A fixed point method for homotopic ℓ₀-minimization with application to MR image recovery

A novel method for highly-undersampled Magnetic Resonance Image (MRI) reconstruction is presented. One of the principal challenges faced in clinical MR imaging is the fundamental linear relation between net exam duration and admissible spatial resolution. Increased scan duration diminishes patient comfort while increasing the risk of susceptibility to motion artifact and limits the ability to depict many physiological events at high temporal rates. With the recent development of Compressive Sampling theory, several authors have successfully demonstrated that clinical MR images possessing a sparse representation in some transform domain can be accurately reconstructed even when sampled at rates well below the Nyquist limit by casting the recovery as a convex ℓ₁-minimization problem. While ℓ₁-based techniques offer a sizeable advantage over Nyquist-limited methods, they nonetheless require a modest degree of over-sampling above the true theoretical minimum sampling rate in order to guarantee the achievability of exact reconstruction. In this work, we present a reconstruction model based on homotopic approximation of the ℓ₀ quasi-norm and discuss the ability of this technique to reconstruct undersampled MR images at rates even lower than are achievable than with ℓ₁-minimization and arbitrarily close to the true minimum sampling rate. A semi-implicit numerical solver is presented for efficient numerical computation of the reconstruction process and several examples depicting the capability for accurate MRI reconstructions from highly-undersampled K-space data are presented.