  This article considers the detection of point sources in two dimensional
astronomical images. The detection scheme we propose is based on peak
statistics. We discuss the example of the detection of far galaxies in Cosmic
Microwave Background experiments throughout the paper, although the method we
present is totally general and can be used in many other fields of data
analysis. We assume sources with a Gaussian profile --that is a fair
approximation of the profile of a point source convolved with the detector beam
in microwave experiments-- on a background modeled by a homogeneous and
isotropic Gaussian random field characterized by a scale-free power spectrum.
Point sources are enhanced with respect to the background by means of linear
filters. After filtering, we identify local maxima and apply our detection
scheme, a Neyman-Pearson detector that defines our region of acceptance based
on the a priori pdf of the sources and the ratio of number densities. We study
the different performances of some linear filters that have been used in this
context in the literature: the Mexican Hat wavelet, the matched filter and the
scale-adaptive filter. We consider as well an extension to two dimensions of
the biparametric scale adaptive filter (BSAF). The BSAF depends on two
parameters which are determined by maximizing the number density of real
detections while fixing the number density of spurious detections. For our
detection criterion the BSAF outperforms the other filters in the interesting
case of white noise.
