Future ground-based direct imaging of exoplanets depends critically on high-contrast coronagraph and wave-front manipulation. A coronagraph is designed to remove most of the unaberrated starlight. Because of the wave-front error, which is inherit from the atmospheric turbulence from ground observations, a coronagraph cannot deliver its theoretical performance, and speckle noise will limit the high-contrast imaging performance. Recently, extreme adaptive optics, which can deliver an extremely high Strehl ratio, is being developed for such a challenging mission. In this publication, we show that barely taking a long-exposure image does not providemuch gain for coronagraphic imaging with adaptive optics. We further discuss a speckle subtraction and suppression technique that fully takes advantage of the high contrast provided by the coronagraph, as well as the wave front corrected by the adaptive optics. This technique works well for coronagraphic imaging with conventional adaptive optics with a moderate

Strehl ratio, as well as for extreme adaptive optics with a high Strehl ratio. We show how to substrate and suppress speckle noise efficiently up to the third order, which is critical for future ground-based high-contrast imaging. Numerical simulations are conducted to fully demonstrate this technique.