The space of ideal signals in parallel magnetic resonance imaging is a Reproducing Kernel Hilbert Space (RKHS) of vector-valued functions which is characterized by a kernel derived from the receive sensitivities. Parallel imaging in k-space can be expressed as approximation in this space. This novel formulation yields insights about sampling in k-space which go beyond what is possible with the traditional g-factor analysis.

**References:**

Vivek Athalye, Michael Lustig, and Martin Uecker. Parallel Magnetic Resonance Imaging as Approximation in a Reproducing Kernel Hilbert Space, Inverse Problems, 31:045008 (2015) preprint arXiv:1310.7489 [physics.med-ph]

ESPIRiT is a new algorithm for autocalibrated parallel MRI, which combines the robustness of GRAPPA with the speed and flexibility of a SENSE-based reconstruction. Implementations of ESPIRiT calibration and reconstruction are available in our reconstruction toolbox. The algorithm is related to multi-channel multi-variate spectral estimation.

**References:**

Martin Uecker, Patrick Virtue, Shreyas S Vasanawala, and Michael Lustig. ESPIRiT Reconstruction Using Soft SENSE. Annual Meeting ISMRM, Salt Lake City 2013, In Proc. Intl. Soc. Mag. Reson. Med 21; 127 (2013)

Martin Uecker, Peng Lai, Mark J. Murphy, Patrick Virtue, Michael Elad, John M. Pauly, Shreyas S. Vasanawala, and Michael Lustig. ESPIRiT - An Eigenvalue Approach to Autocalibrating Parallel MRI: Where SENSE meets GRAPPA. Magnetic Resonance in Medicine, 71:990-1001 (2014)

**Reference:**

Martin Uecker, Thorsten Hohage, Kai Tobias Block, and Jens Frahm, Image Reconstruction by Regularized Nonlinear Inversion - Joint Estimation of Coil Sensitivities and Image Content, Magnetic Resonance in Medicine 60 (3): 674-682 (2008) [source code]

Florian Knoll, Christian Clason, Martin Uecker, and Rudolf Stollberger, Improved Reconstruction in Non-Cartesian Parallel Imaging by Regularized Nonlinear Inversion, Annual Meeting ISMRM Honolulu 2009, In Proc. Intl. Soc. Mag. Reson. Med. 17: 2721 (2009)

Martin Uecker, Shuo Zhang, and Jens Frahm, Nonlinear Inverse Reconstruction for Real-time MRI of the Human Heart Using Undersampled Radial FLASH, Magnetic Resonance in Medicine 63 (6): 1456-1462 (2010) [source code]

Florian Knoll, Christian Clason, Kristian Bredies, Martin Uecker, and Rudolf Stollberger, Parallel Imaging with Nonlinear Reconstruction using Variational Penalties, Magnetic Resonance in Medicine 67:34-41 (2012) [source code]

**References:**

Kai Tobias Block, Martin Uecker, and Jens Frahm, Undersampled Radial MRI with Multiple Coils. Iterative Image Reconstruction Using a Total Variation Constraint, Magnetic Resonance in Medicine 57 (6): 1086-1098 (2007)

Martin Uecker, Kai Tobias Block, and Jens Frahm, Nonlinear Inversion with L1-Wavelet Regularization - Application to Autocalibrated Parallel Imaging, ISMRM Annual Meeting, Toronto 2008, In Proc. Intl. Soc. Mag. Reson. Med. 16: 1479 (2008)

Florian Knoll, Christian Clason, Kristian Bredies, Martin Uecker, and Rudolf Stollberger, Parallel Imaging with Nonlinear Reconstruction using Variational Penalties, Magnetic Resonance in Medicine 67:34-41 (2012) [source code]

Martin Uecker, Peng Lai, Mark J. Murphy, Patrick Virtue, Michael Elad, John M. Pauly, Shreyas S. Vasanawala, and Michael Lustig. ESPIRiT - An Eigenvalue Approach to Autocalibrating Parallel MRI: Where SENSE meets GRAPPA. Magnetic Resonance in Medicine, 71:990-1001 (2014) [source code]

**References:**

Kai Tobias Block, Martin Uecker, and Jens Frahm, Iterative Reconstruction for R2 Mapping Based on Radial Fast Spin-Echo MRI, ISMRM Annual Meeting, Toronto 2008, In Proc. Intl. Soc. Mag. Reson. Med. 16: 1432 (2008)

Kai Tobias Block, Martin Uecker, and Jens Frahm, Model-based Iterative Reconstruction for Radial Fast Spin-Echo MRI, IEEE Transactions on Medical Imaging 28:1759-1769 (2009)

Tilman J Sumpf, Martin Uecker, Susann Boretius, and Jens Frahm, Model-based Nonlinear Inverse Reconstruction for T2 Mapping Using Highly Undersampled Spin-Echo MRI, Journal of Magnetic Resonance Imaging, 34:420-428 (2011) [source code]

Tilman J. Sumpf, Amir Moussavi, Martin Uecker, Susann Boretius, and Jens Frahm, Effects of Phase Alternations in Nonlinear Inverse T2 Reconstructions from Undersampled Data, Annual Meeting ISMRM, Melbourne 2012, In Proc. Intl. Soc. Mag. Reson. Med 20: 2400 (2012)

Tilman J. Sumpf, Andreas Petrovic, Martin Uecker, Florian Knoll, and Jens Frahm. Fast T2 Mapping with Improved Accuracy Using Undersampled Spin-echo MRI and Model-based Reconstructions with a Generating Function, IEEE Transactions on Medical Imaging, 33:2213-2222 (2014) arxiv:1405:3574 [physics.med-ph]