Learning Task-Optimal Registration Cost Functions for Localizing Cytoarchitecture and Function in the Cerebral Cortex
Image registration is typically formulated as an optimization problem with multiple tunable, manually set parameters. We present a principled framework for learning thousands of parameters of registration cost functions, such as a spatially-varying tradeoff between the image dissimilarity and regularization terms. Our approach belongs to the classic machine learning framework of model selection by optimization of cross-validation error. This second layer of optimization of cross-validation error over and above registration selects parameters in the registration cost function that result in good registration as measured by the performance of the specific application in a training data set. Much research effort has been devoted to developing generic registration algorithms, which are then specialized to particular imaging modalities, particular imaging targets and particular postregistration analyses. Our framework allows for a systematic adaptation of generic registration cost functions to specific applications by learning the "free" parameters in the cost functions. Here, we consider the application of localizing underlying cytoarchitecture and functional regions in the cerebral cortex by alignment of cortical folding. Most previous work assumes that perfectly registering the macro-anatomy also perfectly aligns the underlying cortical function even though macro-anatomy does not completely predict brain function. In contrast, we learn 1) optimal weights on different cortical folds or 2) optimal cortical folding template in the generic weighted sum of squared differences dissimilarity measure for the localization task. We demonstrate state-of-the-art localization results in both histological and functional magnetic resonance imaging data sets.
1424 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 7, JULY 2010
Learning Task-Optimal Registration Cost Functions
for Localizing Cytoarchitecture and Function
in the Cerebral Cortex
B. T. Thomas Yeo*, Mert R. Sabuncu, Tom Vercauteren, Daphne J. Holt, Katrin Amunts, Karl Zilles,
Polina Golland, and Bruce Fischl
Abstract—Image registration is typically formulated as an opti-
mization problem with multiple tunable, manually set parameters.
We present a principled framework for learning thousands of pa-
rameters of registration cost functions, such as a spatially-varying
tradeoff between the image dissimilarity and regularization terms.
Our approach belongs to the classic machine learning framework
of model selection by optimization of cross-validation error. This
Manuscript received October 24, 2009; revised April 21, 2010; accepted
April 22, 2010. Date of publication June 07, 2010; date of current version June
30, 2010. This work was supported in part by the NAMIC (NIH NIBIB NAMIC
U54-EB005149), in part by the NAC (NIH NCRR NAC P41-RR13218), in
part by the mBIRN (NIH NCRR mBIRN U24-RR021382), in part by the
NIH NINDS R01-NS051826 Grant, in part by the NSF CAREER 0642971
Grant, in part by the National Institute on Aging (AG02238), in part by the
NCRR (P41-RR14075, R01 RR16594-01A1), in part by the NIBIB (R01
EB001550, R01EB006758), in part by the NINDS (R01 NS052585-01), and in
part by the MIND Institute. Additional support was provided by The Autism
& Dyslexia Project funded by the Ellison Medical Foundation. The work
of B. T. Thomas Yeo was supported by the A*STAR, Singapore. Asterisk
indicates corresponding author.
*B. T. T. Yeo is with the Computer Science and Artiﬁcial Intelligence
Laboratory, Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:
P. Golland is with the Computer Science and Artiﬁcial Intelligence
Laboratory, Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:
M. R. Sabuncu is with the Computer Science and Artiﬁcial Intelligence Lab-
oratory, Department of Electrical Engineering and Computer Science, Massa-
chusetts Institute of Technology, Cambridge, MA 02139 USA and also with
the Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts Gen-
eral Hospital, Harvard Medical School, Charlestown, MA 02129 USA (e-mail:
T. Vercauteren is with Mauna Kea Technologies, 75010 Paris, France (e-mail:
D. J. Holt is with the Massachusetts General Hospital Psychiatry Depart-
ment, Harvard Medical School, Charlestown, MA 02139 USA (e-mail: dholt@
K. Amunts is with the Department of Psychiatry and Psychotherapy, RWTH
Aachen University and the Institute of Neuroscience and Medicine, Research
Center Jülich, 52425 Jülich, Germany (e-mail: firstname.lastname@example.org).
K. Zilles is with the Institute of Neuroscience and Medicine, Research Center
Jülich and the C.&O. Vogt-Institute for Brain Research, University of Düssel-
dorf, 52425 Jülich, Germany (e-mail: email@example.com).
B. Fischl is with the Computer Science and Artiﬁcial Intelligence Laboratory,
Department of Electrical Engineering and Computer Science, Massachusetts In-
stitute of Technology, Cambridge, MA 02139 USA, and also with the Athinoula
A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital,
Harvard Medical School, Charlestown, MA 02129 USA, and also with the De-
partment of Radiology, Harvard Medical School and the Divison of Health Sci-
ences and Technology, Massachusetts Institute of Technology, Cambridge, MA
02139 USA (e-mail: ﬁschl@nmr.mgh.harvard.edu).
Color versions of one or more of the ﬁgures in this paper are available online
Digital Object Identiﬁer 10.1109/TMI.2010.2049497
second layer of optimization of cross-validation error over and
above registration selects parameters in the registration cost
function that result in good registration as measured by the per-
formance of the speciﬁc application in a training data set. Much
research effort has been devoted to developing generic registration
algorithms, which are then specialized to particular imaging
modalities, particular imaging targets and particular postregistra-
tion analyses. Our framework allows for a systematic adaptation
of generic registration cost functions to speciﬁc applications by
learning the “free” parameters in the cost functions. Here, we
consider the application of localizing underlying cytoarchitecture
and functional regions in the cerebral cortex by alignment of
cortical folding. Most previous work assumes that perfectly reg-
istering the macro-anatomy also perfectly aligns the underlying
cortical function even though macro-anatomy does not completely
predict brain function. In contrast, we learn 1) optimal weights
on different cortical folds or 2) optimal cortical folding template
in the generic weighted sum of squared differences dissimilarity
measure for the localization task. We demonstrate state-of-the-art
localization results in both histological and functional magnetic
resonance imaging data sets.
Index Terms—Cross validation error, functional magnetic res-
onance imaging (fMRI), histology, ill-posed, leave one out error,
local maxima, local minima, model selection, objective function,
parameter tuning, registration parameters, regularization, space
of local optima, tradeoff.
IN medical image analysis, registration is necessary to es-
tablish spatial correspondence across two or more images.
Traditionally, registration is considered a preprocessing step
[Fig. 1(a)]. Images are registered and are then used for other
image analysis applications, such as voxel-based morphometry
and shape analysis. Here, we argue that the quality of image
registration should be evaluated in the context of the applica-
tion. In particular, we propose a framework for learning the
parameters of registration cost functions that are optimal for
a speciﬁc application. Our framework is therefore equivalent
to classic machine learning approaches of model selection by
optimization of cross-validation error , , .
Image registration is typically formulated as an optimization
problem with a cost function that comprises an image dissim-
ilarity term and a regularization term [Fig. 1(a)]. The param-
eters of the cost function are frequently determined manually
by inspecting the quality of the image alignment to account
0278-0062/$26.00 © 2010 IEEE
Authorized licensed use limited to: MIT Libraries. Downloaded on July 22,2010 at 11:10:18 UTC from IEEE Xplore. Restrictions apply.
YEO et al.: LEARNING TASK-OPTIMAL REGISTRATION COST FUNCTIONS FOR LOCALIZING CYTOARCHITECTURE 1425
Fig. 1. Traditional and proposed frameworks for image registration. indicates a collection of images. In image registration, we seek a deformation for
each image . The resulting deformations are then used for other applications, such as segmentation or group analysis. The registration cost function
typically contains multiple parameters, such as the tradeoff parameter and the template . Changes in these parameters alter the deformations and thus the
outcomes of downstream applications. In our framework (b), we assume a training data set, which allows us to evaluate the quality of the registration as measured by
the application performance (or cross-validation error metric) for each training subject. This allows us to pick the best parameters that result in good registration
as measured by . Subsequent new subjects are registered using these learned parameters.
Fig. 2. Examples of ambiguities in image registration, which can potentially be resolved by taking the application at hand into account. (a) Postcentral sulci with
different topology. (b) BAs overlaid on cortical surfaces.
for the characteristics (e.g., resolution, modality, signal-to-noise
ratio) of the image data. During this process, the ﬁnal task is
rarely considered in a principled fashion. Furthermore, the vari-
ability of the results due to these tunable parameters is rarely re-
ported in the literature. Yet, recent work has shown that taking
into account the tradeoff between the regularization and simi-
larity measure in registration can signiﬁcantly improve popula-
tion analysis  and segmentation quality , .
In addition to improving the performance of applications
downstream, taking into account the end-goal of registration
could help resolve ambiguities and the ill-posed nature of
1) The variability of the folding pattern in the human cerebral
cortex is well-documented (see e.g., ). Fig. 2(a) shows
postcentral sulci of two different subjects. Note the differ-
ences in topology between the two sulci. When matching
cortical folds, even neuroanatomical experts disagree on
whether to join the ends of the broken sulcus or to break
up the uninterrupted sulcus.
2) In population studies of human brain mapping, it is
common to align subjects into a single coordinate system
by aligning macroanatomy or cortical folding patterns.
The pooling of functional data in this common coordinate
system boosts the statistical power of group analysis and
allows functional ﬁndings to be compared across different
studies. However, substantial cytoarchitectonic , ,
 and functional , –, ,  variability is
widely reported. One reason for this variability is certainly
misregistration of the highly variable macroanatomy.
However, even if we perfectly align the macroanatomy,
the underlying function and cellular architecture of the
cortex will not be aligned because the cortical folds do
not completely predict the underlying brain function ,
. To illustrate this, Fig. 2(b) shows nine Brodmann
areas (BAs) projected onto the cortical surfaces of two
different subjects, obtained from histology. BAs deﬁne
cytoarchitectonic parcellation of the cortex closely related
to brain function . Here, we see that perfectly aligning
the inferior frontal sulcus [Fig. 2(b)] will misalign the
superior end of BA44 (Broca’s language area). If our
goal is to segment sulci and gyri, perfect alignment of the
cortical folding pattern is ideal. However, it is unclear that
perfectly aligning cortical folds is optimal for function
Authorized licensed use limited to: MIT Libraries. Downloaded on July 22,2010 at 11:10:18 UTC from IEEE Xplore. Restrictions apply.
1426 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 7, JULY 2010
In this paper, we propose a task-optimal registration framework
that optimizes parameters of any smooth family of registration
cost functions on a training set, with the aim of improving the
performance of a particular task for a new image [Fig. 1(b)]. The
key idea is to introduce a second layer of optimization over and
above the usual registration. This second layer of optimization
assumes the existence of a smooth cost function or cross-valida-
tion error metric [ in Fig. 1(b)] that evaluates the performance
of a particular task given the output of the registration step for
a training data set. The training data provides additional infor-
mation not present in a test image, allowing the task-speciﬁc
cost function to be evaluated during training. For example, if the
task is segmentation, we assume the existence of a training data
set with ground truth segmentation and a smooth cost function
(e.g., Dice overlap measure) that evaluates segmentation accu-
racy. If the registration cost function employs a single param-
eter, then the optimal parameter value can be found by exhaus-
tive search . With multiple parameters, exhaustive search is
not possible. Here, we establish conditions for which the space
of local minima is locally smooth and demonstrate the optimiza-
tion of thousands of parameters by gradient descent on the space
of local minima, selecting registration parameters that result in
good registration local minima as measured by the task-speciﬁc
cost function in the training data set.
We validate our framework on two datasets. The ﬁrst dataset
consists of 10 ex vivo brains with the BAs of each subject ob-
tained via histology ,  and mapped onto the cortical sur-
face representation of each subject obtained via MRI . The
second dataset consists of 42 in vivo brains with functional re-
gion MT+ (V5) deﬁned using functional magnetic resonance
imaging (fMRI). Here, our task is deﬁned to be the localiza-
tion of BAs and MT+ in the cortical surface representation via
the registration of the cortical folding pattern. While it is known
that certain cytoarchitectonically or functionally-deﬁned areas,
such as V1 or BA28, are spatially consistent with respect to
local cortical geometry, other areas, such as BA44, present a
challenge for existing localization methods , . We learn
the weights of the weighted sum of squared differences (wSSD)
family of registration cost functions and/or estimate an optimal
macroanatomical template for localizing the cytoarchitectural
and functional regions using only the cortical folding pattern.
We demonstrate improvement over existing methods .
B. Related Work
An alternative approach to overcome the imperfect corre-
lation between anatomy and function is to directly use the
functional data for establishing across-subject functional cor-
respondence , . However, these approaches require
extra data acquisition (such as fMRI scans) of all future test
subjects. In contrast, our method aims to learn the relationship
between macro-anatomy and function (or cytoarchitectonics)
in a training data set containing information about both
macro-anatomy and function (or cytoarchitectonics). We use
this information to localize function (or cytoarchitectonics) in
future subjects, for which only macro-anatomical information
Our approach belongs to the class of “wrapper methods”
for model or feature selection in the machine learning litera-
ture , . In particular, our model selection criterion of
application-speciﬁc performance is equivalent to the use of
cross-validation error in various model selection algorithms
, , . Unlike feature selection methods that operate
in a discrete parameter space, we work in a continuous param-
eter space. Consequently, standard algorithms in the “wrapper
methods” literature do not apply to this problem.
Instead, our resulting optimization procedure borrows
heavily from the mathematical ﬁeld of continuation methods
. Continuation methods have been recently introduced to
the machine learning community for computing the entire path
of solutions of learning problems (e.g., SVM or Lasso) as a
function of a single regularization parameter , , .
For example, the cost function in Lasso  consists of the
tradeoff between a least-squares term and a regularization
term. Least-angles regression (LARS) allows one to compute
the entire set of solutions of Lasso as a function of the tradeoff
parameter . Because we deal with multiple (thousands of)
parameters, it is impossible for us to compute the entire solution
manifold. Instead, we trace a path within the solution manifold
that improves the task-speciﬁc cost function. Furthermore,
registration is not convex (unlike SVM and Lasso), resulting in
several theoretical and practical conundrums that we have to
overcome, some of which we leave for future work.
The wSSD similarity measure implicitly assumes an indepen-
dent Gaussian distribution on the image intensities, where the
weights correspond to the precision (reciprocal of the variance)
and the template corresponds to the mean of the Gaussian distri-
bution. The weights can be set to a constant value ,  or a
spatially-varying variance can be estimated from the intensities
of registered images . However, depending on the wSSD
regularization tradeoff, the choice of the scale of the variance
is still arbitrary . With weaker regularization, the training
images will be better aligned, resulting in lower variance esti-
Recent work in probabilistic template construction resolves
this problem by either marginalizing the tradeoff under a
Bayesian framework  or by estimating the tradeoff with
the minimum description length principle . While these
methods are optimal for “explaining the images” under the
assumed generative models, it is unclear whether the estimated
parameters are optimal for application-speciﬁc tasks. After
all, the parameters for optimal image segmentation might be
different from those for optimal group analysis. In contrast, Van
Leemput  proposes a generative model for image segmenta-
tion. The estimated parameters are therefore Bayesian-optimal
for segmentation. When considering one global tradeoff pa-
rameter, a more direct approach is to employ cross-validation
of segmentation accuracy and to perform an exhaustive search
over the values of the tradeoff parameter . This is infeasible
for multiple parameters.
By learning the weights of the wSSD, we implicitly optimize
the tradeoff betweeen the dissimilarity measure and regulariza-
tion. Furthermore, the tradeoff we learn is spatially varying. Pre-
vious work on learning a spatially varying regularization prior
suffers from the lack of ground truth (nonlinear) deformations.
Authorized licensed use limited to: MIT Libraries. Downloaded on July 22,2010 at 11:10:18 UTC from IEEE Xplore. Restrictions apply.
YEO et al.: LEARNING TASK-OPTIMAL REGISTRATION COST FUNCTIONS FOR LOCALIZING CYTOARCHITECTURE 1427
For example, , ,  assume that the deformations ob-
tained from registering a set of training images can be used to es-
timate a registration regularization to register new images. How-
ever, a change in the parameters of the registration cost function
used by these methods to register the training images would lead
to a different set of training deformations and thus a different
prior for registering new images. Furthermore, the methods are
inconsistent in the sense that the learned prior applied on the
training images will not result in the same training deformations
While there has been efforts in obtaining ground truth
human-annotated deformation ﬁelds , the images consid-
ered typically have well-deﬁned correspondences, rather than
for example, the brain images of two different subjects. As
suggested in the previously presented examples (Fig. 2), the
concept of “ground truth deformations” may not always be
well-deﬁned, since the optimal registration may be a function
of the application at hand. In contrast, image segmentation is
generally better deﬁned in the sense that ground truth segmen-
tation is usually known. Our problem therefore differs from
recent work on learning segmentation cost functions , ,
. In this paper, we avoid the need for ground truth defor-
mations by focusing on the application of registration-based
segmentation, where ground truth segmentations are better
deﬁned and available. However, our framework is general and
can be applied whenever a postregistration application can be
well quantiﬁed by a smooth application-speciﬁc performance
This paper is organized as follows. In the next section, we in-
troduce the task-optimal registration framework. We specialize
the framework to align hidden labels in Section III. We present
localization experiments in Section IV and discuss outstanding
issues in Section V. This paper extends a previously presented
conference article  and contains detailed derivations, dis-
cussions and experiments that were omitted in the conference
1) We present a framework for learning the parameters of
registration cost functions with respect to speciﬁc appli-
cations. We present an algorithm sufﬁciently efﬁcient for
optimizing thousands of parameters.
2) We specialize the framework for the alignment of hidden
labels, which are not necessarily well-predicted by local
3) We apply the framework to localizing cytoarchitectural and
functional regions using only the cortical folding pattern
and demonstrate improvements over existing localization
II. TASK-OPTIMAL FRAMEWORK
In this section, we present the task-optimal registration frame-
work for learning the parameters of a registration cost function.
Given an image , let denote a smooth registration cost
function, with parameters and a spatial transformation .For
where is the template image, is the tradeoff between the
image dissimilarity measure and the regularization on the trans-
formation , denotes the deformed and resampled image .
is therefore also a function of the image , which we suppress
for conciseness. The optimal transformation minimizes the
cost function for a given set of parameters
We emphasize that is a function of since a different set of
parameters will result in a different solution to (2.2) and thus
will effectively deﬁne a different image coordinate system.
The resulting deformation is used to warp the input image
or is itself used for further tasks, such as image segmentation
or voxel-based morphometry. We assume that the task perfor-
mance can be measured by a smooth cost function (or cross-val-
idation error metric) , so that a smaller value of cor-
responds to better task performance. is typically a function
of additional input data associated with a subject (e.g., manual
segmentation labels if the task is automatic segmentation), al-
though we suppress this dependency in the notation for concise-
ness. This auxiliary data is only available in the training set;
cannot be evaluated for the new image.
Given a set of training subjects, let denote the so-
lution of (2.2) for training subject for a ﬁxed set of parame-
ters and denote the task performance for training
subject using the deformation and other information
available for the th training subject. A different set of param-
eters would lead to different task performance .
We seek the parameters that generalize well to a new sub-
ject: registration of a new subject with yields the transfor-
mation with a small task-speciﬁc cost . One
approach to solve this functional approximation problem 
is regularized risk minimization. Let denote regulariza-
tion on and deﬁne
Regularization risk minimization seeks
The optimization is difﬁcult because while we assume to be
smooth, the input to is itself the local minimum of an-
other nonlinear cost function . Furthermore, evaluating the cost
function for only one particular set of parameters requires
performing different registrations!
A. Characterizing the Space of Local Minima
In this section, we provide theoretical characterizations of the
optimization problem in (2.4). If is deﬁned strictly to
be a global registration optimum, then is clearly not a
smooth function of , since a small change in can result in a
big change in the global registration optimum. This deﬁnition is
also impractical, since the global optimum of a nonlinear opti-
mization problem cannot be generally found in practice. Instead,
1428 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 7, JULY 2010
we relax the deﬁnition of to be a local minimum of the
registration cost function for ﬁxed values of . Here, we derive
conditions in which is locally a smooth function of ,so
we can employ gradient descent to optimize (2.4).
Let denote a local minimum of the registration cost
function for a ﬁxed . Suppose we perturb by an in-
ﬁnitestimally small , so that is no longer the registra-
tion local minimum for . We consider two repre-
sentations of this change in local minimum.
Additive deformation models arise when the space of defor-
mations is a vector space, such as the space of displacement
ﬁelds or positions of B-spline control points. At each iteration
of the registration algorithm, deformation updates are added to
the current deformation estimates. The additive model is general
and applies to many non-convex, smooth optimization problems
outside of registration. Most registration algorithms can in fact
be modeled with the additive framework.
In some registration algorithms, including that used in this
paper, it is more natural to represent deformation changes
through composition rather than additions , , . For
example, in the diffeomorphic variants of the demons algo-
rithm , , , the diffeomorphic transformation is
represented as a dense displacement ﬁeld. At each iteration,
the transformation update is restricted to be a one parameter
subgroup of diffeomorphism parameterized by a stationary
velocity ﬁeld. The diffeomorphic transformation update is then
composed with, rather than added to, the current estimate of the
transformation, thus ensuring that the resulting transformation
1) Addition Model: Let
denote the new locally optimal deformation for
the updated set of parameters . The following proposi-
tion characterizes the existence and uniqueness of
as is varied. In particular, we show that under some mild
conditions, is a well-deﬁned smooth function in
the neighborhood of . In the remainder, we use
, , and to denote the corresponding partial derivatives.
Proposition 1: If the Hessian1is positive deﬁ-
nite at , then there exists an , such that for all
, a unique continuous function ex-
ists with . Furthermore, has the same order
of smoothness as .
Proof: We deﬁne the vector-valued function
. Since is a local minimum of ,we
At , the Hessian matrix
is positive deﬁnite by the assumption of the proposition and
is therefore invertible. By the Implicit Function Theorem ,
there exists an , such that for all , there is
a unique continuous function , such that
and . Further-
more, has the same order of smoothness as .
1Here, we assume that the transformation is ﬁnite dimensional, such as the
parameters of afﬁne transformations, positions of spline control points or dense
displacement ﬁelds deﬁned on the voxels or vertices of the image domain.
Because the Hessian of is smooth and the eigenvalues of
a matrix depend continuously on the matrix , there exists a
small neighborhood around in which the eigen-
values of are all greater than 0. Since both sufﬁcient
conditions for a local minimum are satisﬁed (zero gradient and
positive deﬁnite Hessian), is indeed a
new local minimum close to .
Observe that the conditions in Proposition 1 are stronger than
those of typical nonlinear optimization problems. In particular,
we do not just require the cost functions and to be smooth,
but also that the Hessian be positive deﬁnite at the
local minimum. At , by deﬁnition, the Hessian
is positive semi-deﬁnite, so the positive deﬁnite
condition in Proposition 1 should not be too restrictive. Unfor-
tunately, degeneracies may arise for local minima with a sin-
gular Hessian. For example, let be the 1 2 vector and
. Then for any value of , there is an in-
ﬁnite number of local minima corresponding to .
Furthermore, the Hessian at any local minimum is singular. In
this case, even though is inﬁnitely differentiable, there is an
inﬁnite number of local minima near the current local minimum
, i.e., is not a well-deﬁned function and
the gradient is not deﬁned. Consequently, the parameters of
local registration minima whose Hessians are singular are also
local minima of the task-optimal optimization (2.4). The proof
of Proposition 1 follows the ideas of the Continuation Methods
literature . We include the proof here to motivate the more
complex composition of deformations model.
2) Composition Model: Let be the registration local
minimum at and denote an update transformation
parameterized by , so that corresponds to the identity
transform. For example, could be a stationary , ,
, nonstationary  velocity ﬁeld parameterization of dif-
feomorphism, positions of spline control points  or simply
displacement ﬁelds . In the composition model,
is a local minimum if and only if there exists an , such
that for all values of
Let denote the new locally optimal
deformation for the new parameters . In general, there
might not exist a single update transformation
that leads to a new local minimum under a perturbation of the
parameters , so that there is no correponding version of Propo-
sition 1 for the general composition model. However, in the spe-
cial case of the composition of diffeomorphisms model ,
,  employed in this paper, the following proposition
characterizes the existence and uniqueness of as
Proposition 2: If the Hessian is
positive deﬁnite at , then there exists an , such that
for all , a unique continuous function
exists, such that is the new local minimum for pa-
rameters and . Furthermore,
has the same order of smoothness as .
Proof: The proof is a more complicated version of Propo-
sition 1 and so we leave the details to Appendix A.
Just like in the case of the additive deformation model, the
parameters of local registration minima that do not satisfy the
YEO et al.: LEARNING TASK-OPTIMAL REGISTRATION COST FUNCTIONS FOR LOCALIZING CYTOARCHITECTURE 1429
conditions of Proposition 2 are also local minima of the task-
optimal optimization (2.4). In the next section, we derive exact
and approximate gradients of .
B. Optimizing Registration Parameters
We now discuss the optimization of the regularized task per-
1) Addition Model: In the previous section, we showed that
at with a positive deﬁnite Hessian, is
a smooth well-deﬁned function such that
is the new local minimum at . Therefore, we can com-
pute the derivatives of with respect to at , allowing
us to traverse a curve of local optima, ﬁnding values of
that improve the task-speciﬁc cost function for the training
images. We ﬁrst perform a Taylor expansion of at
where we dropped the term .For
, the left-hand side is equal to 0 and we can write
Therefore, by taking the limit , we get
Equation (2.8) tells us the direction of change of the local min-
imum at . In practice, the matrix inversion in (2.8)
is computationally prohibitive for high-dimensional warps .
Here, we consider a simpliﬁcation of (2.8) by setting the Hes-
sian to be the identity
Since is the direction of gradient descent of the cost func-
tion (2.2), we can interpret (2.9) as approximating the new local
minimum to be in the same direction as the change in the direc-
tion of gradient descent as is perturbed.
Differentiating the cost function in (2.4), using the chain rule,
We note the subscript on indicates the dependency of the
registration cost function on the th training image.
2) Composition Model: In the previous section, we have
shown that at , assuming the conditions of Propo-
sition 2 are true, is a smooth well-deﬁned function
such that is the new local minimum.
Therefore, we can compute the derivatives of with respect to
. As before, by performing a Taylor expansion, we obtain
Appendix B provides the detailed derivations. Differentiating
the cost function in (2.4), using the chain rule, we get
Once again, the subscript on indicates the dependency of
the registration cost function on the th training image.
Algorithm 1 summarizes the method for learning the task-op-
timal registration parameters. Each line search involves eval-
uating the cost function multiple times, which in turn re-
quires registering the training subjects, resulting in a compu-
tationally intensive process. However, since we are initializing
from a local optimum, for a small change in , each registration
Algorithm 1. Task-Optimal Registration
Data: A set of training images
Result: Parameters that minimize the regularized task
performance [see (2.4)]
Step 1. Given current values of , estimate
, i.e., perform
registration of each training subject .
Step 2. Given current estimates ), compute the
gradient using either
1) Eq. (2.12) via in (2.9) for the addition
2) Eq. (2.16) via in (2.14) for the
Step 3. Perform line search in the direction opposite to
1430 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 7, JULY 2010
Fig. 3. Illustration of the differences between our approach and the pairwise registration approach. In our approach, we use training images and labels to learn an
optimal cost function that is optimal for aligning the labels of the training and template subjects. This cost function is then used to register and predict the hidden
label in a new subject. (a) Pairwise registration without training using ground truth labels. (b) Task-optimal registration framework.
Since nonlinear registration is dependent on initialization, the
current estimates , which were initialized from pre-
vious estimates, might not be achievable when initializing the
registration with the identity transform. The corresponding pa-
rameters might therefore not generalize well to a new sub-
ject, which are typically initialized with the identity transform.
To put this more concretely, suppose our current estimates of
and the registration local minima are ( , ). Next,
we perform the gradient decent step and update w accordingly.
For argument’s sake, let our new estimates of and the registra-
tion local minima be ( , ). Note that this
particular value of is achieved by initializing the
registration with . Had we initialized the registration with
the identity transform (such as for a new subject), then
might instead be equal to 2.1, with possibly poorer application
performance than ( , ). To avoid this form of
overﬁtting, after every few iterations, we reregister the training
images by initializing with the identity transform, and verify that
the value of is better than the current best value of com-
puted with initialization from the identity transform.
The astute reader will observe that the preceding discussion
on “Addition Model” makes no assumptions speciﬁc to the task-
optimal registration problem. The framework can therefore also
be applied to learn the cost functions in other applications that
are formulated as nonlinear optimization problems solved by
III. LEARNING WSSD FOR HIDDEN LABEL ALIGNMENT
We now instantiate the task-optimal registration framework
for localizing hidden labels in images. We demonstrate schemes
for either 1) learning the weights of the wSSD family of registra-
tion cost functions or 2) estimating an optimal template image
for localizing these hidden labels. We emphasize that the op-
timal template is not necessarily the average of the training im-
ages, since the goal is not to align image intensities across sub-
jects, but to localize the hidden labels.
Suppose we have a set of training images with some
underlying ground truth structure manually labeled or obtained
from another imaging modality (e.g., Brodmann areas from his-
tology mapped onto cortical surface representations). We de-
ﬁne our task as localizing the hidden structure in a test image.
In the traditional pairwise registration approach [Fig. 3(a)], a
single training subject is chosen as the template. After pairwise
registration between the template and test images, the ground
truth label of the template subject is used to predict that of
the test subject. The goal of predicting the hidden structure in
the test subject is typically not considered when choosing the
training subject or registration algorithm. For hidden labels that
are poorly predicted by local image intensity (e.g., BA44 dis-
cussed in Section I-A), blind alignment of image intensities lead
to poor localization.
In contrast, we pick one training subject as the initial template
and use the remaining training images and labels [Fig. 3(b)] to
learn a registration cost function that is optimal for aligning the
labels of the training and template subjects—perfect alignment
of the labels lead to perfect prediction of the labels in the training
subjects by the template labels. After pairwise registration be-
tween the template and test subject using the optimal registra-
tion cost function, the ground truth label of the template subject
is used to predict that of the test subject.
We limit ourselves to spherical images (i.e., images deﬁned
on a unit sphere), although it should be clear that the discus-
sion readily extends to volumetric images. Our motivation for
using spherical images comes from the representation of the
human cerebral cortex as a closed 2-D mesh in 3-D. There has
been much effort focused on registering cortical surfaces in 3-D
, , , , . Since cortical areas—both structure
and function—are arranged in a mosaic across the cortical sur-
face, an alternative approach is to warp the underlying spher-
ical coordinate system , , , , , , ,
. Warping the spherical coordinate system establishes cor-
respondences across the surfaces without actually deforming the
surfaces in 3-D. We assume that the meshes have already been
spherically parameterized and represented as spherical images:
a geometric attribute is associated with each mesh vertex, de-
scribing local cortical geometry.
A. Instantiating Registration Cost Function
To register a given image to the template image ,we
deﬁne the following cost function:
YEO et al.: LEARNING TASK-OPTIMAL REGISTRATION COST FUNCTIONS FOR LOCALIZING CYTOARCHITECTURE 1431
where transformation maps a point on the sphere to
another point . The ﬁrst term corresponds to the
wSSD image similiarity. The second term is a percentage metric
distortion regularization on the transformation where is
a predeﬁned neighborhood around vertex and is the orig-
inal distance between the neighbors .
The weights ’s are generalizations of the tradeoff param-
eter , allowing for a spatially-varying tradeoff between the
image dissimilarity term and regularization: a higher weight
corresponds to placing more emphasis on matching the tem-
plate image at spatial location relative to the regularization.
The parameterization of the weights as ensures nonnegative
In this work, we consider either learning the weights or
the template for localizing BA labels or functional labels
by aligning cortical folding pattern. Since the weights of the
wSSD correspond to the precision of the Gaussian model, by
learning the weights of wSSD, we are learning the precision of
the Gaussian model and hence the uncertainty of the sulcal ge-
ometry. Optimizing leads to placing nonuniform importance
on matching different cortical folds with the aim of aligning
the underlying cytoarchitectonics or function. For example, sup-
pose there is a sulcus with functional regions that appear on
either side of the sulcus depending on the subject. The algo-
rithm may decide to place low weight on the “poorly predictive”
sulcus. On the other hand, optimizing corresponds to learning
a cortical folding template that is optimal for localizing the un-
derlying cytoarchitectonics or functional labels of the training
subjects. In the case of the previously mentioned “unpredic-
tive sulcus,” the algorithm might learn that the optimal cortical
folding template should not contain this sulcus.
We choose to represent the transformation as a composi-
tion of diffeomorphic warps parameterized by a stationary
velocity ﬁeld, so that , , .
We note that our choice of regularization is different from the
implicit hierarchical regularization used in Spherical Demons
 since the Demons regularization is not compatible with our
derivations from the previous section. Instead of the efﬁcient
2-Step Spherical Demons algorithm, we will use steepest de-
scent. The resulting registration algorithm is still relatively fast,
requiring about 15 min for registering full-resolution meshes
with more than 100k vertices, compared with 5 min of computa-
tion for Spherical Demons on a Xeon 2.8-GHz single processor
In general, a smooth stationary velocity ﬁeld parameter-
izes a diffeomorphism via a stationary ODE:
with an initial condition . The solution
at is denoted as , where we
have dropped the time index. A solution can be computed efﬁ-
ciently using scaling and squaring . This particular choice of
representing deformations provides a computationally efﬁcient
method of achieving invertible transformations, which is a de-
sirable property in many medical imaging applications. In our
case, the velocity ﬁeld is a tangent vector ﬁeld on the sphere
and not an arbitrary 3-D vector ﬁeld.
B. Optimizing Registration Cost Function
To register subject to the template image for a ﬁxed set
of parameters , let be the current estimate of . We seek
an update transformation parameterized by a stationary
Let be the velocity vector tangent to vertex , and
be the entire velocity ﬁeld. We adopt the techniques in the
Spherical Demons algorithm  to differentiate (3.1) with re-
spect to , evaluated at . Using the fact that the differential
of at is the identity , i.e., ,
we conclude that a change in velocity at vertex does not
affect for up to the ﬁrst order derivatives.
Deﬁning to be the 1 3 spatial gradient of the
warped image at and to be the 3 3 Ja-
cobian matrix of at , we get the 1 3 derivative
We can perform gradient descent of the registration cost func-
tion using (3.2) to obtain , which can be used to eval-
uate the regularized task performance to be described in the
next section. We also note that (3.2) instantiates within
the mixed derivatives term in the task-optimal gradient (2.16)
for this application.
C. Instantiating Regularized Task Performance
We represent the hidden labels in the training subjects as
signed distance transforms on the sphere . We con-
sider a pairwise approach, where we assume that the template
image has a corresponding labels with distance transform
and set the task-speciﬁc cost function to be
A low value of indicates good alignment of the hidden label
maps between the template and subject , suggesting good pre-
diction of the hidden label.
We experimented with a prior that encourages spatially con-
stant weights and template, but did not ﬁnd that the regulariza-
tion lead to improvements in the localization results. In partic-
ular, we considered the following smoothness regularization on
the registration parameters depending on whether we are opti-
mizing for the weights or the template :
1432 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 7, JULY 2010
A possible reason for this lack of improvement is that the rereg-
istration after every few line searches already helps to regularize
against bad parameter values. Another possible reason is that the
above regularization assumes a smooth variation in the relation-
ship between structure and function, which may not be true in
reality. Unfortunately, the relationship between macro-anatom-
ical structure and function is poorly understood, making it difﬁ-
cult to design a more useful regularization that could potentially
improve the results. In the experiments that follow, we will dis-
card the regularization term of the registration parameters (i.e.,
set ). We also note that is typically set to
0 in machine learning approaches of model selection by opti-
mization of cross-validation error , , .
D. Optimizing Task Performance
Tooptimize the task performance over the set of parameters
, we have to instantiate the task-optimal gradient speciﬁed in
(2.16). We ﬁrst compute the derivative of the task-speciﬁc cost
function with respect to the optimal update . Once again, we
represent as the collection , where is a velocity vector
at vertex . Deﬁning to be the 1 3 spatial
gradient of the warped distance transform of the th subject
at , we get the 1 3 derivative
Weight Update: To update the weights of the wSSD,
we compute the derivative of the registration local minimum
update with respect to the weights. Using the approximation
in (2.14), we obtain the 3 1 derivative of the velocity update
with respect to the weights of the wSSD cost function
Here if and is zero otherwise. Since (3.10)
is in the same direction as the ﬁrst term of the gradient descent
direction of registration [negative of (3.2)], increasing will
improve the intensity matching of vertex of the template.
Substituting (3.10) and (3.6) into (2.16) provides the gradient
for updating the weights of the wSSD cost function.
Template Update: To update the template image used for
registration, we compute the 3 1 derivative of the velocity
update with respect to the template
Since (3.14) is in the same direction as the ﬁrst term of the gra-
dient descent direction of registration [negative of (3.2)], when
is larger than , increasing the value of
will warp vertex of the template further along the direction of
increasing intensity in the subject image. Conversely, if
is smaller than , decreasing the value of will
warp vertex of the template further along the direction of de-
creasing intensity in the subject image. Substituting (3.14) and
(3.6) into (2.16) provides the gradient for updating the template
used for registration. Note that the template subject’s hidden la-
bels are considered ﬁxed in template space and are not modiﬁed
We can in principle optimize both the weights and the
template . However, in practice, we ﬁnd that this does not
lead to better localization, possibly because of too many de-
grees-of-freedom, suggesting the need to design better regular-
ization of the parameters. A second reason might come from
the fact that we are only using an approximate gradient rather
than the true gradient for gradient descent. Previous work 
has shown that while using an approximate gradient can lead to
reasonable solutions, using the exact gradient can lead to sub-
stantially better local minima. Computing the exact gradient is
a challenge in our framework. We leave exploration of efﬁcient
means of computing better approximations of the gradient to fu-
We now present experiments on localizing BAs and fMRI-de-
ﬁned MT+ (V5) using macro-anatomical cortical folding in two
different data sets. For both experiments, we compare the frame-
work with using uniform weights  and FreeSurfer .
A. BA Localization
We consider the problem of localizing BAs in the surface
representations of the cortex using only cortical folding pat-
terns. In this study, ten human brains were analyzed histolog-
ically postmortem using the techniques described in  and
. The histological sections were aligned to postmortem MR
with nonlinear warps to build a 3-D histological volume. These
volumes were segmented to separate white matter from other
tissue classes, and the segmentation was used to generate topo-
logically correct and geometrically accurate surface represen-
tations of the cerebral cortex using a freely available suite of
tools . Six manually labeled BA maps (V1, V2, BA2, BA44,
BA45, MT) were sampled onto the surface representations of
each hemisphere, and errors in this sampling were manually
corrected (e.g., when a label was erroneously assigned to both
banks of a sulcus). A morphological close was then performed
on each label to remove small holes. Finally, the left and right
hemispheres of each subject were mapped onto a spherical coor-
dinate system . The BAs on the resulting cortical represen-
tations for two subjects are shown in Fig. 2(b). We do not con-
sider BA4a, BA4p, and BA6 in this paper because they were not
histologically mapped by the experts in two of the ten subjects
in this particular data set (even though they exist in all human
As illustrated in Fig. 2(c) and discussed in multiple studies
, , , we note that V1, V2, and BA2 are well-predicted
YEO et al.: LEARNING TASK-OPTIMAL REGISTRATION COST FUNCTIONS FOR LOCALIZING CYTOARCHITECTURE 1433
by local cortical geometry, while BA44, BA45, and MT are not.
For all the BAs however, a spherical morph of cortical folding
was shown to improve their localization compared with only
Talairach or nonlinear spatial normalization in the Euclidean
3-D space . Even though each subject has multiple BAs, we
focus on each structure independently. This allows for an easier
interpretation of the estimated parameters, such as the optimal
template example we provide in Section IV-A3. A clear future
direction is to learn a registration cost function that is jointly
optimal for localizing multiple cytoarchitectural or functional
We compare the following algorithms.
1) Task-Optimal. We perform leave-two-out
cross-validation to predict BA location. For each test
subject, we use one of the remaining nine subjects as
the template subject and the remaining eight subjects
for training. When learning the weights of the wSSD,
the weights are globally initialized to 1 and
the template image is ﬁxed to the geometry of the
template subject. When learning the cortical folding
template , the template image is initialized to that of
the template subject and the weights are globally
set to 1.
Once the weights or template are learned, we use them
to register the test subject and predict the BA of the
test subject by transferring the BA label from the
template to the subject. We compute the symmetric
mean Hausdorff distance between the boundary of
the true BA and the predicted BA on the cortical
surface of the test subject—smaller Hausdorff distance
corresponds to better localization . The symmetric
mean Hausdorff distance between two curves is deﬁned
as follows. For each boundary point of the ﬁrst curve,
the shortest distance to the second curve is computed
and averaged. We repeat by computing and averaging
the shortest distance from each point of the second
curve to the ﬁrst curve. The symmetric mean Hausdorff
distance is obtained by averaging the two values. We
consider all 90 possibilities of selecting the test subject
and template, resulting in a total of 90 trials and 90
mean Hausdorff distances for each BA and for each
2) Uniform-Weights. We repeat the process for the
uniform-weight method that ﬁxes the template
to the geometry of the template subject, and sets all
the weights to a global ﬁxed value without
training. We explore 14 different values of global
weight , chosen such that the deformations range
from rigid to ﬂexible warps. For each BA and each
hemisphere, we pick the best value of leading
to the lowest mean Hausdorff distances. Because
there is no cross-validation in selecting the weights,
the uniform-weight method is using an unrealistic
oracle-based version of the strategy proposed in .
3) FreeSurfer. Finally, we use FreeSurfer  to register
the 10 ex vivo subjects to the FreeSurfer Buckner40
atlas, constructed from the MRI of 40 in vivo subjects
Fig. 4. FreeSurfer’s atlas-based registration approach. Training and test sub-
jects are registered to an atlas. The BA of a training subject can then be used to
predict that of the test subject.
. Once registered into this in vivo atlas space, for the
same 90 pairs of subjects, we can use the BAs of one ex
vivo subject to predict another ex vivo subject. We note
that FreeSurfer also uses the wSSD cost function, but
a more sophisticated regularization that penalizes both
metric and areal distortion. For a particular tradeoff
between the similarity measure and regularization, the
Buckner40 template consists of the empirical mean
and variance of the 40 in vivo subjects registered
to template space. We use the reported FreeSurfer
tradeoff parameters that were used to produce prior
state-of-the-art BA alignment .
We note that both the task-optimal and uniform-weights
methods use a pairwise registration framework, while
FreeSurfer uses an atlas-based registration framework. Under
the atlas-based framework, all the ex vivo subjects are regis-
tered to an atlas (Fig. 4). To use the BA of a training subject to
predict a test subject, we have to compose the deformations of
the training subject to the atlas with the inverse deformation of
the test subject to the atlas. Despite this additional source of
error from composing two warps, it has been shown that with
carefully constructed atlases, using the atlas-based strategy
leads to better registration because of the removal of template
bias in the pairwise registration framework , , , ,
, , .
We run the task-optimal and uniform-weights methods on a
low-resolution subdivided icosahedron mesh containing 2562
vertices, whereas FreeSurfer results were computed on high-res-
olution meshes of more than 100k vertices. In our implementa-
tion, training on eight subjects takes on average 4 h on a standard
PC (AMD Opteron, 2GHz, 4GB RAM). Despite the use of the
low-resolution mesh, we achieve state-of-the-art localization ac-
curacy. We also emphasize that while training is computation-
ally intensive, registration of a new subject only requires one
minute of processing time since we are working with low-reso-
1) Quantitative Results: Fig. 5 displays the mean and stan-
dard errors from the 90 trials of leave-two-out. On average, task-
optimal template performs the best, followed by task-optimal
weights. Permutation tests show that task-optimal template out-
performs FreeSurfer in ﬁve of the six areas, while task-optimal
1434 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 7, JULY 2010
Fig. 5. Mean Hausdorff distances over an entire range of harmonic energy for BA44, BA45, and MT. First row corresponds to left hemisphere. Second row
corresponds to right hemipshere. indicates that task-optimal template is statistically signiﬁcantly better than FreeSurfer. indicates that task-optimal weights is
statistically signiﬁcantly better than FreeSurfer. Statistical threshold is set at 0.05, FDR corrected with respect to the 24 statistical tests performed in this section.
FreeSurfer is not statistically better than either of the task-optimal methods in any of the Brodmann areas. (a) Left BA44 . (b) Left BA45 . (c) Left MT.
(d) Right BA44 . (e) Right BA45 . (f) Right MT .
weights outperforms FreeSurfer in four of the six areas after cor-
rections for multiple comparisons (see Fig. 5 for more details).
For the Broca’s areas (BA44 and BA45) and MT, this is not
surprising. Since local geometry poorly predicts these regions,
by taking into account the ﬁnal goal of aligning BAs instead of
blindly aligning the cortical folds, our method achieves better
BA localization. FreeSurfer and the uniform-weights method
have similar performance because a better alignment of the cor-
tical folds on a ﬁner resolution mesh does not necessary improve
the alignment of these areas.
Since local cortical geometry is predictive of V1, V2, and
BA2, we expect the advantages of our framework to vanish.
Surprisingly, as shown in Fig. 6, task-optimal template again
achieve signiﬁcant improvement in BAs alignment over the uni-
form-weights method and FreeSurfer. Task-optimal weights is
also signiﬁcantly better than the uniform-weights method, but
only slightly better than FreeSurfer. Permutation tests show that
task-optimal template outperforms FreeSurfer in ﬁve of the six
areas, while task-optimal weights is outperforms FreeSurfer in
three of the six areas after corrections for multiple compar-
isons (see Fig. 6 for more details). This suggests that even when
local geometry is predictive of the hidden labels and anatomy-
based registration achieves reasonable localization of the labels,
tuning the registration cost function can further improve the task
performance. We also note that in this case, FreeSurfer performs
better than the uniform-weights method on average. Since local
cortical folds are predictive of these areas, aligning cortical folds
on a higher resolution mesh yields more precise alignment of the
cortical geometry and of the BAs.
We note that the FreeSurfer Buckner40 atlas utilizes 40
in vivo subjects consisting of 21 males and 19 females of a
wide-range of age. Of these, 30 are healthy subjects whose ages
range from 19 to 87. 10 of the subjects are Alzheimer’s patients
with age ranging from 71 to 86. The average age of the group
is 56 (see  for more details). The T1-weighted scans were
acquired on a 1.5T Vision system (Siemens, Erlangen Ger-
many), with the following protocol: two sagittal acquisitions,
FOV , matrix , resolution mm,
TR ms, TE ms, Flip angle ,TI ms
and TD ms. Two acquisitions were averaged together
to increase the contrast-to-noise ratio. The histological data set
includes ﬁve male and ﬁve female subjects, with age ranging
from 37 to 85 years old. The subjects had no previous his-
tory of neurologic or psychiatric diseases (see  for more
details). The T1-weighted scans of the subjects were obtained
on a 1.5T system (Siemens, Erlangen, Germany) with the
following protocol: ﬂip angle 40 ,TR ms, TE ms and
resolution mm. While there are demographic
and scanning differences between the in vivo and ex vivo data
sets, the performance differences between FreeSurfer and the
task-optimal framework cannot be solely attributed to this
difference. In particular, we have shown in previous work that
FreeSurfer’s results are worse when we use an ex vivo atlas
for registering ex vivo subjects (see [81,Table III]). Further-
more, FreeSurfer’s results are comparable with that of the
uniform-weights baseline algorithm, as well as previously pub-
lished results , where we have checked for gross anatomical
misregistration. We emphasize that since the goal is to optimize
Brodmann area localization, the learning algorithm might take
into account the idiosyncrasies of the registration algorithm
in addition to the relationship between macro-anatomy and
cytoarchitecture. Consequently, it is possible that the perfor-
mance differences are partly a result of our algorithm learning
a registration cost function with better local minima, thus
avoiding possible misregistration of anatomy.
2) Qualitative Results: Fig. 7 illustrates representative lo-
calization of the BAs for FreeSurfer and task-optimal template.
We note that the task-optimal boundaries (red) tend to be in
better visual agreement with the ground truth (yellow) bound-
aries, such as the right hemisphere BA44 and BA45.
YEO et al.: LEARNING TASK-OPTIMAL REGISTRATION COST FUNCTIONS FOR LOCALIZING CYTOARCHITECTURE 1435
Fig. 6. Mean Hausdorff distances over an entire range of harmonic energy for V1, V2, and BA2. First row corresponds to left hemisphere. Second row corresponds
to right hemisphere. indicates that task-optimal template is statistically signiﬁcantly better than FreeSurfer. indicates that task-optimal weights is statistically
signiﬁcantly better than FreeSurfer. Statistical threshold is set at 0.05, FDR corrected with respect to the 24 statistical tests performed in this section. FreeSurfer
is not statistically better than either of the task-optimal methods in any of the Brodmann areas. (a) Left V1 . (b) Left V2 . (c) Left BA2 . (d) Right V1 .
(e) Right V2 . (f) Right BA2.
Fig. 7. Representative BA localization in 90 trials of leave-two-out for FreeSurfer and task-optimal template. Yellow indicates ground truth boundary. Green
indicates FreeSurfer prediction. Red indicates Task-Optimal prediction. The representative samples were selected by ﬁnding subjects whose localization errors are
close to the mean localization errors for each BA. Furthermore, for a given BA, the same subject was selected for both methods to simplify the comparison.
3) Interpreting the Template: Fig. 8 illustrates an example of
learning a task-optimal template for localizing BA2. Fig. 8(a)
shows the cortical geometry of a test subject together with its
BA2. In this subject, the central sulcus is more prominent than
the postcentral sulcus. Fig. 8(b) shows the initial cortical ge-
ometry of a template subject with its corresponding BA2 in
black outline. In this particular subject, the postcentral sulcus
is more prominent than the central sulcus. Consequently, in the
uniform-weights method, the central sulcus of the test subject is
incorrectly mapped to the postcentral sulcus of the template, so
1436 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 7, JULY 2010
Fig. 8. Template estimation in the task-optimal framework improves localization of BA2. (a) Cortical geometry of test subject with corresponding BA2 (in green).
(b) Initial cortical geometry of template subject with corresponding BA2 (in black). In (b), we also show the BA2 of the test subject (in green) after registration to
the intial template. (c) Final cortical geometry of template subject after task-optimal training. BA2 of the test subject (in green) after registration to the task-optimal
template demonstrates signiﬁcantly better alignment with the BA2 of the template subject.
that BA2 is misregistered. Fig. 8(b) also shows the BA2 of the
test subject (green) overlaid on the cortical geometry of the tem-
plate subject after registration to the initial template geometry.
During task-optimal training, our method interrupts the geom-
etry of the postcentral sulcus in the template because the unin-
terrupted postcentral sulcus in the template is inconsistent with
localizing BA2 in the training subjects. The ﬁnal template is
shown in Fig. 8(c). We see that the BA2 of the subject (green)
and the task-optimal template (black) are well-aligned, although
there still exists localization error in the superior end of BA2.
In the next section, we turn our attention to a fMRI data set.
Since the task-optimal template performed better than the task-
optimal weights, we will focus on the comparison between the
task-optimal template and FreeSurfer.
B. fMRI-MT+ Localization
We now consider the application of localizing fMRI-de-
ﬁned functional areas in the cortex using only cortical folding
patterns. Here, we focus on the so-called MT+ area localized
in 42 in vivo subjects using fMRI. The MT+ area deﬁned
functionally is thought to include primarily the cytoarchitec-
tonically-deﬁned MT and a small part of the medial superior
temporal (MST) area (hence the name MT+). The imaging
paradigm involved subjects viewing an alternating 16 s blocks
of moving and stationary concentric circles. The structural
scans were processed using the FreeSurfer pipeline , re-
sulting in spherically parameterized cortical surfaces , .
The functional data were analyzed using the general linear
model . The resulting activation maps were thresholded
by drawing the activation boundary centered around the vertex
with maximal activation. The threshold was varied across sub-
jects in order to maintain a relatively ﬁxed ROI area of about
120 mm ( 5%) as suggested in . The subjects consist
of 10 females and 32 males, with age ranging from 21 to 58
years old. 23 of the 42 subjects are clinically diagnosed with
schizophrenia, while the other 19 subjects are healthy controls.
Imaging took place on a 3T MR scanner (Siemens Trio) with
echoplanar (EP) imaging capability. Subjects underwent two
conventional high-resolution 3-D structural scans, constituting
a spoiled GRASS (SPGR) sequence (128 sagittal slices, 1.33
mm thickness, TR ms, TE ms, Flip angle ,
voxel size mm). Each functional run lasted 224 s
during which -weighted echoplanar (EP) images were ac-
quired (33 3-mm-thick slices, mm voxel size) using
a gradient echo (GR) sequence (TR ms; TE ms;
Flip angle ). To maximize training data, no distinction is
made between the healthy controls and schizophrenia patients.
1) Ex Vivo MT Prediction of In Vivo MT+: In this experi-
ment, we use each of the 10 ex vivo subjects as a template and
the remaining nine subjects for training a task-optimal template
for localizing MT. We then register each task-optimal template
to each of the 42 in vivo subjects and use the template subject’s
MT to predict that of the test subjects’ MT+. The results are
420 Hausdorff distances for each hemisphere. For FreeSurfer,
we align the 42 in vivo subjects to the Buckner40 atlas. Once
registered in this space, we can use MT of the ex vivo subjects
to predict MT+ of the in vivo subjects.
Fig. 9 reports the mean and standard errors of the Hausdorff
distances for both methods on both hemispheres. Once again,
we ﬁnd that the task-optimal template signiﬁcantly outperforms
the FreeSurfer template ( for both hemispheres). We
note that the errors in the in vivo subjects (Fig. 9) are signif-
icantly worse than those in the ex vivo subjects (Fig. 5). This
is not surprising since functionally deﬁned MT+ is slightly dif-
ferent from cytoarchitectonically deﬁned MT. Furthermore, the
ex vivo surfaces tend to be noisier and less smooth than those
acquired from in vivo subjects . Since our framework at-
tempts to leverage domain speciﬁc knowledge about MT from
the ex vivo data, one would expect these mismatches between
the data sets to be highly deterimental to our framework. In-
stead, FreeSurfer appears to suffer more than our framework.
2) In Vivo MT Prediction of In Vivo MT+: To understand
the effects of the training set size on localization accuracy, we
perform cross-validation within the fMRI data set. For each
randomly selected template subject, we consider 9, 19, or 29
training subjects. The resulting task-optimal template is used
to register and localize MT+ in the remaining 32, 22, or 12
test subjects, respectively. The cross-validation trials were re-
peated 100, 200, and 300 times, respectively, resulting in a total
of 3200, 4400, and 3600 Hausdorff distances. This constitutes
thousands of hours of computation time. For FreeSurfer, we per-
form a pairwise prediction of MT+ among the in vivo subjects
after registration to the Buckner40 atlas, resulting in 1722 Haus-
dorff distances per hemisphere.
Fig. 10 reports the mean and standard errors of the Haus-
dorff distances for FreeSurfer and task-optimal template on both
hemispheres. We see that the FreeSurfer alignment errors are
now commensurate with the ex vivo results (Fig. 5). However,
YEO et al.: LEARNING TASK-OPTIMAL REGISTRATION COST FUNCTIONS FOR LOCALIZING CYTOARCHITECTURE 1437
Fig. 9. Mean Hausdorff distances using ex vivo MT to predict MT+ in in vivo
scans. Permutation testing shows that the differences between FreeSurfer and
task-optimal template are statistically signiﬁcant .
the task-optimal template still outperforms FreeSurfer (
for all cases). We also note that the accuracy of MT+ local-
ization improves with the size of the training set. The resulting
localization error with a training set of 29 subjects is less than
7 mm for both hemispheres. For all training set sizes, the lo-
calization errors are also better than the ex vivo MT experiment
V. D ISCUSSION AND FUTURE WORK
The experiments in the previous section demonstrate the fea-
sibility of learning registration cost functions with thousands of
degrees-of-freedom from training data. We ﬁnd that the learned
registration cost functions generalize well to unseen test sub-
jects of the same (Sections IV-A and IV-B2), as well as dif-
ferent imaging modality (Section IV-B1). The almost linear im-
provement with increasing training subjects in the fMRI-deﬁned
MT+ experiment (Fig. 10) suggests that further improvements
can be achieved (in particular in the histological data set) with a
larger training set. Unfortunately, histological data over a whole
human hemisphere is difﬁcult to obtain, while fMRI localiza-
tion experiments tend to focus on single functional areas. There-
fore, a future direction of research is to combine histological
and functional information obtained from different subjects and
imaging modalities during training.
Since our measure of localization accuracy uses the mean
Hausdorff distance, ideally we should incorporate it into our
task-speciﬁc objective function instead of the SSD on the
distance transform representing the BA. Unfortunately, the
resulting derivative is difﬁcult to compute. Furthermore, the
gradient will be zero everywhere except at the BA boundaries,
causing the optimization to proceed slowly. On the other hand,
it is unclear how aligning the distance transform values far from
the boundary helps to align the boundary. Since distance trans-
form values far away from the boundary are larger, they can
dominate the task-speciﬁc objective function . Consequently,
we utilize the distance transform over the entire surface to
compute the gradient, but only consider the distance transform
within the boundary of the template BA when evaluating the
task performance criterion .
The idea of using multiple atlases for segmentation has
gained recent popularity , , , , , . While
we have focused on building a single optimal template, our
method can complement the multiatlas approach. For example,
one could simply fuse the results of multiple individually-op-
timal templates for image segmentation. A more ambitious task
would be to optimize for multiple jointly-optimal templates for
In this work, we select one of the training subjects as the
template subject and use the remaining subjects for training.
The task-speciﬁc cost function evaluates the localization of
the hidden labels via the template subject. During training (ei-
ther for learning the weights or template in the registration cost
function), the Brodmann areas of the template subject are held
constant. Because the ﬁxed Brodmann areas are speciﬁc to the
template subject, the geometry of the template subject should
in fact be the best and most natural initialization. It does not
make sense to use the geometry of another subject (or average
geometry of the training subjects) as initialization for the tem-
plate subject’s Brodmann areas, especially since the geometry
of this other subject (or average geometry) is not registered
to the geometry of the template subject. However, the use of
a single subject’s Brodmann (or functional) area can bias the
learning process. An alternative groupwise approach modiﬁes
the task-speciﬁc cost function to minimize the variance of the
distance transforms across training subjects after registration.
In this case, both the template geometry and Brodmann (func-
tional) area are estimated from all the training subjects and dy-
namically updated at each iteration of the algorithm. The av-
erage geometry of the training subjects provided a reasonable
template initialization. However, our initial experiments in the
ex vivo data set do not suggest an improvement in task perfor-
mance over the pairwise formulation in this paper.
While this paper focuses mostly on localization of hidden
labels, different instantiations of the task-speciﬁc cost func-
tion can lead to other applications. For example, in group
analysis, the task-speciﬁc cost function could maximize differ-
ences between diseased and control groups, while minimizing
intra-group differences, similar to a recent idea proposed for
discriminative Procrustes alignment .
In this paper, we present a framework for optimizing the pa-
rameters of any smooth family of registration cost functions,
such as the image dissimilarity-regularization tradeoff, with re-
spect to a speciﬁc task. The only requirement is that the task
performance can be evaluated by a smooth cost function on an
available training data set. We demonstrate state-of-the-art lo-
calization of Brodmann areas and fMRI-deﬁned functional re-
gions by optimizing the weights of the wSSD image-similarity
measure and estimating an optimal cortical folding template. We
believe this work presents an important step towards the auto-
matic selection of parameters in image registration. The gen-
erality of the framework also suggests potential applications to
other problems in science and engineering formulated as opti-
1438 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 7, JULY 2010
Fig. 10. Plot of mean hausdorff errors for MT+ from cross-validation of the fMRI data set using either FreeSurfer or in vivo trained task-optimal template. For the
task-optimal framework, we tried different number of training subjects. Test errors decrease as we go from 9 to 19 to 29 training subjects. Once again, permutation
testing shows that the differences between FreeSurfer and task-optimal template are statistically signiﬁcant .
PROOF OF PROPOSITION 2
In this appendix, we prove Proposition 2: If the Hessian
is positive deﬁnite at , then
there exists an , such that for all , a unique
continuous function exists, such that
is the new local minimum for parameters and
. Furthermore, has the same order of
smoothness as .
In the next section, we ﬁrst prove that the Hessian
is equal to the mix-deriva-
under the composition of diffeomorphisms model , ,
. We then complete the proof of Proposition 2.
A. Proof of the Equivalence Between the Hessian
and Mix-Derivatives Matrix for the Composition of
We will only provide the proof for when the image is deﬁned
in so as not to obscur the main ideas behind the proof. To
extend the proof to a manifold (e.g., ), one simply need to
extend the notations and bookkeeping by the local parameter-
izing the velocity ﬁelds and using coordinate charts. The
same proof follows.
Let us deﬁne some notations. Suppose the image and there
are voxels. Let be the rasterized coordinates of the
voxels. For conciseness, we deﬁne for the ﬁxed parameters
Therefore, is a function from to . Under the compo-
sition of diffeomorphisms model, is the diffeomorphism
parameterized by the stationary velocity ﬁeld deﬁned on the
voxels, so that is a function from to .To
make the dependence of on explicit, we deﬁne
and so is a function from to . In other words,
we can rewrite
Now that we have gotten the notations out of the way, we will
now show that
Hessian: We ﬁrst compute the Jacobian via the chain
From the above equation, we can equivalently write down the
th component of the Jacobian
where and denote the th and th components of and
, respectively. Now, we compute the th component of the
Hessian using the product rule
YEO et al.: LEARNING TASK-OPTIMAL REGISTRATION COST FUNCTIONS FOR LOCALIZING CYTOARCHITECTURE 1439
Because is the identity matrix and the Jaco-
derivative is zero at local minimum), we get , and
so the second term in (A.10) is zero.
To simplify the ﬁrst term of (A.10), we once again use the
fact that is the identity matrix, and so the summand
is zero unless and . Consequently, (A.10) simpliﬁes
Mix-Derivatives Matrix: We ﬁrst compute the Jaco-
bian via the chain rule
From the above equation, we can equivalently write down the
th component of the Jacobian
Now, we compute the th component of the mix-
derivatives matrix using the product rule
Like before, we have , and so the second term
is zero. Because is the identity, is zero unless
. Since , is also equal to zero
unless . Therefore, we get
B. Completing the Proof of Proposition 2
We now complete the proof of Proposition2. Let
. Since ,
where the last equality comes from the deﬁnition of
being a local minimum for the composition model.
Since the mix-derivatives matrix is invert-
ible by the positive-deﬁnite assumption of this proposition, by
the Implicit Function Theorem, there exists an , such
that for all , there is a unique continuous func-
tion , such that and
. Furthermore, has the same order of
smoothness as .
Then is positive deﬁnite at by the assumption
of the proposition. By the smoothness of derivatives and conti-
nuity of eigenvalues, there exists a small neighborhood around
in which the eigenvalues of are all greater
than zero. Therefore, does indeed de-
ﬁne a new local minimum close to .
COMPUTING THE DERIVATIVE
To compute , we perform a Taylor expansion
and rearranging the terms for , we get
The authors would like to thank P. Parillo for discussion on
the optimization aspects of this paper. The authors would also
like to thank C. Brun, S. Durrleman, T. Fletcher, and W. Mio
for helpful feedback on this work.
 S. Allassonniere, Y. Amit, and A. Trouvé, “Toward a coherent statis-
tical framework for dense deformable template estimation,” J. R. Stat.
Soc., Series B, vol. 69, no. 1, pp. 3–29, 2007.
1440 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 7, JULY 2010
 E. Allgower and K. Georg, Introduction to Numerical Continuation
Methods. Philadelphia, PA: SIAM, 2003.
 K. Amunts, A. Malikovic, H. Mohlberg, T. Schormann, and K. Zilles,
“Brodmann’s areas 17 and 18 brought into stereotaxic space—Where
and how variable?,” NeuroImage, vol. 11, pp. 66–84, 2000.
 K. Amunts, A. Schleicher, U. Burgel, H. Mohlberg, H. Uylings, and
K. Zilles, “Broca’s region revisited: Cytoarchitecture and intersubject
variability,” J. Comparative Neurol., vol. 412, no. 2, pp. 319–341,1999.
 V. Arsigny, O. Commowick, X. Pennec, and N. Ayache, “A log-eu-
clidean framework for statistics on diffeomorphisms,” in Proc. Int.
Conf. Med. Image Computing Computer Assist. Intervent. (MICCAI),
2006, vol. 4190, LNCS, pp. 924–931.
 B. Avants and J. Gee, “Geodesic estimation for large deformation
anatomical shape averaging and interpolation,” NeuroImage, vol. 23,
pp. 139–150, 2004.
 S. Baker and I. Matthews, “Lucas-Kanade 20 years on: A unifying
framework,” Int. J. Comput. Vis., vol. 56, no. 3, pp. 221–255, 2004.
 M. Beg, M. Miller, A. Trouvé, and L. Younes, “Computing large defor-
mation metric mappings via geodesic ﬂows of diffeomorphisms,” Int.
J. Comput. Vis., vol. 61, no. 2, pp. 139–157, 2005.
 K. Brodmann, Vergleichende Lokalisationslehre der Großhirnrinde in
Ihren Prinzipien Dargestellt auf Grund des Zellenbaues 1909.
 O. Commowick, R. Stefanescu, P. Fillard, V. Arsigny, N. Ayache,
X. Pennec, and G. Malandain, “Incorporating statistical measures of
anatomical variability in atlas-to-subject registration for conformal
brain radiotherapy,” in Proc. Int. Conf. Med. Image Computing and
Computer Assist. Intervent. (MICCAI), 2005, vol. 3750, LNCS, pp.
 A. M. Dale, B. Fischl, and M. I. Sereno, “Cortical surface-based anal-
ysis I: Segmentation and surface reconstruction,” NeuroImage, vol. 9,
pp. 179–194, 1999.
 R. Desikan, F. Segonne, B. Fischl, B. Quinn, B. Dickerson, D. Blacker,
R. Buckner, A. Dale, R. Maguire, B. Hyman, M. Albert, and R. Kil-
liany, “An automated labeling system for subdividing the human cere-
bral cortex on MRI scans into gyral based regions of interest,” Neu-
roImage, vol. 31, no. 3, pp. 968–980, 2006.
 M. Dubuisson and A. Jain, “A modiﬁed Hausdorff distance for object
matching,” in Proc. 12th IAPR Int. Conf. Pattern Recognit., 1994, vol.
1, pp. 566–568.
 S. Durrleman, X. Pennec, A. Trouvé, P. , and N. Ayache, “Inferring
brain variability from diffeomorphic deformations of currents: An in-
tegrative approach,” Med. Image Anal., vol. 12, no. 5, pp. 626–637,
2008, PMID: 18658005.
 I. Eckstein, A. Joshi, C. J. Kuo, R. Leahy, and M. Desbrun, “General-
ized surface ﬂows for deformable registration and cortical matching,”
in Proc. Int. Conf. Med. Image Computing Computer Assist. Intervent.
(MICCAI), 2007, vol. 4791, LNCS, pp. 692–700.
 B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, “Least angle re-
gression,” Ann. Stat., pp. 407–451, 2004.
 T. Evgeniou, M. Pontil, and T. Poggio, “Regularization networks
and support vector machines,” in Advances in Computational Mathe-
matics. Cambridge, MA: MIT Press, 2000, pp. 1–50.
 B. Fischl, N. Rajendran, E. Busa, J. Augustinack, O. Hinds, B. T. Yeo,
H. Mohlberg, K. Amunts, and K. Zilles, “Cortical folding patterns
and predicting cytoarchictecture,” Cerebral Cortex, vol. 18, no. 8, pp.
 B. Fischl, M. Sereno, R. Tootell, and A. Dale, “High-resolution in-
tersubject averaging and a coordinate system for the cortical surface,”
Human Brain Mapp., vol. 8, no. 4, pp. 272–284, 1999.
 B. Fischl, A. Stevens, N. Rajendran, B. T. Yeo, D. Greve, K. Van
Leemput, J. Polimeni, S. Kakunoori, R. Buckner, J. Pacheco, D. Salat,
J. Melcher, M. Frosch, B. Hyman, P. E. Grant, B. R. Rosen, A. van
der Kouwe, G. Wiggins, L. Wald, and J. Augustinack, “Predicting the
location of entorhinal cortex from MRI,” Neuroimage, vol. 47, no. 1,
pp. 8–17, 2009.
 Freesurfer Wiki [Online]. Available: http://surfer.nmr.mgh.har-
 K. Friston, A. Holmes, K. Worsley, J.-P. Poline, C. Frith, and R. Frack-
owiak, “Statistical parametric maps in functional imaging: A general
linear approach,” Human BrainMapp., vol. 2, no. 4, pp. 189–210, 1995.
 X. Geng, G. Christensen, H. Gu, T. Ross, and Y. Yang, “Implicit refer-
ence-based group-wise image registration and its application to struc-
tural and functional MRI,” NeuroImage, vol. 47, no. 4, pp. 1341–1351,
 X. Geng, D. Kumar, and G. Christensen, “Transitive inverse-consistent
manifold registration,” in Proc. Int. Conf. Inf. Process. Med. Imag.,
2005, vol. 3564, LNCS, pp. 468–479.
 B. Glocker, N. Komodakis, N. Navab, G. Tziritas, and N. Paragios,
“Dense registration with deformation priors,” in Proc. Int. Conf. Inf.
Process. Med. Imag., 2009, vol. 5636, LNCS, pp. 540–551.
 A. Guimond, J. Meunier, and J.-P. Thirion, “Average brain models: A
convergence study,” Comput. Vis. Image Understand., vol. 77, no. 2,
pp. 192–210, 2000.
 I. Guyon and A. Elisseeff, “An introduction to variable and feature
selection,” J. Mach. Learn. Res., vol. 3, pp. 1157–1182, 2003.
 T. Hastie, S. Rosset, R. Tibshirani, and J. Zhu, “The entire regulariza-
tion path for the support vector machine,” J. Mach. Learn. Res., vol. 5,
pp. 1391–1415, 2004.
 R. Heckemann, J. Hajnal, P. Aljabar, D. Rueckert, and A. Hammers,
“Automatic anatomical brain MRI segmentation combining label prop-
agation and decision fusion,” NeuroImage, vol. 33, no. 1, pp. 115–126,
 S. Jaume, M. Ferrant, S. Warﬁeld, and B. Macq, “Multiresolution pa-
rameterization of meshes for improved surface-based registration,” in
Proc. SPIE Med. Imag., 2001, vol. 4322, pp. 633–642.
 S. Joshi, B. Davis, M. Jomier, and G. Gerig, “Unbiased diffeomorphic
atlas construction for computational anatomy,” NeuroImage, vol. 23,
pp. 151–160, 2004.
 A. Klein, S. S. Ghosh, B. Avants, B. T. Yeo, B. Fischl, B. Ardekani, J.
C. Gee, J. J. Mann, and R. V. Parsey, “Evaluation of volume-basedand
surface-based brain image registration methods,” Neuroimage, 2010.
 R. Kohavi, “A study of cross-validation and bootstrap for accuracy es-
timation and model selection,” in Int. Joint Conf. Artif. Intell., 1995,
vol. 14, pp. 1137–1145.
 R. Kohavi and G. John, “Wrappers for feature subset selection,” Artif.
Intell., vol. 97, no. 1-2, pp. 273–324, 1997.
 D. Lee, M. Hofmann, F. Steinke, Y. Altun, N. Cahill, and B. Schlkopf,
“Learning the similarity measure for multi-modal 3-D image reg-
istration,” in Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern
Recognit., Jun. 2009, pp. 186–193.
 M. Leventon, W. Grimson, and O. Faugeras, “Statistical shape inﬂu-
ence in geodesic active contours,” in Proc. Int. Conf. Comput. Vis. Pat-
tern Recognit., 2000, pp. 1316–1323.
 C. Liu, W. T. Freeman, E. H. Adelson, and Y. Weiss, “Human-assisted
motion annotation,” in Proc. Int. Conf. Comput. Vis. Pattern Recognit.,
2008, pp. 1–8.
 M. Loog and M. de Bruijne, “Discriminative shape alignment,” in
Proc. Int. Conf. Inf. Process. Med. Imag., 2009, vol. 5636, LNCS, pp.
 O. Lyttelton, M. Boucher, S. Robbins, and A. Evans, “An unbiased it-
erative group registration template for cortical surface analysis,” Neu-
roImage, vol. 34, no. 4, pp. 1535–1544, 2007.
 S. Makrogiannis, R. Verma, B. Karacali, and C. Davatzikos, “A joint
transformation and residual image descriptor for morphometric image
analysis using an equivalence class formulation,” in Proc. Workshop
Math. Methods Biomed. Image Anal., Int. Conf. Comput. Vis. Pattern
Recognit., New York, 2006.
 D. McGonigle, A. Howseman, B. Athwal, K. Friston, R. Frackowiak,
and A. Holmes, “Variability in fMRI: An examination of intersession
differences,” NeuroImage, vol. 11, no. 6, pp. 708–734, 2000.
 C. McIntosh and G. Hamarneh, “Is a single energy functional suf-
ﬁcient? Adaptive energy functionals and automatic initialization,” in
Proc. Int. Conf. Med. Image Computing Computer Assisted Intervent.
(MICCAI), 2007, vol. 4792, LNCS, pp. 503–510.
 A. Moore and M. Lee, “Efﬁcient algorithms for minimizing cross vali-
dation error,” in Proc. 11th Int. Conf. Mach. Learn., 1994, pp. 190–198.
 P. Olver, Applications of Lie Groups to Differential Equations, 2nd
ed. New York: Springer-Verlag, 1993.
 M. Ono, S. Kubick, and C. Abernathey, Atlas of the Cerebral Sulci, 1st
ed. Germany: Georg Thieme Verlag, 1990.
 M. Park and T. Hastie, “ -regularization path algorithm for general-
ized linear models,” J. R. Stat. Soc., Series B, vol. 69, pp. 659–677,
 W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical
Recipes in C: The Art of Scientiﬁc Computing, 2nd ed. Cambridge,
U.K.: Cambridge Univ. Press, 1992.
 A. Qiu and M. Miller, “Cortical hemisphere registration via large defor-
mation diffeomorphic metric curve mapping,” in Proc. Int. Conf. Med-
ical Image Computing Computer Assisted Intervent. (MICCAI), 2007,
vol. 4791, LNCS, pp. 186–193.
 T. Rohﬂing, R. Brandt, R. Menzel, and C. Maurer, Jr., “Evaluation of
atlas selection strategies for atlas-based image segmentation with ap-
plication to confocal microscopy images of bee brains,” NeuroImage,
vol. 21, no. 4, pp. 1428–1442, 2004.
YEO et al.: LEARNING TASK-OPTIMAL REGISTRATION COST FUNCTIONS FOR LOCALIZING CYTOARCHITECTURE 1441
 T. Rohﬂing, D. Russakoff, and C. Maurer, “Performance-based classi-
ﬁer combination in atlas-based image segmentation using expectation-
maximization parameter estimation,” IEEE Trans. Med. Imag., vol. 23,
no. 8, pp. 983–994, Aug. 2004.
 W. Rudin, Principles of Mathematical Analysis. New York: Mc-
 D. Rueckert, L. Sonoda, C. Hayes, D. Hill, M. Leach, and D. Hawkes,
“Non-rigid registration using free-form deformations: Application
to breast MR images,” IEEE Trans. Med. Imag., vol. 18, no. 8, pp.
712–720, Aug. 1999.
 M. R. Sabuncu, S. Balci, M. Shenton, and P. Golland, “Image-driven
population analysis through mixture-modeling,” IEEE Trans. Med.
Imag., vol. 28, no. 9, pp. 1473–1487, Sep. 2009.
 M. R. Sabuncu, B. Singer, B. Conroy, R. Bryan, P. Ramadge, and
J. Haxby, “Function-based inter-subject alignment of the cortical
anatomy,” Cerebral Cortex, vol. 20, no. 1, pp. 130–140, 2010.
 M. R. Sabuncu, B. T. Yeo, K. Van Leemput, B. Fischl, and P. Golland,
“Supervised nonparameteric image parcellation,” in Proc. Int. Con.
Med. Image Computing and Computer Assisted Intervent. (MICCAI),
2009, vol. 5762, LNCS, pp. 1075–1083.
 R. Saxe, M. Brett, and N. Kanwisher, “Divide and conquer: A defense
of functional localizers,” NeuroImage, vol. 30, no. 4, pp. 1088–1096,
 T. Schormann and K. Zilles, “Three-dimensional linear and non-linear
transformations: An integration of light microscopical and MRI data,”
Human Brain Mapp., vol. 6, pp. 339–347, 1998.
 J. Shao, “Linear model selection by cross-validation,” J. Am. Stat.
Assoc., pp. 486–494, 1993.
 D. Shen and C. Davatzikos, “HAMMER: Hierarchical attribute
matching mechanism for elastic registration,” IEEE Trans. Med.
Imag., vol. 21, no. 11, pp. 1421–1439, Nov. 2002.
 Y. Shi, J. Morra, P. Thompson, and A. Toga, “Inverse-consistent sur-
face mapping with laplace-beltrami eigen-features,” in Proc. Int. Conf.
Inf. Process. Med. Imag., 2009, vol. 5636, LNCS, pp. 467–478.
 H.-Y. Shum and R. Szeliski, “Construction of panoramic image mo-
saics with global and local alignment,” Int. J. Comput. Vis., vol. 16, no.
1, pp. 63–84, 2000.
 B. Thirion, G. Flandin, P. Pinel, A. Roche, P. Ciuciu, and J.-B. Poline,
“Dealing with the shortcomings of spatial normalization: Multi-sub-
ject parcellation of fMRI datasets,” Human Brain Mapp., vol. 27, pp.
 B. Thirion, P. Pinel, S. Mériaux, A. Roche, S. Dehaene, and J.-B. Po-
line, “Analysis of a large fMRI cohort: Statistical and methodological
issues for group analyses,” NeuroImage, vol. 35, pp. 105–120, 2007.
 B. Thirion, P. Pinel, A. Tucholka, A. Roche, P. Ciuciu, J.-F. Mangin,
and J. Poline, “Structural analysis of fMRI data revisited: Improving
the sensitivity and reliability of fMRI group studies,” IEEE Trans. Med.
Imag., vol. 26, no. 9, pp. 1256–1269, Sep. 2007.
 P. Thompson and A. Toga, “A surface-based technique for warping
3-dimensional images of the brain,” IEEE Trans. Med. Imag., vol. 15,
no. 4, pp. 1–16, Aug. 1996.
 P. Thompson, R. Woods, M. Mega, and A. Toga, “Mathematical/com-
putational challenges in creating deformable and probabilistic atlases
of the human brain,” Human Brain Mapp., vol. 9, no. 2, pp. 81–92,
 R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. R.
Stat. Soc. Series B (Methodological), pp. 267–288, 1996.
 R. Tootell and J. Taylor, “Anatomical evidence for MT and additional
cortical visual areas in humans,” Cerebral Cortex, vol. 5, pp. 39–55,
 D. Tosun and J. Prince, “Cortical surface alignment using geometry
driven multispectral optical ﬂow,” in Proc. Int. Conf. Inf. Process. Med.
Imag., 2005, vol. 3565, LNCS, pp. 480–492.
 Z. Tu, K. Narr, P. Dollar, I. Dinov, P. M. Thompson, and A. W.
Toga, “Brain anatomical structure segmentation by hybrid discrimina-
tive/generative models,” IEEE Trans. Med. Imag., vol. 27, no. 4, pp.
495–508, Apr. 2008.
 C. Twining, T. Cootes, S. Marsland, V. Petrovic,R. Schestowitz, and C.
Taylor, “A uniﬁed information-theoretic approach to groupwise non-
rigid registration and model building,” in Proc. Int. Conf. Inf. Process.
Med. Imag., 2005, vol. 3565, LNCS, pp. 1611–3349.
 E. Tyrtyshnikov, A Brief Introduction to Numerical Analysis. Boston,
MA: Birkhäuser, 1997.
 D. Van Essen, H. Drury, S. Joshi, and M. Miller, “Functional and struc-
tural mapping of human cerebral cortex: Solutions are in the surfaces,”
Proc. Nat. Acad. Sci., vol. 95, no. 3, pp. 788–795, 1996.
 K. Van Leemput, “Encoding probabilistic brain atlases using bayesian
inference,” IEEE Trans. Med. Imag., vol. 28, no. 6, pp. 822–837, Jun.
 T. Vercauteren, X. Pennec, A. Perchant, and N. Ayache, “Dif-
feomorphic demons: Efﬁcient non-parametric image registration,”
NeuroImage, vol. 45, no. 1, pp. S61–S72, 2009.
 S. K. Warﬁeld, K. H. Zou, and W. M. Wells, “Simultaneous Truth and
Performance Level Estimation (STAPLE): An algorithm for the vali-
dation of image segmentation,” IEEE Trans. Med. Imag., vol. 23, no.
7, p. 903, Jul. 2004.
 T. White, D. O’Leary, V. Magnotta, S. Arndt, M. Flaum, and N. An-
dreasen, “Anatomic and functional variability: The effects of ﬁlter size
in group fMRI data analysis,” NeuroImage, vol. 13, no. 4, pp. 577–588,
 J. Xiong, S. Rao, P. Jerabek, F. Zamarripa, M. Woldorff, J. Lancaster,
and P. Fox, “Intersubject variability in cortical activations during a
complex language task,” NeuroImage, vol. 12, no. 3, pp. 326–339,
 B. T. Yeo, M. Sabuncu, R. Desikan, B. Fischl, and P. Golland, “Effects
of registration regularization and atlas sharpness on segmentation ac-
curacy,” Med. Image Anal., vol. 12, no. 5, pp. 603–615, 2008.
 B. T. Yeo, M. Sabuncu, P. Golland, and B. Fischl, “Task-optimal reg-
istration cost functions,” in Proc. Int. Conf. Med. Image Computing
and Computer Assist. Intervent. (MICCAI), 2009, vol. 5761, LNCS,
 B. T. Yeo, M. R. Sabuncu, T. Vercauteren, N. Ayache, B. Fischl, and P.
Golland, “Spherical demons: Fast diffeomorphic landmark-free surface
registration,”IEEE Trans. Med. Imag., vol. 29, no. 3, pp. 650–668, Mar.
 B. T. Yeo, T. Vercauteren, P. Fillard, J.-M. Peyrat, X. Pennec, P. Gol-
land, N. Ayache, and O. Clatz, “DT-REFinD: Diffusion tensor registra-
tion with exact ﬁnite-strain differential,” IEEE Trans. Med. Imag., vol.
28, no. 12, pp. 1914–1928, Dec. 2009.
 S. K. Zhou and D. Comaniciu, “Shape regression machine,” in Proc.
Int. Conf. Inf. Process. Med. Imag., 2007, vol. 4584, LNCS, pp. 13–25.
 K. Zilles, A. Schleicher, N. Palomero-Gallagher, and K. Amunts,
Quantitative Analysis of Cyto- and Receptor Architecture of the
Human Brain. New York: Elsevier, 2002.