Computational acceleration on graphics processing units (GPUs) can make advanced magnetic resonance imaging (MRI) reconstruction algorithms attractive in clinical settings, thereby improving the quality of MR images across a broad spectrum of applications. At present, MR imaging is often limited by high noise levels, significant imaging artifacts, and/or long data acquisition (scan) times. Advanced image reconstruction algorithms can mitigate these limitations and improve image quality by simultaneously operating on scan data acquired with arbitrary trajectories and incorporating additional information such as anatomical constraints. However, the improvements in image quality come at the expense of a considerable increase in computation. This paper describes the acceleration of an advanced reconstruction algorithm on NVIDIA's Quadro FX 5600. Optimizations such as register allocating the voxel data, tiling the scan data, and storing the scan data in the Quadro's constant memory dramatically reduce the reconstruction's required bandwidth to off-chip memory. The Quadro's special functional units provide substantial acceleration of the trigonometric computations in the algorithm's inner loops, and experimentally-tuned code transformations increase the reconstruction's performance by an additional 20%. The reconstruction of a 3D image with 1283 voxels ultimately achieves 150 GFLOPS and requires less than two minutes on the Quadro, while reconstruction on a quadcore CPU is thirteen times slower. Furthermore, relative to the true image, the error exhibited by the advanced reconstruction is only 12%, while conventional reconstruction techniques incur error of 42%. In short, the acceleration afforded by the GPU greatly increases the appeal of the advanced reconstruction for clinical MRI applications.