We propose a method for resolution enhancement for volumetric images based on a regularized split Bregman approach. We explore the regularization of the shrinkage function based on an approximation of the absolute value function to design a

class of split Bregman methods for Total Variation image restoration. We introduce a hierarchy of regularizations depending on a positive parameter that determines the accuracy in the approximation of the absolute value function by means of rational functions. According to the order of the approximation of the generating rational function used, the degree of smoothness can be dosed for particular image processing applications. The method enhances low-resolution 3D brain MRI images demonstrating good capabilities.