Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complex extension of optical flow and its practical evaluation for undersampled dynamic MRI

Matthias J. Ehrhardt, Marco Mauritz

TL;DR

This work extends the optical-flow motion model to complex-valued images for undersampled dynamic MRI, addressing phase-induced reconstruction artifacts by jointly estimating the image sequence $\rho$ and a complex velocity field $v$ within a variational framework. The forward model $A_t$ maps complex images to undersampled multi-coil measurements, and the regularised objective includes a complex optical-flow term $M(\rho,v)$ regularised with a Huber loss to promote plausible motion while acknowledging imperfect adherence. An efficient optimisation strategy based on block coordinate descent and FISTA, with multi-scale smoothing, enables practical reconstruction; the method is benchmarked against frame-wise and velocity-free baselines, and against a ground-truth velocity scenario. Experiments on two real cardiac datasets and simulated data show substantial PSNR and SSIM gains for the complex-flow approach, particularly in dynamic regions, demonstrating the practical benefit of incorporating complex-valued motion modelling for dynamic MRI. The results suggest the approach is robust to undersampling and highlights avenues for improved priors, 3D extensions, and potential learning-based integrations to further enhance reconstruction fidelity in real-world imaging scenarios.

Abstract

Reconstructing high-quality images from undersampled dynamic MRI data is a challenging task and important for the success of this imaging modality. To remedy the naturally occurring artifacts due to measurement undersampling, one can incorporate a motion model into the reconstruction so that information can propagate across time frames. Current models for MRI imaging are using the optical flow equation. However, they are based on real-valued images. Here, we generalise the optical flow equation to complex-valued images and demonstrate, based on two real cardiac MRI datasets, that the new model is capable of improving image quality.

Complex extension of optical flow and its practical evaluation for undersampled dynamic MRI

TL;DR

This work extends the optical-flow motion model to complex-valued images for undersampled dynamic MRI, addressing phase-induced reconstruction artifacts by jointly estimating the image sequence and a complex velocity field within a variational framework. The forward model maps complex images to undersampled multi-coil measurements, and the regularised objective includes a complex optical-flow term regularised with a Huber loss to promote plausible motion while acknowledging imperfect adherence. An efficient optimisation strategy based on block coordinate descent and FISTA, with multi-scale smoothing, enables practical reconstruction; the method is benchmarked against frame-wise and velocity-free baselines, and against a ground-truth velocity scenario. Experiments on two real cardiac datasets and simulated data show substantial PSNR and SSIM gains for the complex-flow approach, particularly in dynamic regions, demonstrating the practical benefit of incorporating complex-valued motion modelling for dynamic MRI. The results suggest the approach is robust to undersampling and highlights avenues for improved priors, 3D extensions, and potential learning-based integrations to further enhance reconstruction fidelity in real-world imaging scenarios.

Abstract

Reconstructing high-quality images from undersampled dynamic MRI data is a challenging task and important for the success of this imaging modality. To remedy the naturally occurring artifacts due to measurement undersampling, one can incorporate a motion model into the reconstruction so that information can propagate across time frames. Current models for MRI imaging are using the optical flow equation. However, they are based on real-valued images. Here, we generalise the optical flow equation to complex-valued images and demonstrate, based on two real cardiac MRI datasets, that the new model is capable of improving image quality.

Paper Structure

This paper contains 20 sections, 11 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Methods
  4. MRI modelling
  5. Modelling motion for complex-valued images
  6. Optimisation and implementation
  7. Models for comparison
  8. Metrics
  9. Hyperparameter optimisation
  10. Data
  11. Ground truth images
  12. Mask focusing on dynamic parts in images
  13. Simulated data
  14. Undersampling
  15. Experiments and results
  16. ...and 5 more sections

Figures (7)

  • Figure 1: Comparison between real-valued and complex-valued reconstructions in MRI. Left: Complex-valued reconstruction. Right: Real-valued reconstruction. Top row: Magnitude representation. The bottom row shows the same images but as a complex-valued image: The hue represents the normalised phase and the brightness represents the magnitude. The red boxes point to some missing structures in the real-valued image.
  • Figure 2: Each row shows a reconstruction for a different data set. The first column shows a dynamic ground truth (obtained as a reconstruction using \ref{['eq:optimisationFunction']} and fully-sampled data). The second column shows the same for a frame-wise ground truth (obtained from reconstructing each time frame separately (FW model) using fully-sampled data). The last column shows an overlay of the reconstructions with the dynamic masks. Within each column, the left side shows one time slice of the reconstructions, the right side shows time-space images along the red lines.
  • Figure 3: Results for the simulated data. The top row shows the evolution (magnitude) of the simulated ground truth images for the time points $t_0$, $t_2$, $t_4$, $t_6$ (out of $8$ time points). The second row shows the same images but as a complex-valued image: The hue represents the normalised phase and the brightness represents the magnitude. The other rows depict the difference between the magnitudes of the reconstructions and the magnitude of the simulated images on the dynamic mask. Rows 3-5: FW, DT and OF.
  • Figure 4: Reconstructed velocities of one time frame for the simulated data using our proposed complex-valued optical flow method. First row: Ground truth velocities in $x$ and $y$ direction. Second row: Reconstructed velocities in $x$ and $y$ direction
  • Figure 5: Reconstructions for subject 1. Left to right columns: Ground truth image, FW, DT, OF and Cheat-OF. First row shows magnitude of reconstructions. Second row: Difference of reconstruction to ground truth on the mask. Third row: Magnified area and corresponding metrics on magnification.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Remark 2.1: Choices on optical flow model
  • Remark 2.2: Alternative choice for motion model