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Comparing TAMER (TArgeted Motion Estimation and Reduction) reduced modeling to alternating minimization for data consistency based motion mitigation
Melissa W. Haskell1,2, Stephen F. Cauley1,3, and Lawrence L. Wald1,3,4

1A. A. Martinos Center for Biomedical Imaging, Department of Radiology, MGH, Charlestown, MA, United States, 2Graduate Program in Biophysics, Harvard University, Cambridge, MA, United States, 3Harvard Medical School, Boston, MA, United States, 4Harvard-MIT Division of Health Sciences and Technology, MIT, Cambridge, MA, United States

### Synopsis

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.

### Purpose

Artifacts due to patient motion present a costly clinical challenge in MRI1. Many methods to correct patient motion in MRI exist2, but current methods have failed to gain traction in the clinic. Data consistency based retrospective motion correction methods are promising due to their minimal disruptions to sequence and clinical workflows3–5. Prior techniques have shown progress using alternating minimizations, where the optimization alternates between reconstruction of the image volume assuming a fixed motion state, and minimizing the data consistency error assuming a fixed image. This approach assumes a decoupling of the two variable types (motion and voxel values). However, the strong variable coupling often leads to constant switching between objectives, leading to repeated full volume image reconstructions. As an alternative approach to solving the image+motion joint optimization, we have previously introduced a reduced modeling strategy, TArgeted Motion Estimation and Reduction (TAMER)6. In this work, we offer direct comparison between the TAMER reduced modeling strategy to alternating based approaches.

### Methods

TAMER Method: Figure 1 outlines the TAMER (TArgeted Motion Estimation and Reduction) 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 matrix7.

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 optimization5, 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.

### Results

Figure 2 shows the run time reduction when using the TAMER reduced model, which had an overall speedup time of 17x compared to the full model when using ~5% of the image as the target voxels. Fig. 3 shows the improved accuracy in the motion search direction when incorporating targeted voxels. Assuming 13 Newton steps for each non-linear motion parameter optimization, the alternating method search direction had 75% average direction error, while the targeted method error was 22%. At a later iteration, the targeted updates search direction error was 14%, but without targeted updates the search direction error increased to 95%. Fig 4. demonstrates the image quality of the TAMER reduced model reconstruction compared to the current state-of-the-art reconstruction5 (single slice from the 3D volume shown).

### Discussion and Conclusion

Here we have examined the search direction accuracy advantages of using a reduced model over an alternating method, although the final motion mitigation performance appears visually comparable in the two methods. Further work appears to be needed to translate the improved search direction accuracy to more complete motion mitigation.

### Acknowledgements

No acknowledgement found.

### References

1. Andre JB, Bresnahan BW, Mossa-Basha M, et al. Toward quantifying the prevalence, severity, and cost associated with patient motion during clinical MR examinations. J Am Coll Radiol. 2015;12(7):689-695. doi:10.1016/j.jacr.2015.03.007.

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.

### Figures

Tamer Overview: (1) Motion corrupted data is reconstructed assuming no motion occurred to create an initial image estimate, and initial target voxels are found (2) The patient motion trajectory and the target voxels are jointly optimized by minimizing data consistency error. As the optimization progresses, the target voxels are shifted around the image volume until TAMER convergences to the true motion and motion corrected image.

Computational Advantage of TAMER Reduced Model. For each operation in the TAMER reduced model, the relative speedup compared to the full model is shown. The simulation of motion artifacts benefits most from a reduced model implementation. Decreased computation time for each individual objective function call will result in decreased overall computation time for TAMER reconstruction. Overall speedup for the objective function was 16.8x using the reduced model.

Search Direction Improvement Using Targeted Voxels. A. The motion search direction error is plotted as a function of the alternating step (i.e. how many times the image volume has been reconstructed). B. The motion search direction magnitude is plotted as a function of the motion parameter index (i.e. translations and rotations at specific shots), for no voxel updates and targeted voxel updates, at steps 2 (top row) and 11 (bottom row) of the motion optimization. Each method (solid lines) is compared to the full model search direction (dotted lines), and the average percent difference is shown.

In Vivo TAMER Results Compared to State-of-the-Art. (Left) A motion corrupted sagittal brain slice from a neonatal subject (data provided in link at top of figure). (Middle) Motion correction using an alternating reconstruction by modifying the code provided in [5] to be comparable (such as removing slice oversampling) to TAMER. (Right) TAMER motion corrected reconstruction. Bottom right shows data consistency error (not provided for the alternating method).

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)
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