A Theory for Sampling in k-Space - Parallel Imaging as Approximation in a Reproducing Kernel Hilbert Space

We show that parallel imaging can be formulated as an approximation of vector-valued functions in a Reproducing Kernel Hilbert Space (RKHS). This formulation provides new theoretical insights into sampling and reconstruction in k-space. In particular, we derive local bounds for the approximation error and noise amplification maps in k-space. These new metrics complement the existing g-factor maps and explain the effect of different sampling schemes on reconstruction quality. This is demonstrated for several sampling patterns using numerical experiments.

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