Mitigating the eﬀects of imperfect ﬁxel correspondence in Fixel-Based
Robert Elton Smith and Alan Connelly
The Florey Institute of Neuroscience and Mental Health, Melbourne, Australia, Florey Department of Neuroscience and Mental
Health, The University of Melbourne, Melbourne, Australia
A requisite step in performing a Fixel-Based Analysis (FBA) is the determination of "ﬁxel correspondence", which deﬁnes
how discrete ﬁbre elements (ﬁxels) for a particular subject map to the ﬁxels deﬁned in each voxel in template space.
The method used thus far for this purpose - simply selecting the subject ﬁxel that best aligns with the template ﬁxel -
fails to take into consideration the possibility for substantial variations in ﬁxel segmentation across subjects. We
propose a more sophisticated algorithm for determining ﬁxel correspondence, which better accounts for diﬀerences in
ﬁxel segmentation, and demonstrate how this reduces the variance observed in ﬁxel data across healthy controls.
The Fixel-Based Analysis (FBA) framework allows for the data-driven identiﬁcation of eﬀects of interest in white matter quantitative
measures , where statistical inference is both sensitive and speciﬁc to eﬀects in individual ﬁbre bundles within voxels ("ﬁxels") even in
the presence of crossing ﬁbres. One aspect of this framework that does not have an analogue in conventional Voxel-Based Analysis
(VBA) is the necessity to obtain not only spatial alignment of each subject to the template image, but also ﬁxel correspondence
between individual subject data and the ﬁxel template. While this process may seem trivial intuitively, imperfections in this
correspondence due to diﬀerences in ﬁxel segmentation or ﬁbre geometry may contribute signiﬁcant variance to the data. We propose
a method for estimating and accounting for these diﬀerences, reducing the impact of this eﬀect on the outcomes of ﬁxel-based
In the existing publicly-available implementation of FBA , for each template ﬁxel, data are extracted from the subject ﬁxel most
collinear with the template ﬁxel (as long as the angle between them is no greater than 45 degrees by default).
Figure 1 shows an example voxel where the ﬁxel representations vary considerably between subject data and the ﬁxel template. Using
the aforementioned approach, the subject quantitative data would be projected to the template as follows (using nomenclature of
This demonstrates two issues in particular. Firstly, only information from subject ﬁxel s is mapped to template ﬁxel t, and ﬁxel s is
omitted; secondly, both template ﬁxels t and t draw their information from subject ﬁxel s. For measures of ﬁbre density in particular,
these may result in quantitative data for this subject varying substantially from that of other subjects. Note that the angle between t
and s is greater than 45 degrees.
A more appropriate mapping in this case would be:
That is: For template ﬁxel t, information from both ﬁxels s and s contribute to the measure for this subject; for template ﬁxels t and
t, data from subject ﬁxel s must be shared between those template ﬁxels.
In the proposed method, all possible conﬁgurations for the mapping of subject to template ﬁxels are constructed. Here we denote this
mapping as M, with elements M for j in T (number of template ﬁxels in voxel) listing those subject ﬁxels s for i in S (number of subject
ﬁxels in voxel) from which data should be mapped in order to match template ﬁxel t.
The optimal mapping M is selected based on minimization of the following cost function:
and are the ﬁbre densities of template ﬁxel t and remapped subject ﬁxel s' respectively; and are the unit directions
of ﬁxels t and s' respectively; is the frequency with which subject ﬁxel s appears in M; is the inner
product between unit directions and . Each remapped subject ﬁxel s' is deﬁned based on the (weighted) combination of those
subject ﬁxels being mapped to it according to M:
2 1 1
3 4 3
1 1 2 3
j j i
f is minimal when each remapped subject ﬁxel s' is both collinear with, and contains the same total ﬁbre density as, corresponding
template ﬁxel t, and no subject ﬁxels s are omitted from the mapping. This process, demonstrated for the optimal mapping for the data
in Figure 1, is shown in Figure 2.
Eﬀects of the proposed ﬁxel matching algorithm on healthy control data mapped to a ﬁxel template are demonstrated in Figure 3.
By constraining the contribution from each subject to the template based on preservation of ﬁbre volume, the total ﬁbre density within
each voxel in the template image no longer contains erroneous local minima or maxima due to imperfect ﬁxel correspondence (Figure 3;
top row). In addition, by accounting for more complex ﬁxel mappings to the template, the variance in ﬁbre density between healthy
controls is reduced, particularly in deep WM crossing-ﬁbre regions (Figure 3; bottom row).
The proposed approach also enables a number of other potential beneﬁts for FBA; for instance: Omitting individual subject data from
particular ﬁxels, or removing ﬁxels from the statistical analysis mask where ﬁxel correspondence is ill-deﬁned; including a measure of
ﬁxel correspondence complexity within the General Linear Model (GLM).
The reduction in intrinsic data variance achieved by accounting for more complex ﬁxel correspondence between subject and template
space should make future FBA studies more sensitive to pathologies with small eﬀect size, and/or located in regions with complex ﬁbre
We are grateful to the National Health and Medical Research Council (NHMRC) of Australia, and the Victorian Government's Operational
Infrastructure Support Program for their support.
1. Raﬀelt, D. A.; Tournier, J.-D.; Smith, R. E.; Vaughan, D. N.; Jackson, G.; Ridgway, G. R. & Connelly, A. Investigating white matter
ﬁbre density and morphology using ﬁxel-based analysis. NeuroImage, 2016, 144, 58-73
Example voxel where the ﬁxel correspondence between subject and template data is not a simple one-to-one mapping, and hence a
naïve mapping of the nearest subject ﬁxel to each template ﬁxel will be imperfect. Each ﬁxel is individually coloured according to its
direction (axis colouring convention provided in bottom left), and its length & thickness reﬂects the ﬁbre density in that ﬁxel.
Demonstration of proposed ﬁxel matching algorithm, using ﬁxel data from Figure 1. For each template ﬁxel, a remapped subject ﬁxel is
generated using some combination of subject ﬁxels, which are then compared to the template ﬁxel in terms of both direction and
density. In this example, reconstruction is demonstrated for the optimal mapping (M = [s, s], M = , M = [s], M = [s]); in order to
ﬁnd this optimum, the algorithm generates these reconstructed template ﬁxels for all plausible combinations of mapping indices.
Comparison between outcomes of existing ﬁxel correspondence algorithm (left column) and proposed solution (right column), for a
cohort of 28 healthy control subjects. Top row: Total ﬁbre density (FD) in each voxel of the template (mean FD within each template ﬁxel
computed across subjects, then the sum of FD for all ﬁxels within each template voxel). Bottom row: Standard deviation of FD within
each ﬁxel in the template.
1 1 2 2 3 3 4 3