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Unbiased groupwise registration of white matter tractography
Lauren J. O’Donnell1,2,3, William Wells III3, Alexandra J. Golby2,3, and Carl-Fredrik
Westin1,3
Lauren J. O’Donnell: odonnell@bwh.harvard.edu
1Laboratory for Mathematics in Imaging, BWH, Harvard Medical School, Boston MA USA
2Golby Lab, a Surgical Brain Mapping Laboratory, BWH, Harvard Medical School, Boston MA
USA
3Surgical Planning Laboratory, BWH, Harvard Medical School, Boston MA USA
Abstract
We present what we believe to be the first investigation into unbiased multi-subject registration of
whole brain diffusion tractography of the white matter. To our knowledge, this is also the first
entropy-based objective function applied to fiber tract registration. To define the probability of
fiber trajectories for the computation of entropy, we take advantage of a pairwise fiber distance
used as the basis for a Gaussian-like kernel. By employing several values of the kernel’s scale
parameter, the method is inherently multi-scale. Results of experiments using synthetic and real
datasets demonstrate the potential of the method for simultaneous joint registration of
tractography.
Keywords
registration; white matter; tractography; diffusion MRI
1 Introduction
Automated medical or neuroscientific analyses of white matter tractography data, such as
segmentation or labeling, creation of atlases, and measurement of tract statistics, all require
initial alignment or normalization of tractography via some method. This alignment is most
often performed by applying the transformations resulting from an image-based fractional
anisotropy or diffusion tensor registration [18, 4]. However if the eventual goal is modeling
and analysis of white matter tracts, it may be advantageous to register the tracts themselves,
as the quantity being optimized during registration will be closely related to the final goal. In
this work we explore the possibility of driving an unbiased multi-subject registration using
the trajectory data produced by streamline tractography.
In contrast to the proposed approach, to our knowledge all other methods for simultaneous
joint registration of tractography have been based on alignment of pre-defined fiber bundles.
These methods have required a pre-existing tractography segmentation for each subject and
have thus been limited to particular structures: corticospinal tract, forceps major, cingulum
and inferior longitudinal fasciculus [1]; structures resulting from an initial clustering plus
expert labels [19]; left uncinate and front-occipital fasciculi [17]; and the arcuate fasciculus,
corticospinal tract, and middle cerebellar peduncles [3]. So far, methods that have performed
registration using unlabeled fiber tracts from the whole brain, e.g. [9, 21, 12], have been
limited to subject-to-subject (pairwise) registration.
NIH Public Access
Author Manuscript
Med Image Comput Comput Assist Interv
. Author manuscript; available in PMC 2013 April 29.
Published in final edited form as:
Med Image Comput Comput Assist Interv
. 2012 ; 15(0 3): 123–130.
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In addition to tractography-based registration, our current work builds on two other
categories of related work: fiber tract comparison, and groupwise image registration. Work
in fiber tract clustering has led to many different metrics [16, 15, 10], generally based on
distances computed between points along the tracts, often with conversion to fiber affinities
using Gaussian kernels as in our proposed objective function. Tracts have also been
analyzed via many styles of point-wise matching, for example [2, 12, 10, 1]. In the image
registration field, several groups have proposed multiple-subject unbiased and template-
based image registration methods. These include entropy-based congealing methods for 2D
[8] and 3D [20] that find a population central tendency image by minimizing entropy, as
well as methods that estimate a population template image that is the minimum distance (in
the space of diffeomorphisms) from all input images [6, 4].
2 Methods
2.1 Objective function
Our basic approach is to represent a brain or atlas by a probability distribution on
trajectories. A “brain” distribution is constructed as a kernel density estimate from the
tractography, and an “atlas” distribution is constructed as a mixture of the constituent brain
distributions. We then choose the alignment parameters on the collection of brains by
maximizing the “sharpness” of the atlas, or minimizing its entropy.
Given a distance metric
D
between fibers we define the probability of a fiber
f
, given
another fiber
fj
, as
(1)
where the distance is used as the basis for a Gaussian-like kernel with standard deviation
σ
,
and the normalization constant
Z
will be discussed later. Our current choice of
D
is
discussed below, however this can in principle work for any of the many existing fiber
distances from the literature. Next, the probability of a fiber
f
, given the set of all fibers
A
and their transformations
T
(“the atlas”), is defined as
(2)
where all fibers
fj
in
A
contribute to the total probability.
The Shannon entropy
H
of the distribution of fibers is the expected value of the negative
log-probability of the fibers. In this case the set of current transformations
T
has been
applied to the fibers (including the transformations
Ti
and
Tj
currently applied to fibers
fi
and
fj
), and we replace the expected value with the sample average value (using the weak
law of large numbers).
(3)
(4)
We minimize the entropy as our objective function, arriving at a set of transformations
T
.
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(5)
Note that to simplify the concept and the notation above, we have not mentioned the fact
that the fibers come from several brains. This is implicitly handled in that the transformation
Ti
applied to fiber
fi
is the same transformation that is also applied to all other fibers from
that brain. We assume that
Z
is constant for a given value of
σ
, and thus does not contribute
to the optimization.
2.2 Fiber representation and distance function
For simplicity and computational speed, we convert the input variable-length fiber
trajectories to a fixed-length representation (as also proposed by [12, 14]). In practice,
representing each fiber by 5 points (endpoints, midpoint, and two intermediate points) was
empirically found to be effective for registration.
Using this fiber parameterization, we propose a pairwise fiber distance metric
D
that is
related to the Hausdorff distance (the maximum of the minimum distances between pairs of
closest points). We calculate
D
as the maximum distance between pairs of corresponding
points along the fibers (i.e. the first through fifth point pairs). This fiber distance
computation can thus take advantage of matrix subtractions.
D
is a symmetric distance that
is the same between fibers (
fi
,
fj
) and (
fj
,
fi
), eliminating the issue of the classic Hausdorff
measure being a directed distance. Because point ordering along the fiber is not known a
priori (a fiber parameterization can equivalently start from either end),
D
is computed with
both possible orderings and the minimum is chosen.
In practice, this method works very well with relatively nearby and corresponding fibers.
For more distant fibers the point correspondence and distance measure may not be
informative, a known problem with all such fiber distance metrics that have been shown to
capture the local structure but poorly reflect the “true relationship” of distant fibers [16].
Luckily in our case, these uninformative large distance measures are unimportant for
registration. These far-away, dissimilar, or outlier fibers are distant relative to the radius of
interest defined by
σ
and have little effect on the objective function.
2.3 Implementation
We have implemented a full affine registration framework using a coordinate descent
method. The code is a Python package that uses VTK [7], scipy [5], and numpy [5]. For
optimization we use Powell’s simplex-based COBYLA (Constrained Optimization by
Linear Approximation) method [13] in the scipy.optimize toolkit. The affine parameters are
constrained (across subjects) as in other entropy-based congealing methods [8, 20] to avoid
an unnecessary overall rotation or translation of all brains, and to avoid the shrink to a point
solution that artifactually reduces entropy. We require that all translation, rotation, and shear
components sum to 0 over all transforms applied to the data, and that all scale factors
average to 1. The COBYLA package allows definition of expected initial step sizes
ρbeg
,
and determines convergence when final step sizes
ρend
are under a user-provided threshold,
thus we have set these
ρ
parameters empirically using the expected magnitudes of our
transform parameters.
The
σ
parameter of the Gaussian kernel (eq. 1) has been tuned to enable multi-scale
registration. In practice, we run several iterations of optimization, alternating translation and
rotation, with an initial
σ
of 30mm. Next, we decrease
σ
to 10mm, then to 5mm, and
optimize while alternating translation, rotation, scale, and shear. The computation of the
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fiber distances has been implemented in a multiprocessor framework. The distances are
computed between a random sample of fibers from each input subject (typically 200–300),
and another, smaller random sample of these fibers whose size we increase during the
registration process (typically beginning with 25 or more fibers). The smaller random
sample is resampled (all sampling is done without replacement) each time we change the
parameters being optimized. The style of randomly sampling data points at which to
compute the objective function has been successfully employed in many registration
strategies [11, 8, 20] and in fiber clustering [15]. The terms in eq. 5 that result from
comparing fibers from the same brain are neglected.
2.4 Data and processing
N=26 healthy subjects dataset: Diffusion weighted images (DWI) scans were acquired on a
3-T GE system using an echo planar imaging sequence and the following parameters: 51
gradient directions with b=900, eight baseline scans with b=0, TR 17000 ms, TE 78 ms,
FOV 24 cm, 144
×
144 encoding steps, 1.7 mm slice thickness. Artifacts due to eddy currents
and head motion were removed by affine registration of diffusion to baseline images using
FSL’s linear image registration tool (FLIRT). Single-tensor streamline tractography was
seeded in the entire brain of each subject in voxels with anisotropy (linear measure)
>
0.2.
3 Results
We performed three registration experiments: objective function probing, synthetic data
validation, and multi-subject registration.
Experiment 1: objective function probing
We investigated the behavior of the multi-scale objective function under simple rotation,
translation, scaling, and shear. One healthy control subject was chosen, and 2000 trajectories
were randomly sampled without replacement, twice, to generate different fixed and moving
brains. A range of transformations was applied to the moving brain, and the objective
function was computed using all fibers from both brains (see fig. 1). Importantly, results
demonstrate that the objective function is very smooth, and that decreasing
σ
has the desired
behavior of increasing sensitivity to small transformations.
Experiment 2: synthetic data validation
Using as input one healthy control subject, we generated synthetic data as follows: 300
trajectories of length greater than 40mm were randomly sampled from the input dataset, and
a randomly generated transformation was applied to these trajectories, to generate a
“synthetic brain.” The parameters of the random transform were: translation up to
±
20mm
along each axis, scale factor from 0.85 to 1.15 in each axis, and rotation up to
±
20 degrees
around each axis. To enable unambiguous computation of errors in the other parameters,
shear was not included. This procedure was repeated 10 times to generate a dataset of 10
synthetic brains with known ground-truth transformations. The registration pipeline was
applied to the 10 brain dataset (see fig. 2), using three levels of scale:
σ
of 30, 10, and 5mm;
and 3 levels of randomly sampled subset fibers: 25, 50, and 75 fibers. These parameters
were set empirically as a compromise between fast optimization and good convergence.
Errors in the resulting parameters were measured by comparison to the ground truth applied
transforms. The mean absolute errors and their standard deviations in each component were:
1.33
±
1.49
,
1.50
±
1.20
,
2.06
±
2.06 degrees rotation; 0.62
±
0.456
,
0.74
±
0.548
,
2.07
±
0.770 mm translation; and 0.015
±
0.014
,
0.006
±
0.007
,
0.017
±
0.015 scale factor
magnitude. The method cannot recover any mean transformation that may have been applied
(e.g. if all input brains were rotated together by 30 degrees that could not be detected) so any
mean transformations were removed from the ground truth transforms before calculation of
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the errors. The experiment ran for 48.8 minutes, spending the following amount of time at
each level of scale: 6.6 minutes at 30mm, 18.1 minutes at 10mm, and 24 minutes at 5mm.
(The computing environment was a 2x2.26 GHz Quad-Core Mac with 16GB of memory.
Note that reported run times could be improved by coding in C rather than python, and/or
increasing use of multiprocessing, however this initial implementation is a proof of
concept.)
Experiment 3: multi-subject dataset
The proposed registration method was applied to the full (N=26) healthy control multi-
subject dataset (see fig. 3). The registration pipeline used 200 randomly sampled fibers of
length greater than 60mm per subject, three levels of scale:
σ
of 30, 10, and 5mm; and 4
levels of numbers of randomly sampled subset fibers: 25, 50, 75, and 100 fibers. The method
spent 48 minutes at the 30mm scale, 188 minutes at the 10mm scale, and 235 minutes at the
5mm scale. The results demonstrate successful alignment of the brains, as can be
appreciated visually in fig. 3, where the output trajectories look locally similar and parallel,
and the subject colors are generally mixed locally (i.e. trajectories from many subjects are
neighboring).
4 Discussion and Conclusion
We have proposed a probabilistic atlas model for tractography that enables the computation
of the entropy of a collection of fibers, and we have shown that registration by minimizing
this entropy can successfully align the white matter in multiple subjects. Advantages of our
objective function include its smoothness and the fact that any fiber outliers will have little
effect. Optimization of the proposed objective, because it is based on tractography data, has
the potential to enhance downstream tractography modeling and statistical analysis results.
Future work will include code optimization, incorporation of higher-order deformations, and
comparison of the method to other fiber- and image-based registration methods. To our
knowledge, this work represents the first method for groupwise registration of whole-brain
diffusion tractography data.
Acknowledgments
We gratefully acknowledge support from NIH grants R21CA156943, P41RR019703, R01MH074794,
R01MH092862, P41RR013218 and P41EB015902, and CIMIT.
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Fig. 1.
Plots of the objective function under x-translation (−40 to 40mm), rotation about x (−40 to
40 degrees), scaling along x (factors of 0.5 to 1.5), and shear (−40 to 40 degrees skew about
x). The curves represent different values of the multi-scale parameter
σ
: 5mm (top curve,
blue), 10mm, 20mm, and 30mm (lowest curve, cyan).
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Fig. 2.
Results of multi-scale registration of randomly transformed synthetic brain (n=10) dataset
(inferior view). Each subject is shown in a different color. The output brains are shown after
each registration scale (
σ
), demonstrating successful coarse-to-fine registration. The output
brains appear slightly “rotated” relative to standard AC-PC orientation due to some mean
component of the initial random transforms.
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Fig. 3.
Results in N=26 healthy subject dataset demonstrate successful coarse-to-fine registration
(inferior and left views shown). Each subject is shown in a different color.
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