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Multi-Contrast MRI Reconstruction with Structure-Guided Total Variation
Matthias J. Ehrhardt, Marta M. Betcke
TL;DR
Multi-contrast MRI requires lengthy sequential acquisitions, although images from the same anatomy are likely to share structures. This paper uses location- and direction-based total-variation priors guided by another contrast; dTV performs best on average, improving on total variation by about 6dB in PSNR and 8% in SSIM for both contrasts.
Problem
MRI contrasts are acquired sequentially, making multi-contrast scans lengthy, while images from the same anatomy are likely to share structures.
Source: §1.1 · p. 1Method
The paper introduces convex total-variation priors using edge-location or edge-location-and-direction information from another contrast, and solves the resulting problem with ADMM.
Source: §1.3 · p. 3Results
On average, dTV improves on total variation by about 6dB in PSNR and 8% in SSIM for both contrasts.
Source: §5.2.3 · p. 14Takeaways & Limitations
Directional information yields better-defined images and better reconstruction of smooth structures and fine details.
Source: §6 · p. 16Patient results indicate little quantitative improvement; the authors note that using noisy reconstructions as ground truth may make the similarity measure diverge from human perception.
Source: §5.3 · p. 14
Abstract
from arXiv · showhide
Magnetic resonance imaging (MRI) is a versatile imaging technique that allows different contrasts depending on the acquisition parameters. Many clinical imaging studies acquire MRI data for more than one of these contrasts---such as for instance T1 and T2 weighted images---which makes the overall scanning procedure very time consuming. As all of these images show the same underlying anatomy one can try to omit unnecessary measurements by taking the similarity into account during reconstruction. We will discuss two modifications of total variation---based on i) location and ii) direction---that take structural a priori knowledge into account and reduce to total variation in the degenerate case when no structural knowledge is available. We solve the resulting convex minimization problem with the alternating direction method of multipliers that separates the forward operator from the prior. For both priors the corresponding proximal operator can be implemented as an extension of the fast gradient projection method on the dual problem for total variation. We tested the priors on six data sets that are based on phantoms and real MRI images. In all test cases exploiting the structural information from the other contrast yields better results than separate reconstruction with total variation in terms of standard metrics like peak signal-to-noise ratio and structural similarity index. Furthermore, we found that exploiting the two dimensional directional information results in images with well defined edges, superior to those reconstructed solely using a priori information about the edge location.
1 Introduction
Multi-contrast MRI requires lengthy sequential acquisitions, while images of the same anatomy are likely to share structures. The paper introduces location- and direction-based total-variation priors that use another contrast to guide reconstruction.
1.1 Multi-Contrast Magnetic Resonance Imaging
- Multi-contrast MRI images are acquired sequentially, making scans lengthy; shorter acquisition could reduce discomfort and increase patient throughput.The introduction notes that clinical sessions may acquire several contrasts, including T1, T2, and FLAIR.Source: §1.1 · p. 1
1.2 Magnetic Resonance Imaging and Compressed Sensing
- Compressed sensing seeks fewer measurements and exploits prior knowledge, while multi-contrast images of the same anatomy are likely to share structures.The paper identifies shared structure across contrasts as potential reconstruction information despite their different information content.Source: §1.2 · p. 3
1.3 Contributions
- The paper proposes convex total-variation priors using edge location or location and direction from another contrast, solved with ADMM and extended fast-gradient proximal methods.A double split leaves all but one ADMM update in closed form.Source: §1.3 · p. 3
1.4 Related Work
- Unlike directional methods deriving directions from image estimates or inputs, this approach uses a vector field from another contrast; both priors reduce to total variation without side information.The paper also distinguishes its approach from prior spatial weighting by handling a non-smooth prior.Source: §1.4 · p. 3
2 Problem Setting and Notation
The problem setting models MRI reconstruction with real, nonnegative images and an acquisition operator for undersampled measurements. It defines the prior and discrete operators used in the reconstruction formulation.
2 Problem Setting and Notation
- Images are modeled as real-valued and nonnegative under negligible phase; with additive Gaussian noise, the stated MAP reconstruction is equivalent to minimizing the objective.Complex-valued images are explicitly outside the paper’s scope.Source: §2 · p. 4
2.1 Forward Operator for Magnetic Resonance Imaging
- The MRI forward operator combines a Fourier transform, real-to-complex embedding, and sampling; the practical case M ≪ N is generally non-invertible.Here M is the number of measurements and N the number of unknowns.Source: §2.1 · p. 4
- The adjoint operator uses the inverse Fourier transform and the sampling adjoint, which places acquired values at their sample locations.Source: §2.1 · p. 4
2.2 Discrete Gradient
- The prior functional J penalizes undesirable solutions, while the discrete gradient represents neighboring-pixel changes and the divergence is its negative adjoint.The simulations use forward differences for the gradient and matching backward differences for the divergence.Source: §2.2 · p. 4
3 Modelling A Priori Information
The paper extends total variation (TV) with priors drawn from another MRI contrast, using either edge locations or edge directions. A matrix-field framework expresses both variants, with isotropic weighting for location cues and anisotropic weighting for direction cues.
3.1 Total Variation
- Standard TV is convex and preserves edges, but its standard formulation cannot incorporate additional a priori knowledge about the solution.Source: §3.1 · p. 5
3.2.1 A Priori Information on Location of Edges
- Weighted TV uses edge-location weights from another contrast because contrast intensities differ while edge structure is expected to correlate; η sets the edge-magnitude scale and is assumed spatially constant.With uninformative weights, weighted TV reduces to standard TV; for weights between 0 and 1, it does not exceed standard TV.Source: §3.2.1 · p. 5
3.2.2 A Priori Information on Direction of Edges
- Directional TV uses edge directions shared across contrasts modulo sign, projecting the reconstructed image gradient so that, in the limit, only its component orthogonal to the side-information direction is penalized.The sign qualification matters because corresponding contrast intensities can jump in opposite directions.Source: §3.2.2 · p. 5
3.2.3 General Framework
- A matrix field D unifies both priors: it is isotropic for weighted TV and anisotropic for directional TV, with principal directions normal and tangential to the side image’s level sets.Source: §3.2.3 · p. 6
4 Algorithmic Approach
The paper solves the reconstruction problem with a double-split ADMM, separating the forward operator from the structural prior. Its structural-TV proximal step uses an accelerated dual fast-gradient method, while the ADMM updates exploit the MRI operator’s structure.
4.1 Proximal Operator with Fast Gradient Projection
- The structural-TV proximal operator is computed by dualizing its convex minimization and applying an accelerated fast-gradient projection method, since no closed-form proximal operator is available.The method is an instance of FISTA with objective-value convergence rate O(1/k^2); the stated sufficient step size in 2D is s = (8α^2)^-1.Source: §4 · p. 6, §4.1 · pp. 6–7
4.2 Double-Split Alternating Direction Method of Multipliers
- Double splitting recasts the MRI reconstruction as a constrained optimization problem separating the data term from the structural prior, with ADMM updates for the resulting variables.The split associates u and x with the first block and z with the second; the first two updates can be performed in parallel, and the diagonal matrix inversion reduces to component-wise scaling.Source: §4.2 · p. 8
- Each ADMM iteration applies the discrete Fourier transform and its inverse, then evaluates the proximal operator; the method adjusts ρ when primal and dual residuals are too far apart.With constant ρ and absolutely summable proximal errors, the paper states that the algorithm converges to a solution.Source: §4.2 · p. 8
- η = 1e-2 yields the best SSIM in every reported wTV and dTV case for radial BrainWebA data.Source: Figure 2 · p. 9
5.1 Technical Details
The evaluation uses six phantom, simulated, and clinical MRI datasets with varied Fourier sampling patterns and added complex-data Gaussian noise. Reconstructions are assessed with PSNR and SSIM, and α and η are selected to maximize SSIM against ground truth.
5.1.1 Data and Algorithms
- Six datasets span Shepp–Logan phantoms, BrainWeb simulations, and clinical patient MRI, with Cartesian, radial, and spiral sampling and Gaussian noise scaled to 5% of the noise-free signal norm for fully sampled data.The algorithms and datasets are reported as available in supplementary material.Source: §5.1.1 · p. 9
5.1.2 Quality Measures and Parameter Selection
- Reconstruction quality is evaluated with PSNR and SSIM, while α and η are chosen to maximize SSIM against the ground truth.Source: §5.1.2 · p. 9
5.2 Results
The parameter study selects η = 1e-2, while visual and quantitative assessments show benefits from structural priors, especially directional guidance. Across test cases, dTV outperforms TV on average and retains advantages across the displayed regularization choices.
5.2.1 Parameter Estimation
- The selected edge parameter is η = 1e-2, approximately 1% of the maximum side-information gradient; at η ≈ 1, both priors coincide with total variation.Similar parameter behavior was observed across other data sets and sampling patterns.Source: §5.2.1 · p. 9
5.2.2 Visual Assessment
- For BrainWebA with golden-angle radial sampling, directional information from the other contrast significantly improves high-resolution reconstructions.Source: Figure 6 · p. 11
- Visually, wTV can produce patchy images, whereas dTV recovers smooth structures and fine details and shows fewer artifacts than no-prior or TV reconstructions.Both structural priors improve the reconstructions relative to no prior and standard TV.Source: Figure 3 · p. 10, §5.2.2 · p. 14
5.2.3 Quantitative Assessment
- wTV and dTV strongly outperform standard total variation across all test cases, with better results for every displayed regularization-parameter choice.The authors report especially strong improvements for T1-weighted SheppLogan and BrainWeb.Source: §5.2.3 · p. 14
- On average, dTV improves on total variation by about 6dB in PSNR and 8% in SSIM for both contrasts.The differences between methods vary strongly across data sets, sampling schemes, and contrasts.Source: §5.2.3 · p. 14
5.3 Discussion
The discussion reports average gains for dTV but notes that the size of improvement varies across evaluation settings. It also cautions that patient-case metrics may not match visual assessments when noisy reconstructions are used as ground truth.
5.3 Discussion
- For patient data, noisy reconstructed MRI images serve as ground truth, so similarity measures may not match human perception despite visual quality gains.Source: §5.3 · p. 14
- Structural priors improve reconstruction quality not only at the optimal regularization parameter but consistently across all displayed parameter choices.Source: Figure 15 · p. 15
- Although relative method performance depends strongly on the dataset and sampling scheme, dTV outperforms TV across all test cases by around 6 dB in PSNR and 8% in SSIM on average for both contrasts.Source: Figure 16 · p. 15, Table 1 · p. 15
6 Conclusions
The conclusion presents location- and direction-based extensions of total variation that use structural information from another MRI contrast. Numerical tests support benefits in highly undersampled reconstruction, with directional information improving edges, smooth structures, and fine details.
6 Conclusions
- The paper extends total variation with structural priors using edge location alone or edge location and direction from another contrast.Source: §6 · p. 16
- Both priors are convex, allowing the reconstruction problem to be solved with efficient convex-optimization methods.Source: §6 · p. 16
- Numerical tests show structural information benefits highly undersampled MRI reconstruction, while directional information improves edge definition, smooth structures, and fine details.Source: §6 · p. 16
- Future work will extend the framework to sequences with more than two contrasts and to joint reconstruction of multiple contrasts.Source: §6 · p. 16