Melissa W. Haskell^{1,2}, Stephen F. Cauley^{1,3}, and Lawrence L. Wald^{1,3,4}

Retrospective motion
correction techniques offer minimal disruptions to sequences and clinical
workflows. The computational burden of retrospective techniques can be eased
either with alternating minimizations, or true joint estimation but on a
reduced model. We provide computational experiments demonstrating the tightly
coupled nature of the optimization variable types (motion and voxel values)
which hinders the alternating based approaches. The alternating techniques can have
an average search direction error of 75%, vs. 22% with reduced modeling. We demonstrate
a computational speedup of 17x using our reduced model approach, and present* in vivo* imaging results comparing TAMER to
a state-of-the-art alternating minimization.

TAMER Method: Figure 1 outlines the TAMER (**TA**rgeted **M**otion **E**stimation and **R**eduction) method. Motion corrupted *k*-space data is first reconstructed assuming no motion has occurred. Targeted subsets of the image are then jointly optimized with the unknown patient motion parameters by minimizing the data consistency error:

$$[\hat{\boldsymbol{\theta}},\hat{\boldsymbol{x}_t}] = \underset{\boldsymbol{\theta},\boldsymbol{x}_t}{\mathrm{argmin}} ||\boldsymbol{s}_t-\boldsymbol{E}_{\boldsymbol{\theta},t}\boldsymbol{x}_t||_2$$

where $$$\boldsymbol{s}_t$$$ is the acquired signal contribution from the target voxels, $$$\boldsymbol{x}_t$$$ is the vector of targeted voxel values, and $$$\boldsymbol{E}_{\boldsymbol{\theta},t}$$$ is the encoding model for the motion trajectory, $$$\boldsymbol{\theta}$$$. To select the target voxels, the strength of coupling between individual voxels is determined by examining the encoding correlation matrix^{7}.

Computational advantage of reduced model vs. full volume updates: The TAMER reduced model allows for motion transformations to be performed across small regions of the 3D FOV, and restricts the regions that FFTs need to be applied or allows for specific cached DFT matrices to be used. These properties hold as we continuously shift the target voxel pattern across the volume during TAMER. To demonstrate the potential speedup of the overall TAMER algorithm, we have created an implementation of our objective function using cached motion operations. We timed 1000 calls of the objective function using both the full model and the reduced model to determine the speedup.

Search direction accuracy comparison: To investigate the coupling between $$$\boldsymbol{x}$$$ and $$$\boldsymbol{\theta}$$$ during the joint optimization, we compare the search direction accuracy (i.e. optimization gradient) of the two strategies within an alternating optimization of simulated motion corrupted data. The alternating optimization transitions between calculating the image volume assuming fixed motion, and then a non-linear optimization is performed to update the motion parameters assuming a fixed image. The coupling of the optimization variables will dictate how many steps of the non-linear motion search can be accurately performed before the next full volume update. During this motion optimization we calculate: (1) the full model search direction (i.e. what direction the motion parameter search would proceed if the entire volume were updated), (2) the search direction if no voxel values are updated (standard for alternating approaches), and (3) the search direction when updating a small targeted subset of voxels. The search direction accuracy was determined at each optimization step by calculating the percent error relative to the full model.

*In vivo* imaging experiments: We compare the reconstruction quality of the TAMER reduced model vs. a state-of-the-art alternating optimization^{5}, using the data and code provided at https://github.com/mriphysics/multiSliceAlignedSENSE/releases/tag/1.0.1. Both in-plane and through-plane motion effects were corrected for multiple slices.

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2. Zaitsev M, Maclaren J, Herbst M. Motion artifacts in MRI: A complex problem with many partial solutions. J Magn Reson Imaging. 2015;42(4):887-901. doi:10.1002/jmri.24850.

3. Odille F, Vuissoz P-A, Marie P-Y, Felblinger J. Generalized Reconstruction by Inversion of Coupled Systems (GRICS) applied to free-breathing MRI. Magn Reson Med. 2008;60(1):146-157. doi:10.1002/mrm.21623.

4. Cordero-Grande L, Teixeira RPAG, Hughes EJ, Hutter J, Price AN, Hajnal J V. Sensitivity Encoding for Aligned Multishot Magnetic Resonance Reconstruction. IEEE Trans Comput Imaging. 2016;2(3):266-280. doi:10.1109/TCI.2016.2557069.

5. Cordero-Grande L, Hughes EJ, Hutter J, Price AN, Hajnal J V. Three-dimensional motion corrected sensitivity encoding reconstruction for multi-shot multi-slice MRI: Application to neonatal brain imaging. Magn Reson Med. 2017;0. doi:10.1002/mrm.26796.

6. Haskell M, Cauley S, Wald L. TArgeted Motion Estimation and Reduction (TAMER): Data Consistency Based Motion Mitigation using a Reduced Model Joint Optimization. Proc 24th Annu Meet ISMRM, Singapore. 2016:1849.

7. Haskell MW, Cauley SF, Wald LL. Retrospective motion correction of head rotations in 2D RARE brain images using TArgeted Motion Estimation and Reduction (TAMER). In: Proceedings of the 25th Annual Meeting of ISMRM, Honolulu, HI. 2017:1305.