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Deep learning for undersampled MRI reconstruction
Chang Min Hyun, Hwa Pyung Kim, Sung Min Lee, Sungchul Lee, Jin Keun Seo
TL;DR
Undersampled MRI reconstruction is ill-posed because reduced k-space data leave fewer equations than image unknowns, while conventional regularization struggles to express complex MR-image structure. The paper learns a U-net reconstruction function from paired undersampled and fully sampled images, using uniform phase-encoding subsampling with added low frequencies. Experiments show high-quality reconstructions, while the approach remains bounded by the ambiguity of uniformly undersampled data and the need for manifold constraints.
Problem
Undersampled MRI reconstruction is ill-posed because the measurement system has fewer equations than unknowns and lacks an explicit reconstruction formula without a strong solution-manifold constraint.
Method
The paper learns a U-net mapping from undersampled or folded MR images to fully sampled reconstructions using paired training data and complex MR-image prior knowledge.
Results
The learned function provides better performance than existing methods and produces sharp, natural-looking reconstructions while suppressing folding artifacts.
Takeaways & Limitations
The experiments indicate that the learned function captures anatomical geometry and small anomalies, and can work on previously untrained CT images.
Takeaways & Limitations
Uniform subsampling can make distinct MR images produce identical undersampled data, so a few added low frequencies are needed for separability and localization.
Abstract
from arXiv · showhide
This paper presents a deep learning method for faster magnetic resonance imaging (MRI) by reducing k-space data with sub-Nyquist sampling strategies and provides a rationale for why the proposed approach works well. Uniform subsampling is used in the time-consuming phase-encoding direction to capture high-resolution image information, while permitting the image-folding problem dictated by the Poisson summation formula. To deal with the localization uncertainty due to image folding, very few low-frequency k-space data are added. Training the deep learning net involves input and output images that are pairs of Fourier transforms of the subsampled and fully sampled k-space data. Numerous experiments show the remarkable performance of the proposed method; only 29% of k-space data can generate images of high quality as effectively as standard MRI reconstruction with fully sampled data.
1. Introduction
The paper addresses long MRI scans caused by time-consuming phase-encoding steps and proposes learning reconstruction functions from undersampled and fully sampled image pairs. Its strategy combines uniform phase-encoding subsampling with a few low-frequency samples and uses a U-net to map aliased images to anti-aliased images.
- MRI scan time is roughly proportional to the number of phase-encoding steps, motivating undersampling to shorten acquisition.Skipping phase-encoding lines can introduce aliasing artifacts when the Nyquist criterion is violated.
- Compressive sensing reconstructs images with sparsity regularization but can limit preservation of diagnostically important fine details and noise-like textures.The regularized formulation balances data residual and sparsity through a parameter λ.
- The proposed deep learning approach reverses regularized least squares by learning f: x 7→y from many undersampled and fully sampled training pairs.The learned manifold constraint captures complex prior knowledge about MR images.
- Uniform phase-encoding subsampling preserves detailed features in folded images while maximizing the rate of skipped k-space data.The strategy is explained using a separability condition and the Poisson summation formula.
- A few low-frequency samples are added to learn overall MR-image structure and address anomaly-location uncertainty from uniform sampling.The paper reports high performance for the proposed method.
- The U-net learns a mapping from aliased images produced by undersampled data to anti-aliased reconstructed images.The general strategy treats the undersampled image as the input and the fully sampled reconstruction as the target.
2. Method
The method combines uniform phase-encoding subsampling with added low-frequency data and a U-net reconstruction pipeline with k-space correction. The added frequencies address location ambiguity caused by folding, while correction preserves measured k-space values.
- 2.1. Subsampling Strategy: Uniform phase-encoding subsampling produces folded images that retain detailed structure but can make distinct images indistinguishable.For factor-2 sampling, different anomaly locations can yield identical subsampled k-space data under the Poisson summation formula.
- 2.1. Subsampling Strategy: Adding a few low-frequency k-space lines restores location information and can satisfy the separability condition needed for learning.The strategy uses factor-4 uniform sampling with 25% of k-space data plus about 4% low-frequency data, totaling approximately 29%.
- 2.1. Subsampling Strategy: Patient and simulation examples show that added low frequencies resolve duplicated lesion locations and suppress undersampling artifacts while preserving morphology.With added low frequencies, tumors are localized at the bottom rather than appearing at both top and bottom.
- 2.2. Image Reconstruction Function: The reconstruction function zero-pads unmeasured k-space, transforms it into a folded image, and feeds that image to a trained U-net.Training pairs are formed from subsampled k-space inputs and fully sampled ground-truth MR images.
- 2.2. Image Reconstruction Function: The U-net estimates the missing k-space region, after which Fourier transformation and replacement with original measurements perform k-space correction.The final image is obtained by inverse Fourier transformation and taking the absolute value.
3. Result
The proposed method suppresses aliasing in undersampled MRI and produces sharp reconstructions, with quantitative evaluation supporting both the U-net and k-space correction.
- Five test brain images show that the method suppresses fourfold folding artifacts while producing sharp, natural-looking reconstructions.
- Figure 6 compares ground truth, aliased input, U-net output, k-space-corrected output, and corresponding difference images.
- K-space correction removes residual folding artifacts that remain after the U-net reconstruction.
- 400 test images from 8 patients were evaluated using mean-squared error (MSE) and structural similarity index (SSIM) after training on 1,400 images from 30 patients.
- Quantitative results support the effectiveness of both the U-net and k-space correction, particularly the correction step.
4. Discussion and Conclusion
The discussion frames deep learning as a feasible way to reconstruct ill-posed undersampled MRI and reports expressive representations, while identifying practical and theoretical boundaries.
- Deep learning provides a feasible way to capture MRI image structure when the undersampled inverse problem has fewer equations than unknowns.
- Regular subsampling at factor 4 failed the separability condition because wrap-around artifacts obscured the locations of small objects.
- Adding low frequencies practically guaranteed separability, after which the authors used a U-net to recover folded images.
- The learned reconstruction function appears to capture anatomical geometry and small anomalies, and also worked on previously unseen CT images.
- The method was limited to 256×256 images because 512×512 inputs caused out-of-memory problems on a four-GPU system.
Appendix A. Minimum-norm solution of the underdetermined system
The appendix explains that the underdetermined MRI system has infinitely many solutions and that the minimum-norm solution may not resemble a head MRI image.
- The minimum-norm solution minimizes ∥y∥_ℓ2 subject to the undersampled Fourier constraint S ◦ F y = x.
- The underdetermined system admits infinitely many image solutions for the same undersampled data.
- The minimum-norm solution is improperly chosen because it does not look like a head MRI image.
Appendix B. Performance of the proposed method with different reduction factors
The experiments vary the uniform subsampling rate ρ and the number L of added low-frequency phase-encoding lines. Reconstruction remains good at high ρ when L = 12, while L = 0 fails and L = 12 performs excellently at ρ = 4.
- Reduction factors ranged from R = 3.37 to R = 5.81 across experiments varying ρ and L.ρ is the uniform subsampling rate along the phase-encoding direction, while L is the number of added low-frequency lines.
- With L = 12, the proposed method provides good reconstruction even at ρ = 8.Figure B1 fixes L = 12 while varying ρ from 4 to 8.
- At ρ = 4, L = 0 violates the separability condition and the proposed method fails.Increasing L to 1 lets the network start learning unfolding, while L = 12 provides excellent reconstruction capability.
- At ρ = 4, the proposed method with L = 12 provides excellent reconstruction capability.
Appendix C. The reconstruction process
Appendix C presents the reconstruction process intuitively using a simplified version of the U-net.
- The appendix presents the reconstruction process intuitively.
- The reconstruction process uses a simplified version of the U-net.
- The appendix uses the simplified U-net to illustrate the reconstruction process.