Rapid Imaging using Undersampled Radial Trajectories and L1 Reconstruction

Introduction: Compressed Sensing (CS) is an emerging field that suggests that data compression can be implicitly incorporated into the data acquisition process [1,2,3]. CS is regarded by some as an alternative to Shannon’s Nyquist sampling theory. The Nyquist sampling theory states that the number of samples required to perfectly reconstruct a signal is determined by its bandwidth. In contrast, the CS theory states that by using nonlinear algorithms based on convex optimization, certain class of signals can be reconstructed perfectly from what appears to be highly incomplete data. More specifically, the CS theory states that sparse or compressible signals can be recovered from a small number of random linear measurements. These results are of practical significance to MR imaging, since MR imaging is performed using linear measurements of the object (in k-space) and the MR images often have sparse (or compressible) representations (e.g. using wavelets or finite differences). The main difficulty in applying these initial theoretical results to MRI lay in performing random sampling in k-space. Since such random sampling is difficult for MRI hardware, recent reports have suggested some alternative strategies for randomizing the measurements. In [5], a rapid imaging method based on randomly perturbed and undersampled spirals was introduced. In [6], a randomly undersampled 3DFT trajectory was used for rapid imaging. Random ordering of the phase encodes in time was used in [7] to randomly undersample k-t space in dynamic cardiac imaging. However, recent developments in CS theory suggest that such random sampling may not always be necessary [4]. More specifically, these recent results indicate that CS methods can yield exact recovery if the sparsity basis and the measurement system obey the uniform uncertainty principle and are incoherent [4]. In this work, we illustrate that it is possible to use the CS theory principles with radial trajectories as suggested in [3]. Theory: Let f denote the object being imaged, M the measurement matrix, and g the measurements. In MRI, M is an undersampled Fourier matrix and g is the measured k-space data such that Mf g = . Furthermore, let (f) Ψ denote the projection of the object f onto a