Multiplicative noise removal is of momentous significance in coherent imaging systems and various image processing applications. This paper proposes a new nonconvex variational model for multiplicative noise removal under the Weberized total variation (TV) regularization framework. Then, we propose and investigate another surrogate strictly convex objective function for Weberized TV regularization-based multiplicative noise removal model. Finally, we propose and design a novel way of fast alternating optimizing algorithm which contains three subminimizing parts and each of them permits a closed-form solution. Our experimental results show that our algorithm is effective and efficient to filter out multiplicative noise while well preserving the feature details.