It is faster and therefore cheaper to acquire magnetic resonance images (MRI) with higher in-plane resolution than through-plane resolution. The low resolution of such acquisitions can be increased using post-processing techniques referred to as super-resolution (SR) algorithms. SR is known to be an ill-posed problem. Most state-of-the-art SR algorithms rely on the presence of external/training data to learn a transform that converts low resolution input to a higher resolution output. In this paper an SR approach is presented that is not dependent on any external training data and is only reliant on the acquired image. Patches extracted from the acquired image are used to estimate a set of new images,where each image has increased resolution along a particular direction. The final SR image is estimated by combining images in this set via the technique of Fourier Burst Accumulation. Our approach was validated on simulated low resolution MRI images,and showed significant improvement in image quality and segmentation accuracy when compared to competing SR methods. SR of FLuid Attenuated Inversion Recovery (FLAIR) images with lesions is also demonstrated.