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Product of Gaussian Mixture Diffusion Model for non-linear MRI Inversion

Laurenz Nagler, Martin Zach, Thomas Pock

TL;DR

The paper tackles non-linear multi-coil MRI inversion by jointly reconstructing the image and coil sensitivities using a lightweight, interpretable product of Gaussian mixture diffusion model (PoGMDM) as the image prior, paired with smoothness priors on coil sensitivities. It presents a probabilistic framework that factors the posterior into a Gaussian likelihood and diffusion priors, enabling posterior mean and pixel-wise uncertainty via sampling, while maintaining fast inference. The method demonstrates robustness to varying sampling trajectories and contrast, with competitive performance and a significantly smaller parameter count compared to large black-box models. Experiments on fastMRI knee and CORPD/CORPDFS data highlight the approach’s practical potential for fast, uncertainty-aware MRI reconstruction, albeit with some hyperparameter tuning requirements and room for improving joint priors on $(X,\Sigma)$.

Abstract

Diffusion models have recently shown remarkable results in magnetic resonance imaging reconstruction. However, the employed networks typically are black-box estimators of the (smoothed) prior score with tens of millions of parameters, restricting interpretability and increasing reconstruction time. Furthermore, parallel imaging reconstruction algorithms either rely on off-line coil sensitivity estimation, which is prone to misalignment and restricting sampling trajectories, or perform per-coil reconstruction, making the computational cost proportional to the number of coils. To overcome this, we jointly reconstruct the image and the coil sensitivities using the lightweight, parameter-efficient, and interpretable product of Gaussian mixture diffusion model as an image prior and a classical smoothness priors on the coil sensitivities. The proposed method delivers promising results while allowing for fast inference and demonstrating robustness to contrast out-of-distribution data and sampling trajectories, comparable to classical variational penalties such as total variation. Finally, the probabilistic formulation allows the calculation of the posterior expectation and pixel-wise variance.

Product of Gaussian Mixture Diffusion Model for non-linear MRI Inversion

TL;DR

The paper tackles non-linear multi-coil MRI inversion by jointly reconstructing the image and coil sensitivities using a lightweight, interpretable product of Gaussian mixture diffusion model (PoGMDM) as the image prior, paired with smoothness priors on coil sensitivities. It presents a probabilistic framework that factors the posterior into a Gaussian likelihood and diffusion priors, enabling posterior mean and pixel-wise uncertainty via sampling, while maintaining fast inference. The method demonstrates robustness to varying sampling trajectories and contrast, with competitive performance and a significantly smaller parameter count compared to large black-box models. Experiments on fastMRI knee and CORPD/CORPDFS data highlight the approach’s practical potential for fast, uncertainty-aware MRI reconstruction, albeit with some hyperparameter tuning requirements and room for improving joint priors on .

Abstract

Diffusion models have recently shown remarkable results in magnetic resonance imaging reconstruction. However, the employed networks typically are black-box estimators of the (smoothed) prior score with tens of millions of parameters, restricting interpretability and increasing reconstruction time. Furthermore, parallel imaging reconstruction algorithms either rely on off-line coil sensitivity estimation, which is prone to misalignment and restricting sampling trajectories, or perform per-coil reconstruction, making the computational cost proportional to the number of coils. To overcome this, we jointly reconstruct the image and the coil sensitivities using the lightweight, parameter-efficient, and interpretable product of Gaussian mixture diffusion model as an image prior and a classical smoothness priors on the coil sensitivities. The proposed method delivers promising results while allowing for fast inference and demonstrating robustness to contrast out-of-distribution data and sampling trajectories, comparable to classical variational penalties such as total variation. Finally, the probabilistic formulation allows the calculation of the posterior expectation and pixel-wise variance.
Paper Structure (12 sections, 12 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 12 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Generative Modeling and Diffusion
  4. Inverse Problems and Diffusion Priors
  5. Methods
  6. Image Reconstruction Algorithm
  7. Model Architecture and Implementational Details
  8. Experimental Data
  9. Comparison and Evaluation
  10. Results
  11. Reconstruction
  12. Conclusion

Figures (3)

  • Figure 1: The learned overcomplete model with Gaussian mixture experts. Below each filter the weight $\gamma_k$ is shown. The first row shows the vertical cone and the second row is the horizontal cone of the shearlet system. The first five entries correspond to the shearing for the first scale and the second five for the second scale. The colours indicate the diffusion time $\sqrt{2.0t} = 0.0 , 0.025 , 0.05 , 0.1 , 0.2 $.
  • Figure 2: Qualitative comparison for against . The first row shows results on data, the second on data. The zoom shows an image region and the corresponding absolute error and the pixel-wise variance in the right column (0.0 [ scale=0.32, colormap=examplesamples of colormap = (8 of inferno), colorbar horizontal,point meta max=0.2,colorbar style=ticks=none, ] 0.25).
  • Figure 3: nullspace residuals for Cartesian subsampling. In each block, the measured data has 8% (left) and 4% (right) respectively.