  The detection of point-sources in microwave maps is a critical step in the
analysis of the Cosmic Microwave Background (CMB) data. If not properly
removed, these sources have adverse effects on the estimation of the
power-spectrum and/or the test of Gaussianity of the CMB component. The
techniques to extract point sources from an observed image used up to now do
not show a general consensus about their real performance and properties. Their
characteristics have been studied essentially through numerical simulations
based on semi-empirical models of the CMB and the Galactic foreground. Such
models often have different levels of sophistication and/or are based on
different physical assumptions (e.g. the number of Galactic components and
level of the noise). Moreover, the application of a given technique to a set of
data (either simulated or experimental) requires the tuning of one or more
parameters that unavoidably is a subjective operation. Hence, a reliable
comparison is difficult. What is missing is a statistical analysis of the
properties of the proposed methodologies. This is the aim of the present paper.
The statistical properties of the detection techniques in the context of two
different criteria, i.e. the Neyman-Pearson criterion and the maximization of
the signal-to-noise ratio, are analyzed through an analytical approach. Various
examples, techniques and the conditions under which they work satisfactorily
are shown together with their limits and implementation details. Exploiting
some a priori information, it is possible to develop simple algorithms with
performances similar to those of more sophisticated but complex techniques. In
this respect, a detection algorithm, useful in future ground-based experiments
as, for example the Atacama Large Millimeter/submillimeter Array, is presented.
