Rapid MR Imaging with "Compressed Sensing" and Randomly Under-Sampled 3DFT Trajectories

M. Lustig, D. L. Donoho, J. M. Pauly Electrical Engineering, Stanford University, Stanford, CA, United States, Statistics, Stanford University, Stanford, CA, United States Introduction Recently a rapid imaging method was proposed [1] that exploits the fact that sparse or compressible signals, such as MR images, can be recovered from randomly under-sampled frequency data [1,2,3]. Because pure random sampling in 2D is impractical for MRI hardware, it was proposed to use randomly perturbed spirals to approximate random sampling. Indeed, pure 2D random sampling is impractical, however, randomly undersampling the phase encodes in a 3D Cartesian scan (Fig. 1) is practical, involves no overhead, is simple to implement and is purely random in two dimensions. Moreover, scan-time reduction in 3D Cartesian scans is always an issue. We provide a method to evaluate the effective randomness of a randomly under-sampled trajectory by analyzing the statistics of aliasing in the sparse transform domain. Applying this method to MR angiography, where images are truly sparse, we demonstrate a 5fold scan time reduction, which can be crucial in time-limited situations or can be used for time resolved imaging Theory Medical images in general, and specifically angiograms, often have a sparse representation using a linear transform (wavelets, DCT, finite differences, etc.)[1]. Under-sampling the Fourier domain results in aliasing. When the under-sampling is random, the aliasing is incoherent and acts as additional noise interference in the image, but more importantly, as incoherent interference of the sparse transform coefficients. Therefore, it is possible to recover the sparse transform coefficients using a non-linear reconstruction scheme [1-4] and consequently, recover the image itself. The interference in the sparse domain is a generalization of a point-spread function (PSF) and is computed by I(n,m)= where xn is the n th