Diffraction-limited imaging with Extremely Large ground-based Telescopes is a major challenge of modern astrophysics. But it requires the correction of the wavefront corrugations due to the atmospheric turbulence. I present a global optimization algorithm to solve the non-linear inverse problem of phase retrieval from a speckle image, and I show the benefit of polychromatic diversity (images at different wavelengths) to deal with more degrees of freedom. In the Polychromatic Laser Guide Star framework, I address the problem of assessing the chromatic differential tip-tilt with more accuracy than centroïds. The benefit of the full wavefront joint estimation from the images is evidenced in the case the PLG Star is unresolved. Otherwise I develop an estimator of the star's differential movement, based on the bichromatic cross-spectrum. Finally, I present an optimal real-time data processing of a photon counting detector with an inverse problem formulation (gain of 4 in spatial resolution).