Total variation minimization has been extensively researched for image denoising and sparse view reconstruction. These methods show superior denoising performance for simple images with little texture, but result in texture information loss when applied to more complex images. It could thus be beneficial to use other regularizers within medical imaging. We propose a general regularization method, based on a split-Bregman approach. We show results for this framework combined with a total variation denoising operator, in comparison to ASD-POCS. We show that sparse-view reconstruction and noise regularization is possible. This general method will allow us to investigate other regularizers in the context of regularized CT reconstruction, and decrease the acquisition times in µCT.