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#### Using the GRAPPA operator and the generalized sampling theorem to reconstruct undersampled non-Cartesian data

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##### Citation

Seiberlich, N., Breuer FA, Ehses, P., Moriguchi H, Blaimer M, Jakob, P., & Griswold, M. (2009).
Using the GRAPPA operator and the generalized sampling theorem to reconstruct undersampled non-Cartesian data.*
Magnetic Resonance in Medicine,* *61*(3), 705-715. doi:10.1002/mrm.21891.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-C597-B

##### Abstract

As expected from the generalized sampling theorem of Papoulis, the use of a bunched sampling acquisition scheme in conjunction with a conjugate gradient (CG) reconstruction algorithm can decrease scan time by reducing the number of phase-encoding lines needed to generate an unaliased image at a given resolution. However, the acquisition of such bunched data requires both modified pulse sequences and high gradient performance. A novel method of generating the “bunched” data using self-calibrating GRAPPA operator gridding (GROG), a parallel imaging method that shifts data points by small distances in k-space (with Δk usually less than 1.0, depending on the receiver coil) using the GRAPPA operator, is presented here. With the CG reconstruction method, these additional “bunched” points can then be used to reconstruct an image with reduced artifacts from undersampled data. This method is referred to as GROG-facilitated bunched phase encoding (BPE), or GROG-BPE. To better understand how the patterns of bunched points, maximal blip size, and number of bunched points affect the reconstruction quality, a number of simulations were performed using the GROG-BPE approach. Finally, to demonstrate that this method can be combined with a variety of trajectories, examples of images with reduced artifacts reconstructed from undersampled in vivo radial, spiral, and rosette data are shown.