  Purpose: Parallel imaging and compressed sensing reconstructions of large MRI
datasets often have a prohibitive computational cost that bottlenecks clinical
deployment, especially for 3D non-Cartesian acquisitions. One common approach
is to reduce the number of coil channels actively used during reconstruction as
in coil compression. While effective for Cartesian imaging, coil compression
inherently loses signal energy, producing shading artifacts that compromise
image quality for 3D non-Cartesian imaging. We propose coil sketching, a
general and versatile method for computationally-efficient iterative MR image
reconstruction.
  Theory and Methods: We based our method on randomized sketching algorithms, a
type of large-scale optimization algorithms well established in the fields of
machine learning and big data analysis. We adapt the sketching theory to the
MRI reconstruction problem via a structured sketching matrix that, similar to
coil compression, considers high-energy virtual coils obtained from principal
component analysis. But, unlike coil compression, it also considers random
linear combinations of the remaining low-energy coils, effectively leveraging
information from all coils.
  Results: First, we performed ablation experiments to validate the sketching
matrix design on both Cartesian and non-Cartesian datasets. The resulting
design yielded both improved computational efficiency and preserved
signal-to-noise ratio (SNR) as measured by the inverse g-factor. Then, we
verified the efficacy of our approach on high-dimensional non-Cartesian 3D
cones datasets, where coil sketching yielded up to three-fold faster
reconstructions with equivalent image quality.
  Conclusion: Coil sketching is a general and versatile reconstruction
framework for computationally fast and memory-efficient reconstruction.
