In this paper, we present a method for reconstructing images or volumes from a partial set of observations, under the Rayleigh distributed multiplicative noise model, which is the appropriate algebraic model in ultrasound (US) imaging. The proposed method performs a variable splitting to introduce an auxiliary variable to serve as the argument of the total variation (TV) regularizer term. Applying the Augmented Lagrangian framework and using an iterative alternating minimization method lead to simpler problems involving TV minimization with a least squares term. The resulting Gauss Seidel scheme is an instance of the Alternating Direction Method of Multipliers (ADMM) method for which convergence is guaranteed. Experimental results show that the proposed method achieves a lower reconstruction error than existing methods.