The accurate detection and subsequent removal of point-like objects in astronomical images is a frequent problem. Let us mention, for example, the case of stars pervading an extended object or cosmic ray impacts on CCD images. The problem is often complicated by various distortions affecting the point-like objects, e.g. saturations, variable Point Spread Function in the field-of-view, irregular images, which preclude the use of fitting methods. Indeed the classical method of correlation with or without filtering (implemented for instance in the ``inventory" package of MIDAS) does not work well on saturated or irregular images. We propose a new approach to the problem based on the properties of surface curvature which is related to peaks, pits and flats in images and therefore contain the key information to discriminate point-like objects.
The intensity values of an image can be viewed as a two-dimensional surface in a three-dimensional space with the two spatial coordinates forming two of the three dimensions and the intensity axis forming the third dimension. The properties of two-dimensional surfaces embedded in a three-dimensional space are classically studied in differential geometry (Do Carmo 1976; Doneddu 1978; Faux & Pratt 1987). Surface curvature describes how much this surface is bending in a given direction at a particular point; it involves the first and second partial derivatives with respect to the spatial coordinates and thus involves the spatial neighborhood of the point. Surface curvature as an intrinsic tool to analyze images is an old topic in the field of image processing. It has been introduced to detect and characterize peaks, pits, flats (Paton 1974; Peuker & Douglas 1975; Toriwaki & Fukumura 1978; Halarick et al. 1983; Laffey et al. 1985; Nackman 1984), edges (Halarick 1984), and corners (Denger & Billie 1984). Texture analysis and computer vision use it extensively. Curiously, astronomical data processing has ignored this method (as far as we know). It is due, perhaps, to odd side effects induced into the classification process by the frequent very large photometric range of astronomical images. We shall show how to alleviate this difficulty by taking the logarithm of the intensity.
In the sections below, we first recall the definition of the minimum and maximum local curvatures k1 and k2 in a continuous image and then we present the algorithm to calculate these curvatures in a digital image. We then extensively discuss the (k1, k2) diagram in terms of the morphology of objects. Next we explain how to define the neighborhood of point-like objects to be removed and how to restore the background. The method is illustrated on: