3. Discrimination of point-like objects on the (k₁, k₂) diagram

Let us now consider how the above theoretical considerations apply to real astronomical images. Figure 2 (click here) is a sketch of the (k₁, k₂) diagram based on a CCD image which exhibits an extended source, the tail of comet P/Halley, a rich star field and various defects and artefacts. The procedure applied to this image will be detailed in Sect. 5 (click here) as a practical example. The features of this (k₁, k₂) diagram are sufficiently general to localize the various domains of interest and to establish the appropriate criteria for the discrimination.

The main feature of the diagram is a ``butterfly" pattern composed of two wings A and B and a central core C. Both wings pertain to stars, wing A to their top and wing B to their bottom parts. These later parts are fairly asymmetric saddle points leaning to valleys and explaining the closeness of the wing B to the k₂ axis. In fact, the orientation of the wing, as given for instance by the slope k₂/k₁ of their symmetry axis, is a very good estimate of the anisotropy of the surface curvature. For instance, elongated (e.g. trailed) star images will induce a rotation of wing A. The central core C obviously corresponds to the flat parts of the image, essentially the extended object and the background. Finally, two additional features are conspicuous in the diagram, the blobs E and F respectively localized on the k₁ and k₂ axis. They correspond to elongated structures in the original image which involve both ridges (top parts) and valleys (bottom parts). These structures are typically bad columns in the CCD and saturated star images which spill off along the column direction.

It is now a very simple matter to establish criteria for discriminating objects in the original image. These criteria are written in terms on conditions on k₁ and k₂ or on a combination of the two. A mask or several masks are generated and then applied to the original image. Two simple cases may be given as practical examples. Let denote some small positive values of the curvatures. Two criteria may be used to detect the top of the stars

i)

ii)

Conversely the possible criteria for discriminating the extended source and background are

i)

and

ii)

respectively delimiting a square and a circular region in the (k₁, k₂) diagram. Our experience is that the two criteria lead to nearly the same result. For practical matters, the modulus of the curvature k = (k²₁ + k²₂)^1/2 has the advantage of combining the principal curvatures so that a single parameter has to be dealt with. Inspection of the histogram of the k values generally allows a straightforward determination of the border between point-like objects and extended objects.