N images, and preliminary results are presented in last section.

where are the translation parameters, is an additive noise, and is the sampling operateur. If the image is Nyquist sampled, one can reconstruct, via the Fourier domain, a shifted version of the image for any subpixel shift. If we approximate the noise in the image, i.e. a mixture of gaussian (detector) and poissonian noise, as a non-stationnary gaussian noise, then the anti log-likelihood of observing an intensity for the reference intensity and for the hypothesis is given by:

where is the noise variance which can be directly estimated on the image. It is easy to show that, following the two hypothesis of stationnarity of the noise and of periodicity of the reference, the ML estimate of the translation between the two images is the maximum of the linear cross-correlation of the images. When the reference is not known, one has to consider a noisy frame as a reference.

Where includes both the noise in the image used as a reference and the noise in the image to be recentered. Then the anti log-likelihood to be minimize has the same expression as in equation 2 changing into and into :

To find the minimum of this criterion, we used a gradient type adaptive step minimization algorithm, issued from a collaboration of our team with the Groupe des Problemes Inverses at Laboratoire des Signaux et Systemes (GPI 1997). However, one has to notice that the criterion, in the case of unknown reference and considering the real noise variance contains a lot of local minima. This make the minimization difficult and so should decrease the performance of the method in this case.

In the case of an unknown reference, the performance of our method is quite identical considering or not the real noise distribution since the criterion contains a lot of local minima in the first case. It notably outperforms the cross-correlation at low photon level and allows subpixel accuracy as soon as the number of photon per pixel is greater than the variance of the detector noise as the accuracy of the interpolated cross correlation is worst than the pixel.

This allows to compare this image to the one obtain with the space telescope in other bands giving insightfull astophysical interpretations (see Gratadour et al. 2003).

One can show that minimizing on and is equivalent to minimize: on , with:

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