💻 computer science

SuCor: Susceptibility Distortion Correction via Parameter-Free and Self-Regularized Optimal Transport

SuCor is a parameter-free, self-regularized optimal transport method that corrects susceptibility-induced geometric distortions in EPI images by modeling distortion fields as Wasserstein-2 barycentric displacements, achieving superior alignment with structural references and faster processing times compared to FSL TOPUP.

Original authors: Sreekar Chigurupati, Eleftherios Garyfallidis

Published 2026-03-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sreekar Chigurupati, Eleftherios Garyfallidis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to take a perfect photograph of a city skyline, but you are looking through a thick, wavy pane of glass. The glass bends the light, making the buildings look squished, stretched, or shifted to the side. In the world of brain scanning (MRI), this "wavy glass" is caused by the difference between air and tissue in your head, which messes up the magnetic field. This creates a blurry, distorted picture of the brain, making it hard for doctors or researchers to see exactly where things are.

This paper introduces a new tool called SuCor to fix these blurry brain pictures. Here is how it works, explained through simple analogies:

The Problem: The "Stretchy Tape"

When scientists take these brain scans, they usually take two pictures at the same time but with the magnetic field "pushed" in opposite directions.

Picture A: The brain looks stretched to the left.
Picture B: The brain looks stretched to the right.

The goal is to figure out exactly how much the "tape" was stretched in every single spot so they can un-stretch it and reveal the true shape of the brain.

The Old Way (FSL TOPUP): The Slow, Careful Sculptor

The current standard method (called TOPUP) is like a master sculptor trying to fix a warped statue.

How it works: The sculptor looks at the whole statue, guesses where the warp is, makes a tiny adjustment, checks the result, and repeats this process thousands of times.
The downside: It takes a long time (sometimes an hour or more) and requires the sculptor to be very careful about how hard they press (tuning parameters). If they press too hard or too soft, the result isn't perfect.

The New Way (SuCor): The Instant, Smart Organizer

The authors propose SuCor, which uses a mathematical concept called "Optimal Transport." Think of it less like sculpting and more like organizing a line of people.

1. The "Column-by-Column" Strategy

Instead of looking at the whole brain at once, SuCor looks at the image one vertical column of pixels at a time (like looking at a single file of people in a line).

The Analogy: Imagine you have two lines of people. In Line A, everyone is shuffled randomly. In Line B, they are shuffled in the exact opposite way.
The Magic: Because you know Line A and Line B are just the same people shuffled differently, you can instantly calculate exactly how far each person moved to get from Line A to Line B. You don't need to guess or iterate; the math gives you the answer immediately.
The Result: SuCor does this for every single column in the brain image instantly. It's like having a super-fast robot that fixes every vertical slice of the brain in a split second.

2. Smoothing the Rough Edges (The "Bending" Fix)

Because SuCor fixes each column independently, the result might look a bit "striped" or jagged, like a corrugated metal roof.

The Analogy: Imagine you have a sheet of metal that is bent into a wavy shape. You want to smooth it out, but you don't want to flatten it so much that you lose the shape of the mountains underneath.
The Solution: SuCor uses a "bending energy" rule. It smooths out the jagged stripes just enough to make the image look natural, but it stops before it blurs the important details.
The "Auto-Pilot" Feature: The best part? SuCor figures out exactly how much smoothing is needed on its own. It doesn't need a human to say, "Smooth it a little more." It listens to the "noise" in the image and automatically finds the perfect balance.

Why is SuCor Better?

The paper tested SuCor against the old method (TOPUP) using real brain scans from the Human Connectome Project.

Speed: While the old method took 55 minutes, SuCor finished the job in 12 seconds. That's like going from a slow walk to a sprint.
Accuracy: When they compared the corrected brain images to a high-quality 3D model of the brain (the "T1 reference"), SuCor's images matched the real anatomy better than the old method.
- Think of it this way: The old method smoothed the brain so much that it lost some fine details. SuCor kept the fine details (like the wrinkles on a brain) while still fixing the big distortions.

The Trade-off

There is one small catch. The old method (TOPUP) makes the two original pictures (Left-Right and Right-Left) look almost identical to each other. SuCor makes them look slightly different because it is preserving more of the unique, fine details of the brain's shape. However, the authors argue that preserving the true shape of the brain is more important than making the two blurry pictures look identical.

Summary

SuCor is a new, lightning-fast tool that fixes distorted brain scans. Instead of slowly guessing and adjusting like a sculptor, it uses a smart, mathematical "line-up" trick to instantly figure out how to un-stretch the image. It does this automatically, without needing a human to tweak settings, and it produces a clearer, more accurate picture of the brain than the current gold standard.

1. Problem Statement

Susceptibility-Induced Distortions in EPI:
Magnetic susceptibility differences at air-tissue interfaces create spatially varying $B_0$ inhomogeneity. In Echo Planar Imaging (EPI), commonly used for diffusion MRI and fMRI, these inhomogeneities cause off-resonance frequencies to accumulate phase during long readouts. This results in:

Geometric Distortions: Spatial shifts along the phase encoding (PE) direction.
Intensity Modulation: Local signal pile-up (compression) or signal loss (expansion), modeled by the Jacobian determinant of the distortion field.

Limitations of Current Solutions (e.g., FSL TOPUP):
The standard approach, FSL TOPUP, acquires two EPI volumes with opposite PE polarities and estimates a deformation field via nonlinear optimization over a B-spline parameterized field. While effective, it suffers from:

High Computational Cost: Iterative 3D optimization takes tens of minutes to hours.
Parameter Sensitivity: Requires manual tuning of regularization parameters via configuration files.
Over-smoothing: The global smoothness prior (B-splines) may suppress fine-grained local distortion structures, reducing anatomical fidelity.

2. Methodology: SuCor

The authors propose SuCor, a method that reformulates the distortion correction problem using 1D Optimal Transport (OT) and Spectral Regularization.

A. Problem Formulation

Given two EPI volumes with opposite PE polarity, $I_+$ and $I_-$ , and an underlying undistorted image $I_0$ , the forward model accounts for displacement $u(x)$ and intensity modulation (Jacobian terms):
$I_+(x) = I_0(x + u(x)) \cdot (1 + \partial_{PE}u(x))$
$I_-(x) = I_0(x - u(x)) \cdot (1 - \partial_{PE}u(x))$

B. Column-Wise Optimal Transport

The core innovation is treating each column along the PE axis independently.

Observation: For a fixed column, the intensity profiles of $I_+$ and $I_-$ represent the same signal mass redistributed by equal-and-opposite shifts ( $\pm u$ ).
OT Mapping: The displacement field mapping one profile to the other is the Wasserstein-2 optimal transport map.
Closed-Form Solution: For 1D distributions, the optimal map $T^*$ is computed via quantile rearrangement: $T^* = F_b^{-1} \circ F_a$ , where $F$ are cumulative distribution functions.
Displacement Calculation: The raw displacement is derived as the barycentric mean: $u_0(t) = \frac{1}{2}(T^*(t) - t)$ .
Complexity: This step is $O(N \log N)$ per column, allowing independent processing of all columns.

C. Spectral Bending-Energy Regularization

The raw OT field $u_0$ is accurate locally but exhibits "columnar striations" (noise/artifacts) in the cross-PE direction due to independent estimation.

Regularization: The field is regularized in the Fourier domain using a bending-energy penalty (thin-plate energy).
Filter: $\hat{u}_\lambda(k) = \frac{\hat{u}_0(k)}{1 + \lambda |k|^4}$ .
Automatic Parameter Selection (Morozov Discrepancy Principle): Instead of manual tuning, the regularization strength $\lambda$ $λ$ is selected automatically.
1. Estimate noise level $\sigma$ from background voxels using Median Absolute Deviation (MAD).
2. Set a target residual norm $\delta = \tau \cdot \sigma$ (with $\tau=1.5$ ).
3. Use bisection to find $\lambda$ such that $\|u_\lambda - u_0\| = \delta$ .
  This ensures the regularization suppresses noise without over-smoothing genuine signal structure.

D. Final Correction

The corrected images are generated by applying the regularized field $\pm u_\lambda$ to the original volumes via linear interpolation, including Jacobian intensity correction.

3. Key Contributions

Optimal Transport Formulation: First application of 1D Wasserstein-2 OT to model susceptibility distortions, providing an exact, closed-form solution for the displacement map per column.
Parameter-Free & Self-Regularized: Eliminates the need for manual configuration files or grid searches. The regularization strength is determined automatically via the Morozov discrepancy principle.
Computational Efficiency: The algorithm is "embarrassingly parallel" across slices and columns. It processes a full 3D volume in ~12 seconds on a single CPU core, compared to ~55 minutes for TOPUP.
Superior Anatomical Fidelity: By avoiding the global smoothness constraints of B-splines, SuCor preserves fine-grained local distortion structures, leading to better alignment with structural references.

4. Experimental Results

Dataset: Human Connectome Project (HCP) 1200 Young Adult dataset (LR/RL b=0 EPI pairs and co-registered T1).

Metrics & Performance:

Metric	Uncorrected	FSL TOPUP	SuCor
T1 Mutual Information (MI)	0.284	0.317	0.341
LR-RL Normalized Cross-Correlation	0.670	0.964	0.933
LR-RL RMSE	2215	706	948
Runtime	-	55 min	12 sec

Key Findings:

Anatomical Alignment: SuCor achieves a 7.6% higher Mutual Information with the T1 structural reference compared to TOPUP. This indicates superior anatomical localization, particularly in regions with strong distortions (e.g., frontal and temporal lobes).
Trade-off: SuCor shows slightly lower LR-RL consistency (NCC/RMSE) than TOPUP. The authors attribute this to the fact that TOPUP's strong smoothness prior forces the two images to match perfectly, potentially at the cost of anatomical accuracy. SuCor's field captures genuine local variations that TOPUP smooths out.
Field Characteristics: The estimated displacement fields show that SuCor preserves sharper field transitions at tissue boundaries compared to the smoother TOPUP B-spline fields.

5. Significance

Clinical & Research Impact: SuCor offers a rapid, automated solution for distortion correction, making it highly suitable for large-scale studies (like the HCP) and clinical workflows where time and ease of use are critical.
Methodological Shift: It challenges the dominance of iterative nonlinear optimization (B-splines) by demonstrating that 1D optimal transport, when combined with spectral regularization, provides a more faithful representation of local tissue distortions.
Future Directions: The paper highlights a fundamental tension between field smoothness (LR-RL consistency) and correction fidelity (T1 alignment). Future work may explore adaptive regularization to balance these competing objectives further.

In conclusion, SuCor represents a significant advancement in EPI distortion correction, delivering state-of-the-art anatomical alignment with a fraction of the computational cost and zero manual parameter tuning.