Abstract
Diffusion magnetic resonance imaging (dMRI) offers a unique approach to study the structural connectivity of the brain. DMRI allows to reconstruct the 3D pathways of axons within the white matter as a set of polylines (streamlines), called the tractogram. Tractograms of different brains need to be aligned in a common representation space for various purposes, such as group-analysis, segmentation or atlas construction. Typically, such alignment is obtained with affine registration, through which tractograms are globally transformed, with the limit of not reconciling local differences. In this paper, we propose to improve registration-based alignment by what we call mapping. The goal of mapping is to find the correspondence between streamlines across brains, i.e. to find the map of which streamline in one tractogram correspond to which streamline in the other tractogram. We frame the mapping problem as a rectangular linear assignment problem (RLAP), a cornerstone of combinatorial optimization. We adopt a variant of the famous Hungarian method to get the optimal solution of the RLAP. We validate the proposed method with a tract alignment application, where we register two tractograms and, given one anatomical tract, we segment the corresponding one in the other tractogram. On dMRI data from the Human Connectome Project, we provide experimental evidence that mapping, implemented as a RLAP, can vastly improve both the true positive rate and false discovery rate of registration-based alignment, establishing a strong argument in favor of what we propose. We conclude by discussing the limitations of the current approach, which gives perspective for future work.
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Notes
- 1.
Additionally we introduce the technical assumption that \(\vert T_{A}\vert \leq \vert T_{B}\vert \).
- 2.
A partial permutation matrix is a rectangular version of the permutation matrix, i.e. \(P = [p_{ij}]_{ij} \in \{ 0, 1\}^{n\times m}\) and \(\sum _{j=1}^{n}p_{ij} = 1\) but \(\sum _{i=1}^{m}p_{ij} \leq 1\).
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- 4.
- 5.
For simplicity we limit the view to one slice only.
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Acknowledgements
The research was funded by the Autonomous Province of Trento, Call “Grandi Progetti 2012”, project “Characterizing and improving brain mechanisms of attention - ATTEND”.
Data were provided [in part] by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.
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Sharmin, N., Olivetti, E., Avesani, P. (2016). Alignment of Tractograms as Linear Assignment Problem. In: Fuster, A., Ghosh, A., Kaden, E., Rathi, Y., Reisert, M. (eds) Computational Diffusion MRI. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-28588-7_10
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