Fractal Compressive Sensing

TL;DR

This paper presents a novel sampling and reconstruction approach for compressed sensing MRI using a deterministic pseudo-random scheme based on discrete Radon transform projections, leading to improved image quality over traditional methods.

Contribution

It introduces a new sampling scheme called p.frac and a reconstruction method called FFR, combining to form finite compressive sensing (FCS) for MRI, with demonstrated superior performance.

Findings

FCS achieves 3-5dB PSNR gain over 1D Cartesian random sampling.

p.frac sampling scheme is highly incoherent, suitable for CS-MRI.

Experimental results show improved image quality with FCS.

Abstract

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.