While many existing CT noise filtering post-processing techniques optimize minimum mean squared error (MSE)-based quality metrics, it is well-known that the MSE is generally not related to the diagnostic quality of CT images. In medical image quality assessment, model observers (MOs) have been proposed for predicting diagnostic quality in medical images. MOs optimize a task-based quality criterion such as lesion or tumor detection performance. In this paper, we first discuss some of the non-stationary properties of CT noise. These properties will be utilized to construct a multi-directional non-stationary noise model that can be used by MOs. Next, we investigate a new shearlet-based denoising scheme that optimizes a task-based image quality metric for CT background noise. This work makes a connection between multi-resolution sparsity-based denoising techniques on the one hand and model observers on the other hand. The main advantage is that this approach avoids the two-step procedure of MSE-optimized denoising followed by a MO-based quality evaluation (often with contradictory quality goals), while instead optimizing the desired task-based image quality directly. Experimental results are given to illustrate the benefits of the proposed approach.