LieGROG: A Lie algebra approach for calibrating GRAPPA operators
Abstract
Purpose: In multi-coil acquisitions based on non-Cartesian k-space sampling, image reconstruction using GRAPPA Operator based gridding (GROG) can provide advantages with respect to the non-uniform FFT (NUFFT): It waives both the need for calculating a density compensation function, and for transferring back and forth between Cartesian and non-Cartesian points in iterative algorithms. Non-trivial noise propagation and self-calibration limited to specific trajectories, however, have contributed to GROG being rather sparsely applied so far. Theory and Methods: The group structure of the GRAPPA Operator allows estimating its generator as a single joint fit for all k-space dimensions within the associated Lie algebra. Assuming a linear regime, the generator can be fitted using subsequent pairs of sampling points from nearly any arbitrary non-Cartesian read-out. This approach called LieGROG was evaluated against classical GROG calibration and NUFFT/Toeplitz, using simulations with an analytic ground truth and various 2D and 3D MR-scans employing typical non-Cartesian trajectories. Results: Throughout all applications, LieGROG proved to provide fast, robust and straightforward calibration of the GRAPPA Operator. It clearly outperformed the classical GROG calibration approaches in terms of noise amplification. LieGROG matched the results of the NUFFT/Toeplitz-based reference MR reconstructions closely, while being 6 to 189 times faster and reducing peak VRAM by 20 to 45% in the 3D experiments. Conclusion: LieGROG transforms GROG into a more versatile and stable tool, ultimately aligning non-Cartesian imaging with the standard, FFT-based reconstruction pipelines established for Cartesian scans.
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BibTeX
@misc{wech2026liegrog,
title = {{LieGROG: A Lie algebra approach for calibrating GRAPPA operators}},
author = {Tobias Wech},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30927},
eprint = {2610.30927},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30927},
}
RIS
TY - UNPB
TI - LieGROG: A Lie algebra approach for calibrating GRAPPA operators
AU - Tobias Wech
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30927
UR - https://arxiv.org/abs/2610.30927
AB - Purpose: In multi-coil acquisitions based on non-Cartesian k-space sampling, image reconstruction using GRAPPA Operator based gridding (GROG) can provide advantages with respect to the non-uniform FFT (NUFFT): It waives both the need for calculating a density compensation function, and for transferring back and forth between Cartesian and non-Cartesian points in iterative algorithms. Non-trivial noise propagation and self-calibration limited to specific trajectories, however, have contributed to GROG being rather sparsely applied so far. Theory and Methods: The group structure of the GRAPPA Operator allows estimating its generator as a single joint fit for all k-space dimensions within the associated Lie algebra. Assuming a linear regime, the generator can be fitted using subsequent pairs of sampling points from nearly any arbitrary non-Cartesian read-out. This approach called LieGROG was evaluated against classical GROG calibration and NUFFT/Toeplitz, using simulations with an analytic ground truth and various 2D and 3D MR-scans employing typical non-Cartesian trajectories. Results: Throughout all applications, LieGROG proved to provide fast, robust and straightforward calibration of the GRAPPA Operator. It clearly outperformed the classical GROG calibration approaches in terms of noise amplification. LieGROG matched the results of the NUFFT/Toeplitz-based reference MR reconstructions closely, while being 6 to 189 times faster and reducing peak VRAM by 20 to 45% in the 3D experiments. Conclusion: LieGROG transforms GROG into a more versatile and stable tool, ultimately aligning non-Cartesian imaging with the standard, FFT-based reconstruction pipelines established for Cartesian scans.
ER -