Wavelet-Based Reconstruction for Rapid MRI Rafiqul Islam, Andrew J. Lambert and Mark R. Pickering School of Engineering and Information Technology University of New South Wales Canberra, Australia Abstract—In magnetic resonance imaging (MRI), slow data acquisition times often introduce artefacts due to motion and also limit the resolution of the images captured. To address this issue, compressed sensing (CS) techniques have recently been applied to allow under-sampling of the k-space data providing faster acquisition times. To reconstruct the image from the under-sampled measurements, a number of image reconstruction methods have been used. These techniques typically make use of l1-regularization and sparsifying transforms such as the wavelet transform. In this paper, we present a wavelet domain reconstruction method that utilises wavelet regularization with a Gaussian scale mixture (GSM) model prior combined with a Total Variation (TV) constraint in the complex wavelet domain. Our results show that, when compared to the results of previous approaches, the volume reconstructed using our proposed method has superior quality both visually and quantitatively. Index Terms—Magnetic Resonance Imaging, Reconstruction, Gaussian Scale Mixture Model, Wavelet Regularization, Total Variation, Rapid MRI. I. I NTRODUCTION MRI is an imaging technique used in medical settings for capturing images of the human body or parts of the human body for clinical purposes. It is a non-invasive method and is harmless to the patient because it uses strong magnetic fields and non-ionizing radiation in the radio frequency range. During the image acquisition process, spatial resolution is determined by the ”phase-encode” axis and an increase in spatial resolution along the phase encode axis requires an increase in the number of phase-encoded projections. This increases the acquisition time. Hence, reducing the scan time limits the resolution and introduces artifacts. To deal with these problems, applying compressed sensing (CS) image acquisition approaches to MRI offers significant scan time reductions without degrading the image quality. CS is the process of acquiring a signal using sparse sampling schemes and reconstructing this signal from those measurements using non-linear reconstruction methods. In order to recover the signal from the sparsely sampled measurements, the CS approach has three main requirements: (a) the signal must have a sparse representation in a known domain (b) The aliasing artifacts introduced during sampling must be incoherent in the sparsifying domain and (c) a non-linear reconstruction method that is able to enforce both the sparsity of the image representation and consistency with the acquired data. Taking account of those requirements, the theory of CS [1]–[3] states that it is possible to reconstruct the signal by taking fewer measurements from the sparse representation of a signal than is required to satisfy the nyquist criterion. This theory also proved that there is a strong connection between the data sparsity and the regularization approach used during reconstruction. CS in MRI [4] has emerged as a particularly active field and it has been shown that a high quality MRI image can be reconstructed from sparsely sampled k-space data. This allows significantly faster scan times. In order to recover the signal from the sparse measurements, the majority of the CS reconstruction methods utilize a non-linear reconstruction approach with l1- regularization. For example, in [5], [6] the authors utilized a total variation (TV) prior to recover the image. For better optimization, they used a non-linear conjugate descent method. To take advantage of the benefits of wavelet domain sparsity, a wavelet based soft-thresholding method has also been used to recover MRI volumes from sparsely sampled k-space data [7]. In this paper, we address the problem of choosing the best regularization prior for MRI reconstruction using sparsely sampled data. In our proposed method, we adopt a wavelet based Gaussian scale mixture (GSM) model [8] prior in combination with a TV constraint to recover the image from the k-space sparsely-sampled measurements. We utilize a Cartesian sampling grid to sparsely sample the k-space data, which allows greater flexibility and provides low coherence in the sampling trajectories. Variable density random sparse sampling is widely used for Cartesian k-space trajectories. In Cartesian trajectories, random sparse-sampling of the phase-encode axis can significantly improved the imaging speed and has been shown to be superior to other sampling trajectories [4]. Our experimental results show that this approach produces reconstructed images with quality superior to that provided by other previously proposed methods. The remainder of the paper is organized as follows: In Section II, we explain the general approaches of our proposed method. Section III is devoted to the experimental evaluation of the algorithm and finally our conclusions are presented in Section IV. II. RECONSTRUCTION METHODS The MRI data acquisition method using CS can be modelled by the equation: y = F u x + n (1) 978-1-4673-2181-5/12/$31.00 ©2012 IEEE