Bivariate signals occupy a special place, as they can be processed specifically to take account of their geometric properties. They correspond to two-dimensional waves characterized by their polarization state, which reflects the covariance between the two signal components. The aim of this work is to enable the resolution of inverse restoration problems for bivariate signals to benefit from regularization terms relating not only to the signal itself, but also to the covariance between components. This objective leads us to work with the analytical signal and to consider the presence of a quartic term in the cost function. An optimization problem under linear constraints is formulated. An ADMM method is proposed, involving efficient updating steps. Numerical simulations illustrate the good performance and speed of the proposed method