# Improved Riemannian FOD averaging for fiber bundle priors incorporation in FOD-based tractography algorithms

Grégoire Ville, Emmanuel Caruyer, Julie Coloigner

► To cite this version:

Grégoire Ville, Emmanuel Caruyer, Julie Coloigner. Improved Riemannian FOD averaging for fiber bundle priors incorporation in FOD-based tractography algorithms. IST 2025 - International Society for Tractography, Oct 2025, Bordeaux, France. pp.1-1. hal-05425887

HAL Id: hal-05425887

https://hal.science/hal-05425887v1

Submitted on 19 Dec 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Improved Riemannian FOD averaging for fiber bundle priors incorporation in FOD-based tractography algorithms

G. Ville¹, E. Caruyer,¹* J. Coloigner¹*

¹Univ Rennes, Inria, CNRS, Inserm, IRISA UMR 6074, Empenn ERL Rennes, U-1228, France

*These authors contributed equally to this work.

## INTRODUCTION

Guiding tractography algorithms with prior information from anatomy or microstructure can allow for better coverage of white matter bundles with streamlines, without increasing the number of false positives (improved sensitivity-specificity ratio). In [1], these priors are estimated from templates of streamlines and take the form of track orientation distributions (TOD) [2], which once incorporated voxel wise into fiber orientation distributions (FOD), help tractography algorithms to reconstruct the fiber bundles. In a similar approach [3], TOD are mixed with diffusion orientation distribution functions (ODF) thanks to a weighted averaging following a Riemannian framework [4]. The voxel-specific weights allow to adapt the importance given to priors to the fiber configuration and to the confidence in the ODF estimate. Following this work, we proposed an improved way to perform averaging between model and priors, which is also generalized to FOD.

## METHODS

The approach in [3] can be divided in three steps, that we also followed in our work: i) building for each bundle of interest an anatomical atlas from pre-segmented streamlines, ii) estimating the TOD-based priors from this atlas of streamlines and iii) incorporating the TOD into the ODF of the studied subject, once registered in the latter's space, by using the Riemannian weighted averaging framework mentioned above. The latter starts by computing the square root of the input distributions and has been devised for functions whose square integrates to 1 over the sphere [4].

While the averaging implementation from [3] only worked for ODF (diffusion-based), we extended it to include FOD (fiber-based). This required an adaptation, as (square roots of) FOD don't (squarely) integrate to 1 over the sphere, contrary to diffusion ODF. Thus, we normalized the square roots of FOD to make them squarely integrate to 1, by multiplying each of the spherical harmonics (SH) coefficients used to represent them by a constant factor. After averaging, we multiplied the output's SH coefficients by the inverse of this factor, to bring the distribution back to its initial density. Furthermore, we chose to use spherical designs [5] to compute a (point, value) representation of the FOD, useful to get their square root or their square after hand, instead of a simple grid sampling as it was done before. Spherical designs allow error-free replacement of integrals by sums; they are hence useful to find the representation by coefficients of the FOD after computing the square root or the square. Finally, another contribution was to constrain the computations within a white matter (WM) mask, to avoid falsely modifying gray matter (GM) and cerebrospinal fluid (CSF) distributions.

We tried our framework with 105 high resolution healthy subjects from the Human Connectome Project (HCP)<fnref n="2" />. They have a tractogram pre-segmented into fiber bundles, that we used as ground-truth. 100 subjects were used to build TOD-based priors for 6 bundles of interest. The 5 remaining subjects were used for testing. We first estimated FOD using MSMT-CSD [6]; that allowed us to create a WM mask from GM and CSF partial volume maps. We then incorporated the bundle priors into the FOD images thanks to our improved averaging framework. Afterward, we performed tractography with the iFOD2 algorithm [7], using as input both the classical FOD and the output of the averaging, that we call as in [1] enhanced FOD (E-FOD). We generated 1 million streamlines in these two experiments for equal comparison. The tractograms were then filtered with bundle extremities masks from TractSeg [8] to keep only the streamlines of interest for the bundle of study. Finally, we performed a clustering with one cluster using Quickbundles [9], to remove every streamline further than a bundle-specific threshold from the obtained centroid. That allowed us to remove some false positives. We compared this output to ground-truth using a weighted-Dice coefficient and the overlap score (which is the proportion of the ground-truth mask covered by streamlines).

## RESULTS

Fig. 1 shows that we managed to generalize the Riemannian averaging framework to FOD, by preserving the input distributions' density while enhancing the bundle of study. Indeed, in the E-FOD image, the cingulum (CG) is more recognizable and covers a greater surface. The validity of the enhancing can be checked at the voxel scale with Fig. 2: the green peak, corresponding to CG, is more voluminous after priors incorporation, while the second peak is reduced.

Fig. 3 and 4 show that with our priors incorporation, tractography manages to better reconstruct the bundles, mostly at the bundles' extremities which become much better covered, even with a same input number of streamlines. For the right part of the corticospinal tract (CST right), the average overlap between the 5 subjects used for testing rises from 0.813 without priors to 0.947 with priors. For the right part of the superior longitudinal fascicle I (SLF_I right), it rises from 0.691 to 0.877. However, the average weighted-Dice score only goes from 0.648 to 0.651 for CST right and from 0.615 to 0.662 for SLF_I right. It wasn't possible to show all the results by lack of space, but we tried to display the most representative ones.

## DISCUSSION AND CONCLUSION

Thanks to priors incorporation, we managed to get a better overlap and spatial resolution at the bundle scale with the same number of streamlines, as did in [1] and [3], but using a weighted averaging framework generalized to FOD and technically improved (constrained by a WM mask and using spherical designs for sampling). However, most of the time, the weighted-Dice score increases very little or not at all. Fig. 3 gives us an explanation: incorporating the priors leads to estimate streamlines that were not present in the ground-truth, like in the fanning part of CST right. They are hence accounted for false positives and lower the weighted-Dice score. Conversely, the overlap sharply increases since it is not penalized by false positives. But the weighted-Dice may not be the most relevant metric, since it bases its computations on density maps, namely images showing the proportion of streamlines passing through each voxel. And streamline density is known to have its own biases in tractography. Our future plans include using Recobundles [10] to segment the output tractograms of our method into bundles instead of masks from TractSeg, and to add a streamline-based evaluation score. We also aim at evaluating our framework on clinical data and subjects with neurological disorders.

**REFERENCES:** [1] Rheault et al., Neuroimage 186:382-398 (2019). [2] Dhollander et al., Neuroimage 94:312-336 (2014). [3] Durantel et al., PLoS One, 20(3), e0304449 (2025). [4] Goh et al., Neuroimage 56:1181-1201 (2011). [5] Delsarte et al., Geom Dedicata 6:363-388 (1977). [6] Jeurissen et al., Neuroimage 103:411-426 (2014). [7] Tournier et al., Proc. Intl. Soc. Mag. Reson. Med. 18:1670 (2010). [8] Wasserthal et al., Neuroimage 183:239-253 (2018). [9] Garyfallidis et al., Frontiers in Neuroscience, 6, 175 (2012). [10] Garyfallidis et al., Neuroimage 170:283-295 (2018).

<figure><img src="image_2.png" /></figure>

<footnote n="2">²https://www.humanconnectome.org/</footnote>