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Fast Optimal Transport Averaging of Neuroimaging Data
Alexandre Gramfort, Gabriel Peyré, Marco Cuturi
TL;DR
Neuroimaging averages are challenging because brain geometry and between-subject variability complicate group analysis, while common smoothing can blur signals. The paper develops an efficient transport-based method for averaging non-normalized data, and reports focal activations in fMRI and MEG without requiring a smoothing kernel bandwidth.
Problem
Method
The paper computes constrained Kantorovich means for non-normalized data using entropic smoothing and an efficient algorithm.
Source: §1 · p. 2, §3 · p. 6, §5 · p. 10Results
The fMRI and MEG results show activation foci while avoiding data smoothing, with reported findings in right motor cortex, primary visual cortex, and fusiform gyrus.
Source: §5 · p. 11Takeaways & Limitations
The method handles irregular brain geometry using a ground metric and avoids manually selecting a kernel bandwidth in the reported neuroimaging applications.
Source: §5 · p. 11Because the approach is nonlinear, statistical threshold estimation should use non-parametric permutation tests.
Source: §5 · p. 11
Abstract
from arXiv · showhide
Knowing how the Human brain is anatomically and functionally organized at the level of a group of healthy individuals or patients is the primary goal of neuroimaging research. Yet computing an average of brain imaging data defined over a voxel grid or a triangulation remains a challenge. Data are large, the geometry of the brain is complex and the between subjects variability leads to spatially or temporally non-overlapping effects of interest. To address the problem of variability, data are commonly smoothed before group linear averaging. In this work we build on ideas originally introduced by Kantorovich to propose a new algorithm that can average efficiently non-normalized data defined over arbitrary discrete domains using transportation metrics. We show how Kantorovich means can be linked to Wasserstein barycenters in order to take advantage of an entropic smoothing approach. It leads to a smooth convex optimization problem and an algorithm with strong convergence guarantees. We illustrate the versatility of this tool and its empirical behavior on functional neuroimaging data, functional MRI and magnetoencephalography (MEG) source estimates, defined on voxel grids and triangulations of the folded cortical surface.
1 Introduction
Averaging neuroimaging data is difficult because Euclidean averages ignore the geometry of the domain, while brain signals vary across subjects. The paper proposes a model-free geometric approach for averaging weighted data on regular or irregular grids.
1 Introduction
- Euclidean averaging ignores domain geometry, so the average brain image need not represent the brain of an average individual.The paper identifies atlas computation and group analysis of functional imaging as relevant examples.Source: §1 · p. 1
- The proposed approach uses only geometric arguments and requires prior knowledge of a metric between grid locations.It is intended for regular or irregular grids, including voxel grids and triangulated cortical surfaces.Source: §1 · pp. 1–2
- The method extends Wasserstein barycenter ideas to non-normalized data by using generalized transport distances and constraining barycenter mass to the samples’ average mass.The approach is designed for weighted samples on discretized spaces without assumptions on metric regularity.Source: §1 · p. 2
- The paper develops an efficient computation method and demonstrates it with fMRI and MEG data on voxel grids and folded cortical surfaces.The paper links Kantorovich means to Wasserstein barycenters to use entropic smoothing.Source: §1 · p. 2
2 Optimal Transport Between (Un)Normalized Measures
Wasserstein transport compares equal-mass histograms, but neuroimaging data may be non-normalized. This section describes extensions that handle unequal masses, including the Kantorovich distance used by the paper.
Wasserstein Distance for Normalized Histograms
- For equal-mass histograms, p-Wasserstein distance is the pth root of the minimum transport cost under flows whose row and column marginals match the two histograms.The pairwise location distances form the transport cost matrix.Source: §2 · pp. 2–3
Optimal Transport for Unnormalized Measures
- Other unnormalized transport variants relax marginal equalities or penalize mass differences, but one described variant does not yield a metric.The passage reports that EMD-hat is claimed to be a metric when its added constant is large enough.Source: §2 · p. 3
Kantorovich Norms for Signed Measures
- A zero-sum vector’s Kantorovich norm is extended to arbitrary vectors using a penalty for deviation from zero sum, yielding a distance between nonnegative vectors of different masses.The penalty vector must satisfy the stated bounds relative to the metric.Source: §2 · p. 4
Kantorovich Distances for Unnormalized Nonnegative Measures
- Kantorovich’s distance for unnormalized measures can be cast as standard optimal transport by adding a virtual point that buffers mass differences.The distance from each location to the virtual point is set by its virtual cost.Source: §2 · p. 4
- The section assumes nonnegative histograms have mass bounded by a known constant, which can be rescaled to one for finite data collections.The Kantorovich metric is defined on this bounded-mass set.Source: §2 · p. 4
3 Kantorovich Mean of Unnormalized Measures
The paper defines Kantorovich means as constrained averages of nonnegative measures and connects their computation to Wasserstein barycenters. Entropic regularization makes the optimization smooth, while a fixed target mass addresses sensitivity to virtual costs.
3 Kantorovich Mean of Unnormalized Measures
- Kantorovich means minimize the sum of p-Kantorovich distances from a candidate average to the input measures, subject to a target mass.The default target mass is the mean mass of the input histograms.Source: §3 · pp. 5–6
- The constrained problem is equivalent to a Wasserstein barycenter problem on an augmented simplex, but its linear-program formulation has N(d + 1)^2 variables and is intractable for d around or above 10^4.The equivalence motivates using efficient computational approaches.Source: §3 · p. 5
- Entropic regularization yields a unique optimal transport solution and a differentiable objective, so the method replaces OT terms with their smoothed counterparts.The paper builds on the first algorithm in [7], which can incorporate constraints on the barycenter.Source: §3 · p. 5
- Fixing the barycenter’s total mass addresses sensitivity to virtual costs, allowing Δ to be any nonnegative vector, typically set to a large quantile of pairwise distances.Without the constraint, the solution’s mass depends directly on Δ.Source: §3 · pp. 5–6
- The experiments use p = 1 and λ = 100/median(M), corresponding to Earth Mover’s Distance and small entropic regularization.The regularization makes the problem strongly convex; computations of dual optima can be vectorized and run on GPUs.Source: §3 · p. 6
4 Application to the Averaging of Neuroimaging Data
The application evaluates Kantorovich averaging on simulated cortical signals and fMRI/MEG data, comparing its focality with standard or smoothed means. Across these settings, it highlights localized activations without prespecified spatial smoothing.
Application setting
- The experiments focus on spatial averaging, using voxel distances or cortical geodesics; spatiotemporal extension is stated as straightforward if the metric includes time.The method is intended for non-normalized data on discrete domains with a natural distance.Source: §4 · p. 7
Simulation results
- Gaussian smoothing blurred simulated signals beyond the regions of interest and reduced amplitudes, whereas Kantorovich averaging highlighted two focal group-level signals.The simulation used random positive signals in BA45 and MT across 100 subjects.Source: §4 · p. 8, Figure 1 · p. 8
fMRI results
- The constrained-mass Kantorovich mean highlighted a clear active region without kernel smoothing and yielded an average standardized effect around 1.7, versus 0.23 after Gaussian smoothing with 8-mm FWHM.Source: §4 · p. 9
MEG results
- At 190 ms, Kantorovich averaging localized famous-face activity within V1 and contrast activity to the ventral fusiform gyrus.The contrasts compared responses to famous versus scrambled faces in 16 subjects.Source: §4 · pp. 9–10, Figure 3 · p. 10, Figure 4 · p. 10
Application results
- Across variable subjects, Kantorovich means yield focal population activations without spatial smoothing; on 10,024 vertices and 16 subjects, one barycenter took under 1 minute on a Tesla K40 GPU.The method depends on the metric derived from cortical geometry rather than a manually specified smoothing bandwidth.Source: §4 · p. 10
5 Conclusion
The method highlighted activation foci in fMRI and MEG data without smoothing, while avoiding a kernel-bandwidth choice. It extends transport-based barycenter estimation to non-normalized data and complex brain geometry, and its nonlinearity calls for non-parametric permutation tests when estimating statistical thresholds.
5 Conclusion
- The work extends transportation-metric barycenter estimation to non-normalized measures using a smoothed transport problem and an efficient optimization algorithm.Source: §5 · p. 10
- The algorithm handles irregular grids and surfaces using a ground metric, with simple GPU-suited operations computing full-brain barycenters in a few minutes.Source: §5 · p. 11
- The method better identified activation foci in the primary visual cortex and fusiform gyrus in MEG, and showed right motor cortex activation in fMRI, without smoothing.The findings were consistent with previous neuroscience literature and compared favorably with current pipelines that require a kernel bandwidth parameter.Source: §5 · p. 11
- Because the approach is nonlinear, statistical thresholds for barycenters should be estimated with non-parametric permutation tests.Source: §5 · p. 11