*Astron. Astrophys. Suppl. Ser.* **144**, 363-370

**C. Alard ^{1,2}**

1 - DASGAL, 61 avenue de l'Observatoire, F-75014
Paris, France

2 -
Institut d'Astrophysique de Paris, 98bis boulevard Arago, F-75014, France

Received March 31, 1999; accepted March 6, 2000

Image subtraction is a method by which one image is matched against
another by using a convolution kernel, so that they can be differenced
to detect and measure variable objects. It has been demonstrated that
constant optimal-kernel solutions can be derived over small sub-areas
of dense stellar fields. Here we generalize the theory to the case of
space-varying kernels. In particular, it is shown that the CPU cost
required for this new extension of the method is almost the same as for
fitting a constant kernel solution. It is also shown that constant flux
scaling between the images (constant kernel integral) can be imposed in
a simple way. The method is demonstrated with a series of Monte-Carlo
images. Differential PSF variations and differential rotation between
the images are simulated. It is shown that the new method is able to
achieve optimal results even in these difficult cases, thereby
automatically correcting for these common instrumental problems. It is
also demonstrated that the method does not suffer due to problems
associated with under-sampling of the images. Finally, the method is
applied to images taken by the OGLE II collaboration. It is proved
that, in comparison to the constant-kernel method, much larger
sub-areas of the images can be used for the fit, while still
maintaining the same accuracy in the subtracted image. This result is
especially important in case of variables located in low density
fields, like the Huchra lens. Many other useful applications of the
method are possible for major astrophysical problems; Supernova
searches and Cepheids surveys in other galaxies, to mention but
two. Many other applications will certainly show-up, since variability
searches are a major issue in astronomy.

**Key words: **methods: numerical -- methods: statistical -- stars: variables: general --
cosmology: gravitational lensing

- 1 Introduction
- 2 Basic equations
- 3 Solving for space-varying kernel solution with minimum computing time
- 4 Imposing constant flux scaling
- 5 Numerical simulations
- 6 Testing with astronomical images
- 7 Conclusion and summary
- References

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