# Parallel Magnetic Resonance Imaging as Approximation in a Reproducing Kernel Hilbert Space

arXiv: https://arxiv.org/abs/1310.7489 · HTML: https://mlsanity.com/paper/1310.7489/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Oct 2013.

## Built on by (most cited first, 1 of 1)

- [High-Frequency Space Diffusion Models for Accelerated MRI](https://mlsanity.com/paper/2208.05481/) (2022)

## Abstract (arXiv)

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. 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 g-factor noise analysis to both noise amplification and approximation errors. This is demonstrated with numerical examples.