Practical experiments with the surface curvature of astronomical images have soon revealed to us the superiority of dealing with the logarithm of the intensity, , rather than with the intensity itself. The justifications of this transformation are easily understood:

- i)
- The range of curvatures will be narrower on the images than on the intensity images insuring a better detection of faint objects;
- ii)
- Point-like objects such as stars may be viewed, at least in a first
approximation, as having a Gaussian shape. The logarithmic transformation
will generate similar paraboloids (i.e. having the same parameters)
for which the maximum of curvature takes place at their tops. Practically,
the logarithmic transformation of the star images will exhibit a coarse
paraboloidal shape which will be easily localized from the maximum of
curvature.

Let *x* and *y* be the two spatial coordinates and , the log
transform of the intensity value. We now closely follow the formalism
introduced by Peet & Sahota (1985) and specify any point on the surface by
its position vector:

The first fundamental form of the surface is the expression for the element
of arc length of curves on the surface which pass through the point under
consideration.

It is given by:

where

The second fundamental form arises from the curvature of these curves at the point of interest and in the given direction:

where

and

Casting the above expression into matrix form with:

the two fundamental forms become:

Then the curvature of the surface in the direction defined by *V* is given
by:

Extreme values of *k* are given by the solution to the eigenvalue problem:

which gives *k*_{1} and *k*_{2}, the minimum and maximum curvatures, respectively.
*k*_{1} and *k*_{2} are invariant under rigid motions of the surface. Two other
curvatures are often defined, the Gaussian or total curvature *K* = *k*_{1} *k*_{2}
and the mean curvature *M* = (*k*_{1} + *k*_{2})/2 but it turns out that the principal
curvatures *k*_{1} and *k*_{2} are best suited to the detailed characterization
required by our problem.

As curvature is a geometric property of surfaces, we must introduce a
continuous representation of the discrete 2-dimensional array of data in
order to calculate local derivatives and then, local curvatures. We retain
the so-called facet model (Halarick 1984) which considers a square window
of data points (*L* is an odd integer) centered on the pixel of
interest and calculates the 2-dimensional polynomial fit of degree *N*
(), satisfying the criterion of least square errors
(other functional representations may be chosen,
see Laffey et al. 1985). This
polynomial represents the local continuous surface with continuous
derivatives up to order *N*. The window size *L* can, for instance, be
adapted to the
spatial extent of the Point Spread Function (PSF). It is very often the
case in astronomical images that the PSFs are undersampled; then the lowest
size corresponding to *L* = 3 is retained so as to minimize the error
introduced by the facet model. This, in turn, maximizes the chance of
detecting point-like objects but may result in a large number of false
alarms due to local noise (e.g. noise spikes). When the ultimate
goal is to restore the background, a trade-off should be reached
which insures a minimal percentage of valid pixels as discussed in
Sect. 4 (click here). When the PSFs are oversampled, then a larger size is
recommended to limit the false alarms.
For the simple facet model of second order, i.e. a window
implemented in our examples, the surface is approximated by a
quadric

and the practical calculation of *k*_{1} and *k*_{2} is extremely simple.

It is quite convenient to discuss the surface curvature in the (*k*_{1}, *k*_{2})
space as illustrated in Fig. 1 (click here). With the above convention
and only the corresponding half plane is therefore allowed. Also, according
to the above definitions, if a curve on the surface is concave up, the
surface curvature in that direction is positive and vice-versa. The allowed
half-plane of Fig. 1 (click here) is divided in three regions by the *k*_{1} and *k*_{2} axis.
The two axis themselves correspond to parabolic points, ridges on the *k*_{1}
axis (*k*_{1} < *k*_{2} = 0) and valleys on the *k*_{2} axis (*k*_{2} > 0 = *k*_{1}).
The second quadrant *k*_{2} > 0 > *k*_{1} is the region of the saddle points. The
elliptic points are located in the remaining two half-quadrants, distinguished
by their concavity : pits for *k*_{2} > *k*_{1} > 0 and peaks for *k*_{1} < *k*_{2} < 0.
Obviously, a flat is at the origin *k*_{1} = *k*_{2} = 0.

**Figure 1:** Surface curvature visualized in the plan of the two
principal curvatures *k*_{1} and *k*_{2}

**Figure 2:** A practical *k*_{1}, *k*_{2} diagram for an astronomical image.
The regions of interest are related to specific parts or curvatures
of real objects