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Comparing TAMER (TArgeted Motion Estimation and Reduction) reduced modeling to alternating minimization for data consistency based motion mitigation1A. 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
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