A Texture Manifold for Curve-Based Morphometry of the Cerebral Cortex.

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A Texture Manifold for Curve-Based
Morphometry of the Cerebral Cortex
Maxime Boucher?, Alan Evans, and Kaleem Siddiqi
Center for Intelligent Machines, McGill University
{boucher,siddqi}@cim.mcgill.ca
McConnell Brain Imaging Center, McGill University
{alan}@bic.mni.mcgill.ca
Abstract. The cortical surface of the human brain is composed of folds
that are juxtaposed alongside one another. Several methods have been
proposed to study the shape of these folds, e.g., by first segmenting
them on the cortical surface or by analysis via a continuous deformation
of a common template. A major disadvantage of these methods is that,
while they can localize shape differences, they cannot easily identify the
directions in which they occur. The type of deformation that causes a fold
to change in length is quite different from that which causes it to change
in width. Furthermore, these two deformations may have a completely
different biological interpretation. In this article we propose a method to
analyze such deformations using directional filters locally adapted to the
geometry of the folding pattern. Motivated by the texture flow literature
in computer vision we recover flow fields that maintain a fixed angle with
the orientation of folds, over a significant spatial extent. We then trace
the flow fields to determine which correspond to the shape changes that
are the most salient. Using the OASIS database, we demonstrate that
in addition to known regions of atrophy, our method can find subtle but
statistically significant shape deformations.
1Introduction
The human cortical surface is composed of a set of folds that run alongside one
another to form an undulating pattern of crests and troughs. The shape of this
folding pattern evolves during the normal cycle of human brain development un-
der the effect of aging or the presence of diseases or cognitive deficits. A popular
method to capture shape differences is to view brain development as a continuous
deformation of a common template [1–3]. The continuous deformation between
the common template and individual cortical surfaces is typically constrained
to be a diffeomorphism and is obtained using surface registration techniques.
Deformation statistics are then used to find regions of significant tissue growth
or atrophy. Such statistics can determine, for example, if the overall area of a
particular region is larger in one group compared with another.
?Corresponding author: boucher@cim.mcgill.ca

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2Maxime Boucher, Alan Evans, and Kaleem Siddiqi
In addition to diffeomorphism based statistics, which capture local changes
in shape, regularization filters such as the isotropic diffusion kernel on surfaces
[4,5], can yield a more global measure of shape differences. However, anatom-
ical structures in the human brain are typically not isotropic and nor are the
changes they induce on the cortical surface as they deform. As an example Fig.
1 illustrates two distinct deformations of a folding pattern, represented by a set
of parallel curves. In the context of cortical shape analysis it is therefore bene-
ficial to analyze differences using filters that are tuned to specific orientations.
A neuroscientist can then examine whether an observed shape difference in a
group follows a particular direction relative to fold orientation.
(a)(b)
Fig.1. Fig. 1(b) shows two deformations of the folding pattern in Fig. 1(a). On
Fig. 1(b), a deformation parallel to a fold increases its length, and perpendicular
to folds decreases the spacing between folds.
In this paper we abstract the shape of cortical folds as a collection of juxta-
posed curves that locally have similar orientations. We then design shape filters
which maintain a constant angle with neighboring folds. For example, a partic-
ular shape filter can be designed to be sensitive to a deformation parallel to fold
orientation (an angle of 0 degrees), while another can be sensitive to a deforma-
tion that is perpendicular. This permits an analysis which is relative to cortical
fold orientation. The challenge here is the estimation of the curvature to apply
so that each filter maintains this constant angle.
To illustrate the computation of curvature consider the fingerprint example
of Fig. 2(a), which is a pattern formed of juxtaposed intensity ridges similar
to the folds of the cerebral cortex. The output of a directional ridge detector
applied to the slice in blue is shown in Fig. 2(b), as an intensity function of
angle (θ) versus position x1. It is evident that this output aggregates along a
a continuous curve (or submanifold), dubbled a texture flow in the work of [6].
When the exact location of the submanifold of maximum intensity in Fig. 2(b)
is known the submanifold can be characterized as a function of orientation in
space Θ(x1,x2). Here Θ gives the orientation of neighboring curves at a location
(x1,x2) of Fig. 2(a). The curvature of neighboring curves is expressed as the
gradient of Θ as ∇Θ. On the other hand, the slope of the aggregated intensity
curve in Fig. 2(b) is also given by ∇Θ. Our aim in this paper is to determine the
slope of the texture flow, and then generate directional filters using the curvature
implied by this slope.
Inspired by the work of [6] we achieve a globally optimal assignment of cur-
vatures to the cortical surface by using a dictionary of smooth flow fields on the
surface. Each smooth flow field provides a hypothesis for the slope of the texture

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Directional Morphometry of the Cerebral Cortex Using a Texture Manifold3
(a) Fingerprint image (b) Ridge detection on
Fig. 2(a) along the
blue slice
(c) Fig. 2(b) resam-
pledalong
slope v1.
agiven
Fig.2. A texture manifold depicts change in orientation of the neighboring
curves in space
flow on the cerebral cortex. We measure the goodness of fit between each flow
field and the actual texture manifold by using anisotropic filters. We then use re-
laxation labeling to achieve a globally optimum assignment of smooth flow fields
that match the cortical folds on the surface. Directional filters are then generated
by launching streamlines whose curvature is provided by this assignment.
Using the OASIS database [7] we test our framework to determine how the
shape of cortical folds in patients with cognitive impairment is affected. We
are able to identify folds that undergo significant deformation in the temporal
lobe, and also a stretch in length of the Cingulate gyrus below the Praecuneus
area and below the sup frontal cortex in the left hemisphere. These results are
also partially revealed by a statistical analysis based on surface area (see Fig. 5).
However, our results clearly indicate that the increase in surface area comes from
a stretch in sulcal length (see Fig. 6(b)). Whether this stretch in length is due to
structural connectivity loss in the white matter remains to be investigated, but
our findings are corroborated by a visual inspection of shape differences between
the two groups.
2Estimating the Texture Manifold from Folds
We give a brief technical overview of the algorithm before detailing each step.
We refer again to Figure 2 to illustrate the method. Suppose that the output
of the ridge detection in Figure 2(a) can be expressed as an intensity function
I(x1,x2,θ) → R+. We formulate different hypotheses Hiabout the slope of the
texture flow. Suppose, for example, that one such hypothesis Histates that the
texture flow has a slope of (v1,v2). This means that if we resample I as
Ii(x1,x2,θ) = I(x1,x2,θ + v1x1+ v2x2), (1)
then, if this hypothesis is good, the intensity function should aggregates along
an “horizontal” plane (as shown in Fig. 2(c)). A simple way to measure if a
particular hypothesis is a good explanation for the observed texture flow is to
measure how horizontal it is using an anisotropic filter. Suppose that ga,b is
a Gaussian kernel where a expresses the width of the kernel along the spatial

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4Maxime Boucher, Alan Evans, and Kaleem Siddiqi
dimensions (x1,x2) and b expresses the width of the kernel along the orientation
dimension θ as
ga,b(x1,x2,θ) =
?
(2π)3/2a2b
?−1
exp
?
−x2
1+ x2
2a2
2
−θ2
2b2
?
.(2)
A good measure to determine if Hiis a good hypothesis is
fi(x1,x2,θ) =
?∂
∂θIi∗ ga,b
?2
(3)
where ∗ is a convolution. The best hypothesis Hiis the one that locally max-
imizes Equation 3. However, an optimal hypothesis should be a good fit for
several curves in a large neighborhood. We therefore use each hypothesis Hias
labels in the space formed by (x1,x2,θ). We achieve a globally optimum labeling
H∗(x1,x2,θ) by selecting, at each point, the label which is both a good fit to
the intensity function I and is the most similar to the neighboring labels.
The ridge detector for surfaces is presented in Section 2.1, the algorithm to
generate flow hypotheses Hion surfaces is explained in Section 2.2 and the relax-
ation labeling approach to find an optimal assignment is explained in Section 2.3.
Once an optimal assignment is reached, it is possible to generate streamlines
that follow the geometry of the folding pattern, which can then be used to de-
tect shape changes. The algorithm to generate streamlines and use them to test
statistics on surfaces is explained in Section 2.4.
2.1A Ridge Detector on Surfaces
In this paper, we used the principal curvature of the surface to detect ridges and
generate an intensity function I that describes the texture manifold. We present
the intensity function I that gave the best result, however the algorithm applies
for other choices of ridge detectors as well.
Let S be a smooth genus 0 surface and let κ1,κ2,|κ1| ≥ |κ2| be the principal
curvatures of S and u1,u2 their associated vector directions. We associate a
fold with a line of low curvature in one direction with high curvature in the
perpendicular direction. Let S be the unit circle and let vθ∈ S. We define the
probability to observe a fold at a given location u ∈ S for a given orientation θ
using a symmetric Von Mises distribution as
I(u,θ) = n(|κ1(u)| − |κ2(u)|)−1e(|κ1(u)|−|κ2(u)|)(ut
where n(|κ1(u)|−|κ2(u)|) is the normalization factor such that?
with a maximum at uθ = ±u1 and where |κ1| − |κ2| determines the spread
around the maximums.
θu1)2,(4)
SI(u,θ)dθ = 1.
Equation 4 can be seen as the equivalent of a Gaussian distribution for angles,

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Directional Morphometry of the Cerebral Cortex Using a Texture Manifold5
2.2 A Dictionary of Smooth Vector Fields to Model the Texture
Manifold
Unlike in images, it is not possible to hypothesize that the texture flow has a
slope given by v1,v2and then resample the image using these slope parameters
as done in Equation 1. Instead, we use a base flow field hi. Let hi(u) be the
angle of the base flow field at a point u ∈ S. We can use hito resample I as
Ii(u,θ) = I(u,θ + hi(u)) .(5)
Then, we can apply a ridge detection to determine if hi is a good hypothesis
locally of the texture manifold.
The point of the algorithm is to generate several flow field hi, each with a
different curvature. Each flow field hiis generated by placing a source and a sink
on S and then generating a smooth (singularity-free) completion between them.
Specifically, let si= (si,1,si,2) be a source and a sink, respectively. A vector field
is then completed on S/siby finding the minimum of the following functional:
?
We note that hi is a unit vector field defined on S. An example for a hi is
given in Figure 3(a). On this figure, the vector field fans in the vicinity of the
singularity. To generate a full set of hypothesis hi,i = 1,...,N, we distribute
sources and sinks uniformly on the cortical surface such that every location is
offered a possibility to “fan” . An interesting property of our formulation in
Equation 5 is that hican be seen as a baseline. The function Ii(u,0) measures
the likelihood that folds fans away from the singularity used to generate hi, as
shown in Fig. 3(a), while the function Ii(u,π/2) measures the likelihood that
folds rotate around the same singularities, as shown in Fig. 3(b).
The streamline tracing algorithm works as follows. Once it is determined
that hiis a locally optimal hypothesis an initial streamline is launched with an
angle of α with hi(for example α = π/2). We then follow the flow given by the
function hi+ α over the entire region for which hi is the optimal hypothesis.
Examples of streamlines generated by this process are shown in Figure 3(c).
hi= argminh∗
S/si
curl(h∗)2+ div(h∗)2dS.
(6)
2.3Relaxation Labeling of Tangential and Normal Curvature
Till now we have described an algorithm to generate flow fields hion a surface
S. In this section, we describe how to select, from hi, an optimal assignment that
best matches the flow field oberved on a surface. Let H∗(u,θ) ∈ {h1,...,hn}.
We use relaxation labeling [8] to determine which of the possible flows hioffers
the best local fit to the fold lines of the cortical surface. Relaxation labeling is a
framework to find the statistical mode of a distribution given general constraints
to be satisfied. Let pi(S,θ) be the probability that hypothesis hi(S,θ) has the
highest support at any given location. Here, we interpret hi(S,θ) has the hy-
pothesis given by hiwith a flow field with initial angle θ. The pi(S,θ) forms a
probability space, such that pi(S,θ) > 0 and?N
i=1pi(S,θ) = 1 everywhere.