  We present a new processing technique aimed at significantly improving the
angular differential imaging method (ADI) in the context of high-contrast
imaging of faint objects nearby bright stars in observations obtained with
extreme adaptive optics (EXAO) systems. This technique, named "SFADI" for
"Speckle-Free ADI", allows to improve the achievable contrast by means of
speckles identification and suppression. This is possible in very high cadence
data, which freeze the atmospheric evolution. Here we present simulations in
which synthetic planets are injected into a real millisecond frame rate
sequence, acquired at the LBT telescope at visible wavelength, and show that
this technique can deliver low and uniform background, allowing unambiguous
detection of $10^{-5}$ contrast planets, from $100$ to $300$ mas separations,
under poor and highly variable seeing conditions ($0.8$ to $1.5$ arcsec FWHM)
and in only $20$ min of acquisition. A comparison with a standard ADI approach
shows that the contrast limit is improved by a factor of $5$. We extensively
discuss the SFADI dependence on the various parameters like speckle
identification threshold, frame integration time, and number of frames, as well
as its ability to provide high-contrast imaging for extended sources, and also
to work with fast acquisitions.
