title: Parallel Magnetic Resonance Imaging as Approximation in a Reproducing Kernel Hilbert Space.
abstract: In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. To understand and design k-space sampling patterns, a theoretical framework is needed to analyze how well arbitrary sampling patterns reconstruct unsampled k-space using receive coil information. As shown here, reconstruction from samples at arbitrary locations can be understood as approximation of vector-valued functions from the acquired samples and formulated using a Reproducing Kernel Hilbert Space (RKHS) with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between approximation theory and parallel imaging. Theoretical tools from approximation theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional image-domain g-factor noise analysis to both noise amplification and approximation errors in k-space. This is demonstrated with numerical examples.
authors.affiliation: Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720.
authors.forename: Vivek
authors.initials: V
authors.lastname: Athalye
authors.affiliation: Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720.
authors.forename: Michael
authors.initials: M
authors.lastname: Lustig
authors.affiliation: Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720.
authors.forename: Martin
authors.initials: M
authors.lastname: Uecker
keywords: Approximation; Image Reconstruction; Inverse Problems; Magnetic Resonance Imaging; Non-Cartesian Sampling; Reproducing Kernel Hilbert Space