on the basis of X-ray data. In principle, the only requirement to characterize a source as extended is that its measured size must be larger than the instrument point spread function (PSF). However, given the large number density of active galactic nuclei (AGNs) and distant star-forming galaxies, unresolved sources largely outnumber extended ones, especially at low fluxes. The risk of finding faint extended emission encompassing several unrelated unresolved sources is high. Moreover, in the classical Wolter-type design of X-ray mirrors, the PSF varies significantly across the field of view (FOV), introducing more uncertainties. Finally, extended X-ray sources are complex in morphology, because they come in a wide variety of shapes and surface brightness distributions, with scales ranging from a few arcsec to a few arcmin. Therefore, a considerable effort should be made to search and characterize extended sources in X-ray images.

In this paper, we aim at providing a flexible and efficient detection algorithm to identify extended X-ray sources down to a low flux level to exploit the large amount of data in the archive of current X-ray missions ([*Chandra*]{}, [*XMM-Newton*]{}, [*Swift*]{}, [*Suzaku*]{}), and to explore the scientific cases of future X-ray facilities. We mostly focus on the detection of 