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The detection of point-sources in microwave maps is a critical step in theanalysis of the Cosmic Microwave Background (CMB) data. If not properlyremoved, these sources have adverse effects on the estimation of thepower-spectrum and/or the test of Gaussianity of the CMB component. Thetechniques to extract point sources from an observed image used up to now donot show a general consensus about their real performance and properties. Theircharacteristics have been studied essentially through numerical simulationsbased on semi-empirical models of the CMB and the Galactic foreground. Suchmodels often have different levels of sophistication and/or are based ondifferent physical assumptions (e.g. the number of Galactic components andlevel of the noise). Moreover, the application of a given technique to a set ofdata (either simulated or experimental) requires the tuning of one or moreparameters that unavoidably is a subjective operation. Hence, a reliablecomparison is difficult. What is missing is a statistical analysis of theproperties of the proposed methodologies. This is the aim of the present paper.The statistical properties of the detection techniques in the context of twodifferent criteria, i.e. the Neyman-Pearson criterion and the maximization ofthe signal-to-noise ratio, are analyzed through an analytical approach. Variousexamples, techniques and the conditions under which they work satisfactorilyare shown together with their limits and implementation details. Exploitingsome a priori information, it is possible to develop simple algorithms withperformances similar to those of more sophisticated but complex techniques. Inthis respect, a detection algorithm, useful in future ground-based experimentsas, for example the Atacama Large Millimeter/submillimeter Array, is presented.

A theoretical framework for the detection of point-sources in Cosmic Microwave Background maps