  This paper introduces a sparse projection matrix composed of discrete
(digital) periodic lines that create a pseudo-random (p.frac) sampling scheme.
Our approach enables random Cartesian sampling whilst employing deterministic
and one-dimensional (1D) trajectories derived from the discrete Radon transform
(DRT). Unlike radial trajectories, DRT projections can be back-projected
without interpolation. Thus, we also propose a novel reconstruction method
based on the exact projections of the DRT called finite Fourier reconstruction
(FFR). We term this combined p.frac and FFR strategy, finite compressive
sensing (FCS), with image recovery demonstrated on experimental and simulated
data; image quality comparisons are made with Cartesian random sampling in 1D
and two-dimensional (2D), as well as radial under-sampling in a more
constrained experiment. Our experiments indicate FCS enables 3-5dB gain in peak
signal-to-noise ratio (PSNR) for 2-, 4- and 8-fold under-sampling compared to
1D Cartesian random sampling. This paper aims to: Review common sampling
strategies for compressed sensing (CS)-magnetic resonance imaging (MRI) to
inform the motivation of a projective and Cartesian sampling scheme. Compare
the incoherence of these sampling strategies and the proposed p.frac. Compare
reconstruction quality of the sampling schemes under various reconstruction
strategies to determine the suitability of p.frac for CS-MRI. It is
hypothesised that because p.frac is a highly incoherent sampling scheme, that
reconstructions will be of high quality compared to 1D Cartesian phase-encode
under-sampling.
