Centre d'Etude Spatiale des Rayonnements (CESR-CNRS), BP 4346, 31028 Toulouse Cedex 4, France

We propose a new Field-Matching algorithm for astronomical images.
This new method is based on a multiresolution analysis. We tried
two cases: first, we compared the test image with a synthetic
image built from a point source catalog; second, we used a reference
image of the relevant portion of the sky. Structures of images are
obtained at different scales by applying the wavelet transform.
An appropriate thresholding of the wavelet coefficients gives the
significant pixels in the Wavelet Transform Space. In order to
compare the selected coefficients between the test and the reference
images we used a genetic algorithm. We applied this method on
images taken by the automatic TAROT (Rapid Action
Telescope for Transient Object) telescope. The reference data are
taken from the USNO-A2.0 Catalog and from the Digital Sky Survey.
The results are more robust and reliable than those obtained
with the FOCAS algorithm. Moreover, the new algorithm is faster
than FOCAS.

We obtain two structures that we match using a genetic algorithm. It will give us the vertical and horizontal offsets between the two structures and then, the offset between the two original images.

*Mallat's Analysis*
This analysis is a non-redundant one because the amount of data
is divided by two at each scale: this is called a dyadic
analysis. In two dimensions, Mallat's analysis uses three
wavelets which leads to an anisotropic analysis. We obtain the
horizontal, diagonal and vertical details of the image at each
scale. We used the Daubechies wavelet of degree four. However,
the astronomical objects are usually isotropic and without privileged directions.
That is why we use the so called ``à trous'' algorithm (Starck et al. 1995).

*The ``à trous'' Algorithm*
This analysis is isotropic but redundant. The image is smoothed
on the different scales. Because of the redundancy, this
algorithm is not as fast as Mallat's algorithm.

At each scale and for each details image, we have 20 significant pixels. The set we obtain is called the structure of the studied image. We apply one of the algorithms on the test image and on the reference image and we obtain two structures. Finally, we have to match both structures and find the original offset between the two images.

Here we want to find the vertical and horizontal offsets between the two original images. The population of our algorithm is a vertical and horizontal offset pair to apply to one of the structures. The algorithm will converge to the best offset pair. Taking into account the scale of the studied structure, we can find the original offset by multiplying by the relevant factor.

We made a Matlab implementation of the algorithms. For the genetic algorithm, we took a population of 100 offset pairs, the selection function was a tournament. We only used two crossover and two mutation functions. With a convergence time of 40s, the Mallat's analysis is faster than the ``à trous'' algorithm (120s). We then decided to use the anisotropic method. We matched the horizontal and vertical details images of the third scale.

Then, we compare the matching with the DSS images and the one with the USNO-A2.0 catalog images. In Table 1, we give the original vertical and horizontal offset, and those found after the convergence. Finally, we show the results of the FOCAS method (matching with the USNO catalog): we give the number of matched stars and the number of stars found on the image.

For the DSS images, 30 images of 31 are matched. For the only non-matched image, we took the structures of the second scale. The new found offset is (40,64), and the matching is then done. For the USNO images, only 13 of 31 images are matched.

Nevertheless, the new method is faster and more robust than other methods. It does not require a good knowledge of the centroïd coordinates.

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Starck, J.-L., Murtagh, F., & Bijaoui, A. 1995, in ASP Conf. Ser., Vol. 77, Astronomical Data Analysis Software and Systems IV, ed. R. A. Shaw, H. E. Payne, & J. J. E. Hayes (San Francisco: ASP), 279

Lega, E., Bijaoui, A., Alimi, J. M., & Scholl, H. 1996, A&A, 309, 23

Houck, C., Joines, J., & Kay, M. 1995, NCSU-IE TR 95-09

Mallat, S. 1989, IEEE Trans on Pattern Anal. and Math. Intel., 11, 7

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