Compressed Sensing uses low-level statistical properties of MR-images to reconstruct images from undersampled k-space data (1). It takes advantage of the of the image or its wavelet or finite difference domain, a property that distinguishes (MR-)images from noise and can be assumed to apply to any MR image. While much has been done to optimize the reconstruction algorithms, the question of how to ... [Show full abstract] optimally sample k-space for sparse reconstruction has not been addressed before. Here, Bayesian inference is used for a directed search for the optimal k-space trajectories. Using a new convex optimization algorithm for approximate inference, this approach is scaled up to full-size MR images. Error = MAP reconstructions for Cartesian undersampled data for 64 of 256 acquired lines. Four different k-space schemes, each consisting of the central 32 lines plus 32 lines positioned by different scenarios are compared to the fully sampled image (center). The given error is the squared difference between the fully sampled and the undersampled images. The optimized k-space scheme shows significantly better recovery of detail than images reconstructed from other undersampled data.