Page 1

DT-REFinD: Diffusion Tensor Registration With Exact

Finite-Strain Differential

Citation

Yeo, B.T.T. et al. “DT-REFinD: Diffusion Tensor Registration

With Exact Finite-Strain Differential.” Medical Imaging, IEEE

Transactions on 28.12 (2009): 1914-1928. © 2010 Institute of

Electrical and Electronics Engineers.

As Published

http://dx.doi.org/10.1109/TMI.2009.2025654

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Institute of Electrical and Electronics Engineers

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Page 2

1914IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 12, DECEMBER 2009

DT-REFinD: Diffusion Tensor Registration

With Exact Finite-Strain Differential

B.T. Thomas Yeo*, Tom Vercauteren, Pierre Fillard, Jean-Marc Peyrat, Xavier Pennec, Polina Golland,

Nicholas Ayache, and Olivier Clatz

Abstract—In this paper, we propose the DT-REFinD algorithm

for the diffeomorphic nonlinear registration of diffusion tensor

images. Unlike scalar images, deforming tensor images requires

choosing both a reorientation strategy and an interpolation

scheme. Current diffusion tensor registration algorithms that

use full tensor information face difficulties in computing the

differential of the tensor reorientation strategy and consequently,

these methods often approximate the gradient of the objective

function. In the case of the finite-strain (FS) reorientation strategy,

we borrow results from the pose estimation literature in computer

vision to derive an analytical gradient of the registration objective

function. By utilizing the closed-form gradient and the velocity

field representation of one parameter subgroups of diffeomor-

phisms, the resulting registration algorithm is diffeomorphic and

fast. We contrast the algorithm with a traditional FS alternative

that ignores the reorientation in the gradient computation. We

show that the exact gradient leads to significantly better reg-

istration at the cost of computation time. Independently of the

choice of Euclidean or Log-Euclidean interpolation and sum of

squared differences dissimilarity measure, the exact gradient

achieves better alignment over an entire spectrum of deformation

penalties. Alignment quality is assessed with a battery of metrics

includingtensoroverlap,fractionalanisotropy,inverseconsistency

and closeness to synthetic warps. The improvements persist even

when a different reorientation scheme, preservation of principal

directions, is used to apply the final deformations.

Index Terms—Diffeomorphisms, diffusion tensor imaging, fi-

nite-strain (FS), finite-strain differential, preservation of principal

directions, registration, tensor reorientation.

Manuscript received April 20, 2009; revised June 09, 2009. First published

June 23, 2009; current version published November 25, 2009. This work

was supported by the INRIA Associated Teams Program Compu-Tumor

(http://www-sop.inria.fr/asclepios/projects/boston/), by the NAMIC under

Grant NIH NIBIB NAMIC U54-EB005149, and by the NAC under Grant

NIH NCRR NAC P41-RR13218). The work of T. Yeo was supported by the

Agency for Science, Technology, and Research, Singapore. Asterisk indicates

corresponding author.

*B.T. T. Yeo is with the Department of Electrical Engineering and Computer

Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

(e-mail: ythomas@csail.mit.edu).

P. Golland is with the Department of Electrical Engineering and Computer

Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

(e-mail: polina@csail.mit.edu).

T. Vercauteren is with the Mauna Kea Technologies, 75010 Paris, France

(e-mail: tom.vercauteren@maunakeatech.com).

P. Fillard, J.-M. Peyrat, X. Pennec, N. Ayache, and O. Clatz are

with the Asclepios Group, INRIA, 06902 Sophia Antipolis, France

(e-mail:pierre.fillard@sophia.inria.fr;

xavier.pennec@sophia.inria.fr; nicholas.ayache@sophia.inria.fr;

clatz@sophia.inria.fr).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMI.2009.2025654

jean-marc.peyrat@sophia.inria.fr;

olivier.

I. INTRODUCTION

D

diffusion is anisotropic in tissues such as cerebral white matter.

DTIisthereforeapowerfulimagingmodalityforstudyingwhite

matter structures in the brain. The rate and anisotropy of diffu-

sion at each voxel of a diffusion tensor image is summarized

by an order 2 symmetric positive definite tensor, i.e., a posi-

tive definite 3

3 matrix. This is in contrast to scalar values

in traditional magnetic resonance images. The eigenvectors of

the tensor correspond to the three principal directions of diffu-

sion while the eigenvalues measure the rate of diffusion in these

directions.

To study the variability or similarity of white matter struc-

tures across a population or to track white matter changes of a

single subject through time, registration is necessary to estab-

lish correspondences across different diffusion tensor (DT) im-

ages. Registration can be simplistically thought of as warping

one image to match another. For scalar images, such a warp can

be defined by a deformation field and an interpolation scheme.

For DT images however, one also needs to define a tensor reori-

entation scheme. Reorientation of tensors is necessary to warp

a tensor image consistently with the anatomy [3]. There are two

commonlyusedreorientationstrategies:thefinite-strain(FS)re-

orientation and the preservation of principal directions (PPD)

reorientation. In this paper, we derive an exact differential of

FS reorientation strategy and show that incorporating the exact

differential into the registration algorithm leads to significantly

better registration than the common practice of ignoring the re-

orientation when computing the gradient [3]. Their empirical

performance is similar [23], [34], [49].

Many DTI registration algorithms have been proposed [2],

[15], [24], [28], [31], [36], [49], [50]. Because the reorientation

strategies greatly complicate the computation of the gradient of

the registration objective function [49], many of these registra-

tion techniques use scalar values or features that are invariant

to image transformations. This includes the use of fractional

anisotropy [31] and fibers extracted through tractography [50].

Leemans et al. [28] use mutual information to affinely align

the diffusion weighted images from which the DT images are

estimated. Nonlinear fluid registration of DT images based

on information theoretic measures has since been introduced

[16], [39].

Instead of using deformation invariant features, Alexander

and Gee [2] perform elastic registration of tensor images by re-

orienting the tensors after each iteration using PPD reorienta-

tion.Thereorientationisnottakenintoaccountwhencomputing

IFFUSIONtensorimaging(DTI)noninvasivelymeasures

thediffusionof water ininvivo biologicaltissues[8].The

0278-0062/$26.00 © 2009 IEEE

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YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL1915

the gradient of the objective function. Cao et al. [15] propose a

diffeomorphic registration of tensor images using PPD reorien-

tation. The diffeomorphism is parameterized by a nonstationary

velocity field under the large deformation diffeomorphic metric

mapping (LDDMM) framework [11]. An exact gradient of the

PPD reorientation is computed by a clever analytical reformu-

lation of the PPD reorientation strategy. In this paper, we com-

plement the work in [15] by computing the exact gradient of the

FS reorientation.

For a general transformation, such as defined by B-splines or

nonparametric free form displacement field, the FS reorienta-

tion [3] is defined through the rotation component of the defor-

mation field. This rotation is estimated by the polar decomposi-

tion of the Jacobian of the deformation field using principles of

continuum mechanics. The rotation produced by the polar de-

composition of the Jacobian is the closest orthogonal operator

to the Jacobian under any unitary invariant norm [26]. However,

the polar decomposition requires computing the square root of

a positive definite matrix, which replaces the eigenvalues of the

original matrix with their square roots. The dependence of the

rotationmatrixontheJacobianofdeformationisthereforecom-

plex and the gradient of any objective function that involves re-

orientation is hard to compute.

Zhang et al. [48], [49] propose and demonstrate a piecewise

localaffineregistrationalgorithmtoregistertensorimagesusing

FS reorientation. The tensor image is divided into uniform re-

gions and the optimal affine transformation is then estimated

for each such region. The rotation component of the deforma-

tion need notbe estimated as a separatestep.Instead, sincerota-

tion is already explicitly optimized in the affine registration, the

gradient due to FS reorientation can be easily computed. These

piecewise affine transformations are fused together to generate

a smooth warp field. The algorithm is iterated in a multiscale

fashion with smaller uniform regions. Unfortunately, it is un-

clear how much of the optimality is lost in fusing these locally

optimal piecewise affine transformations.

In this paper, we borrow results from the pose estimation lit-

erature in computer vision [20] to compute the analytical differ-

ential of the rotation matrix with respect to the Jacobian of the

displacement field. We propose a diffeomorphic DTI registra-

tion algorithm DT-REFinD, which extends the recently intro-

duced diffeomorphic Demons registration of scalar images [42]

to registration of tensor images. The availability of the exact an-

alytical gradient allows us to utilize the Gauss–Newton method

for optimization. Implemented within the Insight Toolkit (ITK)

framework, registration of a pair of 128

tensor volumes takes 15 min on a Xeon 3.2 GHz single pro-

cessor machine. This is comparable to the nonlinear registration

of scalar images whose runtime might range from a couple of

minutes to hours. DT-REFinD has been incorporated into the

freely available MedINRIA software.1

The diffeomorphic Demons registration algorithm [42] is

an extension of the popular Demons algorithm [37]. It guar-

antees that the transformation is diffeomorphic. The space of

transformations is parameterized by a composition of deforma-

12860 diffusion

1MedINRIA can be downloaded at http://www-sop.inria.fr/asclepios/soft-

ware/MedINRIA.

tions, each of which is parametrized by a stationary velocity

field. Such a representation is similar to that used by the large

deformation diffeomorphic metric mapping (LDDMM) frame-

work [11], [38]. However, unlike LDDMM, the diffeomorphic

Demons algorithm does not seek a geodesic of the Lie group of

diffeomorphism. At each iteration, the diffeomorphic Demons

algorithm seeks the best diffeomorphism to be composed with

the current transformation. Restricting each deformation up-

date to belong to a one parameter subgroup of diffeomorphism

results in a faster algorithm than the typical algorithm based on

the LDDMM framework or algorithms that parameterize the

entire diffeomorphic transformation by a stationary velocity

field [7], [25].

In addition to DT-REFinD, we also propose a simpler and

faster algorithm that ignores the reorientation during the gra-

dient computation. Instead, reorientation is performed after

eachiteration.Thisfasteralgorithmisthereforeadiffeomorphic

variant of the method proposed by Alexander and Gee [2] with

Gauss–Newton optimization. We compare the two algorithms

and show that using the exact gradient results in significantly

better registration at the cost of computation time.

While many methods for interpolating and comparing tensor

images exist [27], [32], we use Euclidean interpolation and

sum-of-squares difference (EUC-SSD) [2], [3], [49], as well

as Log-Euclidean interpolation and sum-of-squares difference

(LOG-SSD) [6], [21]. Regardless of the choice of interpo-

lation and dissimilarity metric, we find the exact gradient

achieves better alignment over an entire range of deforma-

tion regularization. Alignment quality is assessed with a set

of seventeen different metrics including tensor overlap, frac-

tional anisotropy and inverse consistency of the warps. We

also find that the exact gradient method recovers synthetically

generated warps with higher accuracy. Finally, we show that

the improvements persist even when PPD is used to apply the

final deformations.

Weemphasizethatthereisnotheoreticalguaranteethatusing

the true gradient will lead to a better solution. After all, the reg-

istration problem is nonconvex and any solution we find is a

local optimum. In practice however, the experiments show that

taking reorientation into account does significantly improve the

registration results. We believe that the reorientation provides

an additional constraint. The registration algorithm cannot arbi-

trarilypullinafarawayregionformatchingbecausethisinduces

the reorientation of tensors in other regions (cf. the famous “C”

example in large deformation fluid registration [18]). This ad-

ditional constraint acts as a further regularization, leading to a

better solution.

This paper extends a previously presented conference article

[47] and contains detailed derivations, experiments and discus-

sions left out in the conference version. The paper is organized

as follows. The Section II describes the computation of the FS

differential. We then present an overview of the diffeomorphic

DemonsalgorithminSectionIIIanddiscusscertainconventions

and numerical limitations of representing diffeomorphic trans-

formations. We extend the diffeomorphic Demons to tensor im-

ages in Section IV using the exact FS differential. We also pro-

poseasimplerandfasteralgorithmthatignoresthereorientation

during the gradient computation. In Section V, we compare the

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1916IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 12, DECEMBER 2009

two algorithms on a set of 10 DT brain images. Further discus-

sion is provided in Section VI.

To summarize, our contributions are as follows.

1) We derive the exact FS differential.

2) WeincorporatetheFSdifferentialintoafastdiffeomorphic

DT image registration algorithm. We emphasize that the

FS differential is useful, even if one were to use a different

registration scheme with a different model of deformation

or dissimilarity metric.

3) We demonstrate that the use of the exact gradient leads

to better registration. In particular, we show that using the

exact gradient leads to better tensor alignment over an en-

tire range of deformation, regardless of whether we use

LOG-SSD or EUC-SSD in the objective function. We also

show that the exact gradient recovers synthetically gener-

ateddeformationfieldssignificantlybetterthanwhenusing

an approximate gradient that ignores reorientation.

4) OurimplementationallowsforEuclideaninterpolationand

EUC-SSD metric, as well as Log-Euclidean interpolation

and LOG-SSD metric.

II. FINITE-STRAIN DIFFERENTIAL

Deforming a tensor image by a transformation

tensor interpolation followed by tensor reorientation [3]. To

compute a deformed tensor at a voxel

the tensor to get the interpolated tensor

schemes include Euclidean interpolation [3], Log-Euclidean

interpolation [6], affine-invariant framework [10], [22], [29],

[30], [32], Geodesic–Loxodromes [27], or other methods. In

this work, we focus on Euclidean and Log-Euclidean interpola-

tion since they are commonly used and computationally simple.

The FS differential we compute in this section characterizes

tensor reorientation. The following discussion is therefore

independent of the interpolation strategy.

Supposethetransformation mapsapoint tothepoint

Let

be the displacement field associated with the

transformation . Then

involves

, one first interpolates

. Interpolation

.

(1)

Similarly, we denote

representation of transformations, such as splines, one can al-

ways derive the equivalent displacement field representation.

According to the FS tensor reorientation strategy [3] for non-

linear deformation, one first computes the rotation component

of the deformation at the th voxel

. Note that even for parametric

(2)

where

voxel

is the Jacobian of the spatial transformation at the

(3)

where

inthe , and directions.

of the matrix

ment field

in the neighborhood of . Under the identity trans-

formation, i.e., zero displacements,

Because of the matrix inverse in (2), to maintain numerical sta-

bility of the computations, the invertibility of the deformation

(corresponding to

) is important.

The interpolated tensor

the final tensor

are the components of the displacement field

iscalledapolardecomposition

and is therefore a function of the displace-

and.

is then reoriented, resulting in

(4)

For registration based on the FS strategy, it is therefore neces-

sary to compute the differential of rotation

transformation . Using chain rule, this reduces to computing

the differential of rotation

with respect to the Jacobian . Let

betheinfinitesimalchangeintheJacobian .Then,asshown

in Appendix A, the infinitesimal change in the rotation matrix

is computed as follows:

with respect to the

(5)

where

, denotes the 3-D vector cross product,

and is the operator defineddenotes the th column of

as

(6)

This skew-symmetric operator is actually the matrix represen-

tation of cross-product, so that for two vectors

.Itisintroducedtosimplifythenotationinthealreadycom-

plicated (5).

Thedetailedderivation,basedontheposeestimationsolution

[20] is presented in Appendix A. Let

of .Equation(5)tellsusthevariationoftherotation

of the components of the Jacobian

computed by setting the matrix

which is set to 1.

and ,

beth component

interms

. In particular,

in (5) to 0, except for

is

III. BACKGROUND ON DIFFEOMORPHIC REGISTRATION

In this section, we briefly review the diffeomorphic exten-

sion [42] of Thirion’s Demons algorithm [37]. We also discuss

numerical issues related to representing diffeomorphism by ve-

locity fields and optimization methods we use in this paper.

A. Diffeomorphic Demons for Scalar Images

Weconsider themodified Demons objectivefunction [14]for

registering a moving scalar image

to a fixed scalar image

(7)

where is the dense spatial transformation to be optimized,

an auxiliary spatial transformation,

denotes the -norm of a vector (or vector field, depending

on the context). We can think of the fixed image

moving image

as 1-D vectors of length

diagonal matrix that defines the variability observed at a

is

denotes composition and

and warped

voxels.is a

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YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL1917

particular voxel.

The instantiation of these parameters are further discussed in

Section V-B.

This formulation enables a fast and simple optimization that

alternately minimizes the first two terms and the last two terms

of (7). Typically,

to be close and

smooth.Theregularizationcanalsobemodifiedtohandleafluid

model. We note that

and

preted probabilistically as a hierarchical prior on the deforma-

tion

[46].

For the classical Demons algorithm and its variants, the

objective function is optimized over the complete space of

nonparametric spatial transformations [14], [35], [37], [43],

typically represented as displacement fields. Unfortunately,

the resulting deformation might not be diffeomorphic. Instead,

Vercauteren et al. [42] optimize over compositions of diffeo-

morphic deformations, each of which is parametrized by a

stationary velocity field. At each iteration, the diffeomorphic

Demons algorithm seeks the best diffeomorphism parame-

terized by the stationary velocity , to be composed with the

current transformation.

In this case, the velocity field

gebra

andis the diffeomorphism associated with .

The operator

is the group exponential relating the Lie

Group

to its associated Lie algebra

be the solution at time of the following stationary or-

dinary differential equation (ODE):

andare parameters of the cost function.

, encouraging

, encouraging

and

to be

can together be inter-

is an element of the Lie al-

. More formally, let

(8)

We define

(9)

An image

obtained by transforming the coordinate system of

: a point in the deformed coordinate system corre-

sponds to a point

in the old coordinate system.

The above formulation of the Demons objective function fa-

cilitates a fast iterative two-step optimization. We summarize

the diffeomorphic Demons algorithm [42] in Algorithm 1 (see

algorithm at the top of the page). Steps 2(ii) to 2(iv) essentially

optimize the last two terms of (7). We refer the reader to [13],

[14] for a detailed discussion of using convolution kernels to

achieve elastic and fluid regularization. We also note that the

aboveformulationisquitegeneral,andinfactthediffeomorphic

is therefore a deformed version of image

by

Demons algorithm can be extended to non-Euclidean domains,

such as the sphere [46].

B. Numerical Details in Velocity Field Representations

While

and

defined on the entire continuous image domain, in practice,

andare represented by vector fields on a discrete grid of

image points, such as voxels [37], [42] or control points [7],

[11]. From the theories of ODEs, we know that the integral

curves

(or trajectories) of a velocity field

exist and are unique ifis Lipschitz continuous in

continuous in [12]. Uniqueness means that the trajectories do

not cross, implying that the deformation is invertible. Further-

more, we know from the theories of ODEs that a

uous velocity field

produces a

field

. Therefore, a sufficiently smooth velocity field re-

sults in a diffeomorphic transformation.

Since the velocity field

is stationary in the case of the one

parameter subgroup of diffeomorphism [5],

uous (and in fact

) in . A smooth interpolation of

tinuous in the spatial domain and is Lipschitz continuous if we

consideracompactdomain,whichholdssinceweonlyconsider

images that are closed and bounded.

To compute the final deformation of an image, we have to

estimate

at least at the set of image grid points. For ex-

ample, we can compute

smoothly interpolated velocity field

In this case, the estimate becomes arbitrarily close to the true

as the number of integration time steps increases. With

a sufficiently large number of integration steps, we expect the

estimate to be invertible and the resulting transformation to be

diffeomorphic.

The parameterization of diffeomorphism by stationary ve-

locity field is made popular by the use of the fast “scaling and

squaring” approach to computing

integration, the “scaling and squaring” method works by mul-

tiple composition of displacement fields

are technically

and

contin-

continuous deformation

is clearly contin-

is con-

by numerically integrating the

with Euler integration.

[5]. Instead of Euler

...

(10)

While this method is correct in the continuous case, in the

discrete case, composition of the displacement fields requires

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1918IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 12, DECEMBER 2009

interpolation of displacement fields, introducing errors in

the process. In particular, suppose

the true trajectories found by performing an accurate Euler

integration up to time

and

does not exist a trivial interpolation scheme that guarantees

. In practice however, it is widely

reported that “scaling and squaring” tends to preserve invert-

ibility even with rather large deformation [5], [7], [42]. In this

work, we employ trilinear interpolation because it is fast. We

findthatinpractice,thetransformationisindeeddiffeomorphic.

Technically speaking, since we use linear interpolation for the

displacement field, the transformation is only homeomorphic

rather than diffeomorphic. However, we will follow the con-

vention of [5], [7], [42] which call the resulting homeomorphic

transformations diffeomorphisms.

and are

respectively. Then, there

C. Gauss–Newton Nonlinear Least-Squares Optimization

We now focus on the optimization of step 2(i) of the diffeo-

morphic Demons algorithm. We choose

,where

. We “subtract” the identity transformation from the resulting

deformation field so that the identity transformation carries no

penalty. The objective function in step 2(i) can then be written

in a nonlinear least-squares form

(11)

(12)

(13)

where we define

and

. Using Taylor series expansion around

we can write (13) as

,

(14)

To

let

interpret(14)

a

for 3-Dimages

vector

with

of

voxels,

becomponents:

.

Then

block corresponds to a 1

is ablock diagonal matrix, whose

3 matrix

th

(15)

(16)

(17)

(18)

(19)

where

voxel

is the transformation of

and is the identity transformation when the velocity

. In (17), we utilize the fact that the differential of the

exponential map at

is the identity. is the

spatial derivativeof theimage intensityat voxel

moving image

.

Similarly,wecanshowthat

is aidentity matrix. The Gauss–Newton optimiza-

tion method ignores the

leading to the classical linear least-squares problem. In partic-

ular, (14) can then be rewritten as

of thewarped

where

term within the norm in (14),

(20)

(21)

which is a linear least-squares problem. Independently of the

sizeofthematrices,itiseasytosolvetheresultinglinearsystem

since the equations for each voxel can be decoupled

from all other voxels. With the help of the Sherman-Morrison

matrix inversion lemma, no matrix inversion is even needed

to invert the resulting small system of linear equations at each

voxel [42].

We note that the original Demons algorithm [37] replaced

by . This is justified by the fact that at the op-

timum, the gradient of the warped moving image should be al-

most equal to the gradient of the fixed image.

IV. DT-REFIND: TENSOR IMAGE REGISTRATION

A. Diffeomorphic Demons for Vector Images

Before incorporating the FS differential for tensor regis-

tration, let us extend the diffeomorphic Demons algorithm to

vector images. In addition to helping us explain our complete

algorithm, the derivation will also be useful for computing up-

date steps when ignoring tensor reorientation in Section IV-D.

We define a vector image to be an image with a vector of inten-

sities at each voxel. We can treat a vector image like a scalar

image in the sense that each vector component is independent

of the other components. Deformation of a vector image works

just like a scalar image, by treating each component of the

vector separately.

It is fairly straightforward to re-derive the results from the

previous section for vector images. Let

the intensity vector at each voxel. For convenience, we define

to be the intensity vector of the

, andto be the

image intensities

Demons algorithm from the previous section applies exactly to

vector images except that in (20),

blockdiagonalmatrix, where each blockis

particular,the thblockof

of

at voxel

be the dimension of

th voxel,

vector of all

. Then the diffeomorphic

is now a sparse

. In

containsspatialderivatives

... ... ...

(22)

The resulting least-squares linear system

harder to solve than before. However, for each voxel , we only

havetosolvea3

3linearsystemforthevelocityvectorupdate

is slightly

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YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL 1919

. The solution of the system is also more stable as there are

more constraints.

B. DT-REFinD: Diffusion Tensor Registration With Exact FS

Differential

We will now extend the Demons algorithm to DT images.

A DT image is different from a vector image because of the

additional structure present in a tensor. In particular, the space

of symmetric positive definite matrices (tensors) is not a vector

space. When deforming a DT image, reorientation is also nec-

essary. We extend the diffeomorphic Demons registration of

vector images to tensor images.

In this work, we use the modified Demons objective function

(7) and the FS reorientation strategy in our registration. The ob-

jective function in step 2(i) of the Demons algorithm, shown in

(11) for scalars, becomes

(23)

(24)

(25)

Here,

squares difference (EUC-SSD) between the tensor images. In

particular,

can be seen as a

the 3

3 order 2 tensor at each voxel into a column vector.

should be interpreted as the interpolated

tensor image. In practice, since the tensors are symmetric,

we can work with

vectors to represent tensors and

increase the weights of the entries of

nondiagonal entries of the tensors by

tensor is then reoriented using the rotation matrix

voxel and “rasterized” into a column vector. Note that

implicitly dependent on the transformation

is the Euclidean sum of

vector by “rasterizing”

corresponding to the

. Each interpolated

of each

is

. The term

computes the SSD between

each tensor of the fixed image and the corresponding reoriented

and interpolated tensorin thewarped movingimage, bytreating

each tensor as a vector and adding the SSD for all voxels.

Equation (23) can also be interpreted as the LOG-SSD be-

tween tensors if

andare the Log-Euclidean transforms of

the original tensor images, obtained by converting each tensor

in the original image to a log-tensor

is simply a symmetric matrix [6].

terpolated log-tensor image.

terpolated and reoriented log-tensor image, since

. Note that

is then the in-

is the in-

(26)

for any rotation matrix

followed by Log-Euclidean transformation is the same as

Log-Euclidean transformation followed by reorientation. This

is convenient since we can perform a one time Log-Euclidean

transformation of the tensor images to log-tensor images before

registration and convert the final warped log-tensor images to

tensor images at the end of the registration.

In this case,

is a sparse

can interpret

as

In particular, the

th block

. Therefore, tensor reorientation

matrix. One

3 matrices.

is equal to

blocks of 9

, where we remind the readers that

and ,are also voxel indices. Using the

chain rule, the product rule and the fact that the differential of

the exponential map at

is the identity, we get

(27)

(28)

(29)

(30)

where

Recallthat

field

pression of

finite central difference as shown in (31) at the bottom of the

page where

of voxel

in the ,and directions, respectively. Therefore

denotes the -coordinate of

and denotes the

transformation

.,

ings in the ,

anddirections respectively. Using the differ-

ential of

(5) and the expression of

using the chain rule. Appendix B provides the

detailed derivation. This definition of

is the identity transformation when the velocity

isafunctionoftheJacobianofdisplacement

at the voxeland that (3) gives an analytical ex-

. In practice, is defined numerically using

.

are the neighbors

after transformation

-coordinate of

and are the voxel spac-

after

(31), we can compute

implies

(32)

(31)

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1920IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 12, DECEMBER 2009

and for neighborof voxel , we get

(33)

Note that the first and second terms in the above expression are

transpose of each other. Therefore, for

of

is zeros if voxels and

As before,

computed the full gradient of our objective function

, theth block

are not neighbors.

. In summary, we have

(34)

where

a sparse

is a constant diagonal matrix, while

matrix.

is

C. Gauss–Newton Nonlinear Least-Squares Optimization

From the previous sections, we can now write

(35)

(36)

The resulting least-squares problem is harder to solve than be-

fore, since the linear systems of equations cannot be separated

into voxel-specificset of equations. However, the sparsity of the

matrix makes the problem tractable. In practice, we solve the

linear systems of equations using Gmm++, a free generic C++

template library for solving linear sparse systems.2At the finest

resolution, solving the sparse linear system requires about 60 s.

This is the bottleneck of the algorithm. However, due to the fast

convergence of Gauss–Newton method, we typically only need

to solve the linear systems 10 times per multiresolution level.

Theresultingregistrationtakesabout15minonaXeon3.2GHz

single processor machine.

The efficiency of the Demons algorithm for scalar images

comes from separating the optimization into two phases: op-

timization of the dissimilarity measure and optimization of the

regularizationterm.Thisavoidstheneedtosolveanonseparable

system of linear equations when considering the two phases to-

gether. Because of the reorientation in tensor registration, we

have to solve a sparse system of linear equations anyway. In

this case, we could have incorporated the optimization of the

regularizationtermtogetherwiththeoptimizationofthedissim-

ilaritymeasurewithoutmuchlossofefficiency.Inthiswork,we

keep the two phases separate to allow for fair comparison with

the case of ignoring the reorientation of tensors in the gradient

computation (see Section IV-D) by using almost the same im-

plementation. Any improvement must then clearly come from

the use of the true gradient and not from using a one-phase op-

timization scheme versus a two-phase optimization scheme.

2http://home.gna.org/getfem/gmm_intro

D. Classical Alternative: Ignoring the Reorientation of

Tensors

Previous work [2] performs tensor registration by not in-

cluding the reorientation in the gradient computation, but

reorienting the tensors after each iteration using the current

estimated displacement field. To evaluate the utility of the true

gradient, we modify our algorithm to ignore the reorientation

part of the objective function in the gradient computation.

In particular, we can simplify the Gauss–Newton optimiza-

tion in the previous section by setting

and

, effectively

ignoring the effects of the displacement field of a voxel on the

reorientation of its neighbors. Note that

different from before because we directly use the gradient of

the warped and reoriented image. In each iteration, we treat

the tensor like a vector, except when deforming the moving

image. The resulting least-squares problem degenerates to that

in Section IV-A. The algorithm is thus much faster since we

only need to invert a 3

3 matrix per voxel at each iteration.

Registration only takes a few minutes on a Xeon 3.2 GHz single

processor machine.

is slightly

V. EXPERIMENTS

We now compare the DT-REFinD algorithm that uses the

exact FS differential with the classical alternative that uses an

approximate gradient and a basic Demons algorithm that uses

the fixed image gradient.

A. Data and Preprocessing

We use 10 DT images acquired on a Siemens 1.5T scanner

using an EPI sequence, consisting of healthy volunteers with

the following acquisition parameters: echo time

diffusiongradients, imagedimensions

resolution

are kindly contributed by Dr. Ducreux, Bicêtre Hospital, Paris,

France.

We first use morphological operations to extract a foreground

mask from the diffusion weighted (DW) images of each of the

10 DT images. This involves an automatic thresholding of any

single DW image, except the baseline, so that the skull and the

eyes do not interfere in the mask calculation. The threshold

is chosen so that the mask contains the entire brain. This in-

evitably contains some outliers in the background. Then, a se-

quence of erosions with a ball of radius 1 voxel is performed

to remove outliers (3 to 4 iterations are sufficient), which re-

sults in a set of connected components ensured to lie within

the brain. Finally, we use conditional reconstruction to create

thefinal mask.Thisinvolves dilatingtheconnectedcomponents

while intersecting the result with the initial mask, and repeating

this process until convergence. Doing so allows the connected

components to grows within the brain while ensuring the back-

ground outliers are canceled. If holes are still present in the

mask, a hole filling algorithm can be applied (a simple morpho-

logical closing is generally sufficient).

; 25

;image

. These images

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YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL1921

B. Implementation Details

WeperformpairwiseregistrationofDTimagesviaastandard

multiresolution optimization, by smoothing and downsampling

the data for initial registration and using the resulting registra-

tion from a coarser resolution to initialize the registration of a

finer resolution. We find that 10 iterations per multiresolution

levelweresufficientforconvergence.WhencomputingtheSSD

objectivefunction,onlyvoxelscorrespondingtothefixedimage

foreground are included.

There are three main parameters in the algorithm: the diag-

onal variability matrix

, and the tradeoff parameters

. could be, in principle, estimated from a set of diffusion

tensor images via coregistration. Since we deal with pairwise

registration, we set

to be a constant diagonal matrix. Con-

sequently, because the local optimum is determined by the rel-

ative weighting of

,and

identity matrix.

determines the step-size taken at each iter-

ation, which affects the stability of the registration algorithm

[42]. Therefore, we empirically set

iteration is about 2 voxels. The relative values of

termine the width of the kernel used to smooth the deformation

field. Once

is determined, the value of

warp smoothness.

As previously shown [14], it does not make sense to com-

pare two registration algorithms with a fixed tradeoff between

the dissimilarity measure and regularization, especially when

the two algorithms use different dissimilarity measures and/or

regularizations. Furthermore, one needs to be careful with the

tradeoff selection for optimal performance in a given applica-

tion [44].

In this work, we compare the algorithms over a broad range

of kernel sizes. We note that larger kernel sizes lead to more

smoothing and thus smoother warps. Because kernel sizes are

not comparable across the different algorithms we consider, we

use harmonic energy as a more direct measure of warp smooth-

ness. We define the harmonic energy to be the average over all

voxels of the squared Frobenius norm of the Jacobian of the dis-

placement field.Note thattheJacobianofthedisplacementfield

corresponds to the Jacobian of the transformation defined in (3)

without the identity. Therefore lower harmonic energy corre-

sponds to smoother deformation.

and

, we simply setto be the

so that the update at each

andde-

determines the

C. Evaluation Metrics

To assess the alignment quality of two registered DT images,

we use a variety of tensor metrics [2], [9], [21]. Let

fusion tensor. We let

,,be its eigenvalues in descending

order with corresponding eigenvectors

the average eigenvalues. Similarly, let

diffusion tensor with corresponding eigenvalues

eigenvectors

. The following measures are averaged

over the foreground voxels of the fixed image. This in turn al-

lows us to average results across different registration trials by

normalizing for brain sizes.

1) Euclidean Mean Squared Errors (EUC-MSE): squared

Frobenius norm of

2) LogEuclideanMeanSquaredErrors(LOG-MSE):squared

Frobenius norm of

be a dif-

,, . We denote

denote another

, and

.

.

3) 1—Overlap:

Note that the EUC-MSE and LOG-MSE correspond to the SSD

and LOG-SSD dissimilarity metrics we employ during registra-

tion. In additional to these tensor metrics, we also consider the

following scalar measures [1], [9]. The dissimilarity between

two tensors is defined to be the sum of squared differences

between these scalar measures, averaged over the foreground

voxels.

1)

.

.

2) LFA (Logarithmic Anisotropy): FA computed for

3) ADC(ApparentDiffusionCoefficient):

4) VOL (Volume):

5) CL (Linear Anisotropic Diffusion):

6) CP (Planar Ansiotropic Diffusion):

7) CS (Spherical Anisotropic Diffusion):

8) RA (Relative Anisotropy):

.

.

.

.

.

.

.

9) VR (Volume Ratio):

10) DISP (Dispersion):

11)

.

12)

.

13)

.

A question then arises over whether these scalar measures

should be computed after deforming the moving image or

computed on the unwarped moving image and then interpolated

with the deformation field. The latter is attractive because the

result is independent of the tensor reorientation and interpola-

tion strategies. On the other hand, since registration is almost

never an end-goal—the deformed tensor images are presum-

ably used for other tasks, one could argue that it is important

to measure the quality of the deformed tensors. Therefore, in

this work, we consider both strategies. Finally, we define the

average distance between two deformation fields

be

.

.

andto

(37)

where

average difference in the deformation fields obtained by the dif-

ferent methods to evaluate how different the deformation fields

are.Wefind thattheaveragedifferenceinthedeformationfields

between the exact gradient method and the approximate gra-

dientmethodrangesfrom2mmatlowharmonicenergyto5mm

at high harmonic energy (cf. image resolution

). The average difference in the deformation

fields between the fixed image gradient method and the other

twomethodsrangesfrom3mmatlowharmonicenergyto8 mm

at high harmonic energy.

Theaveragedistancecanalsobeusedtomeasureinversecon-

sistency. Without the availability of ground truth deformation,

inverseconsistency[17]canbeusedasanindirectassessmentof

deformation quality. In particular, given a deformation

subject to subject

and from subject

is the number of foreground voxels. We compute the

from

to subject , inverse

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1922 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 12, DECEMBER 2009

Fig. 1. Qualitative comparison between the exact FS gradient and the approximated gradient for registering a pair of subjects using the Log-Euclidean framework

and the same parameters in the registration. (a) Moving image. (b) Fixed image. (c) Registration using the approximated moving image gradient. (d) Registration

using the exact FS gradient. Volumes were slightly cropped for better display. Exact gradient achieves better alignment of fiber tracts with a smoother displacement

field. Tensors in the anterior limb of the internal capsule, highlighted in (b) and (d) are coherently oriented in a north-east direction. However, in (c), the directions

of the tensors are more scattered. Furthermore, the volume of the tensors in (c) is swollen relative to (b) and (d). Numerically, the exact FS gradient has lower SSD

with a smoother deformation field (not shown).

consistency is defined to be the average distance between the

displacement field associated with the composed warp

and a zero displacement field.

D. Qualitative Evaluation

Fig.1showsanexampleregistrationoftwosubjectsfromour

data set. Visually, DT-REFinD results in better tract alignment,

such as the anterior limb of the internal capsule highlighted in

the figure. See figure caption for more discussion. In this partic-

ularexample,DT-REFinDalsoachievesabetterLog-Euclidean

mean-square-error (LOG-MSE) and smoother deformation as

measured by the harmonic energy.

E. Quantitative Evaluation I

To quantitatively compare the performance of the exact FS

gradient,theapproximategradientandthefixedimagegradient,

we consider pairwise registration of the 10 DT images. Since

our registration is not symmetric between the fixed and moving

images, there are 90 possible pairwise registration experiments.

We randomly select 20 pairs of images for pairwise registra-

tion. By swapping the roles of the fixed and moving images,

we obtain 40 pairs of image registration. From our experiments,

we find that the statistics we compute appear to converge after

about 30 pairwise registrations, hence 40 pairwise registrations

are sufficient for our purpose.

Even though we are considering algorithms with the same

dissimilarity measure and regularization (and effectively the

same implementation) but different optimization schemes, we

find that for a fixed-size smoothing kernel, using the exact FS

differential tends to converge to a solution of lower harmonic

energy, i.e., a smoother displacement field. Smaller harmonic

energy implies a smoother deformation, providing evidence

that the reorientation acts as an additional constraint for the

registration problem. To properly compare the algorithms,

we consider smoothing kernels of sizes from 0.5 to 2.0 in

increments of 0.1. In particular, we perform the following

experiment.

For each pair of subjects and for each kernel size

i) Run the diffeomorphic Demons registration algorithm

using Euclidean interpolation and EUC-SSD using:

(a) Exact FS gradient (DT-REFinD).

(b) Approximate gradient by ignoring reorientation.

(c) Fixed image gradient. This is the gradient proposed

in Thirion’s original Demons algorithm [37].

ii) Repeat (i) using Log-Euclidean interpolation and LOG-

SSD.

iii) Use the estimated deformation fields to compute the

tensor and scalar measures discussed in Section V-C

using FS reorientation or PPD reorientation.

iv) Compute the inverse consistency of the deformations

from subject to subject

For a given smoothing kernel and registration strategy,

registering different pairs of images leads to a set of error

metric values corresponding to different harmonic energies. To

average the error metric values across different pairs of images

and to compare registration results among different strategies,

for each registration, we linearly interpolate the dissimilarity

metric (EUC-MSE, LOG-MSE, and so on) over a fixed set of

harmonic energies sampled between 0.03 to 0.3. This allows us

to average the error metric across different pairs of images and

compare different strategies at a given harmonic energy.

1) Tensor Alignment: Fig. 2 shows the error metrics (av-

eraged over 40 pairwise registrations) with respect to the har-

monic energies when using the dissimilarity metric EUC-SSD,

euclidean interpolation and FS reorientation for registration.

The final deformations were applied using FS reorientation.

and from subject to subject .

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YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL 1923

Fig. 2. Comparison of exact FS gradient, approximate gradient and fixed image gradient over an entire spectrum of harmonic energy (?-axis) using EUC-SSD,

Euclidean interpolation and FS reorientation for registration. The final deformations are applied using FS reorientation. We find that the exact FS gradient method

achieves the best performance.

Fig. 3 shows the corresponding plot when applying the final

warps using PPD reorientation. In both cases, we find that

at all harmonic energy levels, the exact FS gradient method

achieves the lowest errors. The approximate gradient method

outperforms the fixed image gradient method.

As mentioned earlier, the scalar measures, such as FA, can

be computed after deforming the moving tensor image or

computed on the unwarped moving image and then deformed.

Figs. 2 and 3 show results based on the former strategy. We

obtain similar results using the latter strategy, but omit them

here for brevity.

The amount of improvement increases as the harmonic ener-

gies increase. In our experiments, a harmonic energy of 0.3 cor-

responds to severe distortion (pushing the limits of the numer-

ical stability of scaling and squaring), while a harmonic energy

of0.03correspondstoverysmoothwarps.Inpreviouswork, we

showed in the context of image segmentation that extreme dis-

tortion causes overfitting, while extremely smooth warps might

resultininsufficientfitting[44].Onlyaconcreteapplicationcan

inform us of the optimal amount of distortion and is the sub-

ject of future studies. For now, we assume a “safe” range for

assessing the algorithm’s behavior to be between harmonic en-

ergies0.1and0.2.FromthevaluesinFigs.2and3,weconclude

that the exact FS gradient provides an improvement of between

5% to 10% over the approximate gradient in this “safe” range

of harmonic energies.

To better appreciate the improvements, Fig. 4 shows the dif-

ference in errors by subtracting the error metric values of the

approximate gradient method from the error metric values of

the exact gradient method when using the dissimilarity metric

EUC-SSD,EuclideaninterpolationandFSreorientationforreg-

istration. The final deformations were applied using FS reorien-

tation. The error bars indicate that the exact gradient method

is statistically significantly better than the approximate gradient

method over the entire range of harmonic energies and all the

error metrics (

for almost entire range of harmonic en-

ergies). We emphasize that the improvements persist even when

we evaluate a different dissimilarity measure or use a different

reorientation strategy from those used during registration.

Similarly, we find that the exact FS gradient method achieves

the lowest errors when using LOG-SSD similarity metric and

Log-Euclidean interpolation for registration, regardless of

whether FS or PPD reorientation was used to apply the final

deformation. We omit the results here in the interest of space.

2) Inverse Consistency: Fig. 5 shows the inverse consistency

errors (averaged over 20 sets of forward and backward pair-

wise registrations) with respect to the harmonic energies. Once

again, we find that all harmonic energy levels, the exact FS gra-

dient method achieves the lowest errors, regardless of whether

EUC-SSD and Euclidean interpolation or LOG-SSD and Log-

Euclidean interpolation were used. Similarly, the approximate

gradient method outperforms the fixed image gradient method.

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1924 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 12, DECEMBER 2009

Fig. 3. Comparison of exact FS gradient, approximate gradient and fixed image gradient over an entire spectrum of harmonic energy (?-axis) using EUC-SSD,

Euclidean interpolation and FS reorientation for registration. The final deformations are applied using PPD reorientation. We find that the exact FS gradient method

achieves the best performance.

F. Quantitative Evaluation II

We perform a second set of experiments to evaluate the algo-

rithm’s ability to recover randomly generated synthetic warps.

Given a DT image, we first generate a set of random warps by

samplingarandomvelocityateachvoxellocationfromaninde-

pendent and identically distributed (I.I.D.) Gaussian. The fore-

ground mask is then used to remove the velocity field from the

backgroundvoxels.Theresultingvelocityfieldissmoothedspa-

tially with a Gaussian filter. We compute the resulting displace-

ment field by “scaling and squaring.” This displacement field is

used to warp the given DT image using Log-Euclidean interpo-

lation. We use either FS or PPD to reorient the tensors. I.I.D.

Gaussian noise is added to the warped DT image.

We pick a single DT image and generate 40 sets of random

warps. We obtain an average displacement of 9.4 mm over the

foreground voxels. The average harmonic energy is 0.15. We

then perform pairwise registration between the DT image and

the warped DT image using LOG-SSD. Once again, we con-

sider a wide range of smoothing kernel sizes. We also compute

the registration error defined to be the average difference be-

tween the ground truth random warps and the estimated defor-

mationfieldspecifiedin(37).Notethatwithoutregistration,i.e.,

under the identity transformation, the average registration error

is 9.4 mm.

Fig.6showstheregistrationerrors(averagedover40trials)of

the three gradients we are considering. From the plots, when the

synthetic warps were applied using FS reorientation, the exact

FS gradient recovers the ground truth warps up to 1.56 mm or

17% error with respect to the average 9.4 mm random warps.

The approximate gradient achieves 2.34 mm or 25% error. Fi-

nally, the fixed image gradient achieves 2.73 mm or 29% error.

Therefore, the exact FS gradient achieves an average of

and 43% reduction in registration er-

rors compared with the approximate gradient and fixed image

gradient, respectively.

When the synthetic warps were applied using PPD reorienta-

tion,theexactFSrecoversthegroundtruthwarpsupto2.80mm

or 30% error with respect to the average 9.4 mm random warps.

The approximate gradient achieves 3.20 mm or 34% error. Fi-

nally, the fixed image gradient achieves 3.50 mm or 37% error.

Therefore, the exact FS gradient achieves an average of 13%

and 20% reduction in registration errors compared with the ap-

proximate gradient and fixed image gradient respectively.

Consistent with the previous experiments, using the exact

FS gradient leads to the lowest registration errors regardless

of whether FS or PPD reorientation were used to apply the

synthetic deformation fields. We note that the registration er-

rors inevitably increase when PPD were used to apply the syn-

thetic deformation field, since we use FS reorientation during

registration.

VI. DISCUSSION AND FUTURE WORK

Since Gauss–Newton optimization allows the use of “big

steps” in the optimization, it might cause the approximate

gradient to be more sensitive to the reorientation. It is possible

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YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL1925

Fig. 4. Comparison of exact FS gradient and approximate gradient over an entire spectrum of harmonic energy (?-axis) using EUC-SSD, Euclidean interpolation

and FS reorientation for registration. The final deformations are applied using PPD reorientation. ? -axis shows the difference in errors obtained by subtracting

the error metrics of the approximate gradient method from the error metrics of the exact gradient method. Negative values imply that the exact gradient method

outperforms the approximate gradient method. The error bars show the statistical variability (and thus significance) of the results.

Fig. 5. Comparison of exact FS gradient, approximate gradient and fixed image gradient over an entire spectrum of harmonic energy (?-axis). ? -axis shows the

inverse-consistency errors averaged over 20 sets of forward and backward pairwise registrations. We find that the exact FS gradient method achieves the lowest

errors.

that other optimization methods, such as the conjugate gradient,

might improve the results of using the approximate gradient,

by allowing for “smaller steps” and reorient after each “small

step.” Possible future work would involve comparing the

exact gradient and approximate gradient under an optimization

framework that takes small steps in the optimization procedure.

However, from optimization theory and from our experience,

Gauss–Newton method requires much fewer iterations to con-

verge than conjugate gradient. Furthermore, conjugate gradient

requires a line search, resulting in many function evaluations.

Function evaluations are quite expensive in our case, because of

the need to reorient and perform “scaling and squaring” of the

velocity field. On the other hand, we find that in practice, line

search is not necessary with Gauss–Newton optimization.

We should also emphasize that ignoring the gradient of the

reorientation term can lead to registration errors that cannot

be recovered regardless of any gradient-based optimization

scheme. For example, consider the registration of a 2-D diffu-

sion tensor image consisting of only horizontal tensors and a

2-D diffusion tensor image consisting of tensors orientated at

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1926IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 12, DECEMBER 2009

Fig. 6. Comparison of exact FS gradient, approximate gradient and fixed image gradient over an entire spectrum of harmonic energy (?-axis) using synthetic

images with ground truth deformations. ? -axis shows the registration errors (in millimeters) averaged over 40 pairwise registrations. We find that the exact FS

gradient method achieves the lowest errors regardless of whether FS or PPD reorientation were used to generate the warps.

a 10 angle. In this case,

approximate gradient and fixed image gradient updates are both

zeros. In contrast, the exact gradient update is nonzero due to

the reorientation.

In our experiments, we show that using the exact gradient re-

sultsinbetterinverseconsistencythantheapproximategradient.

However, the inverse consistency is not perfect. Incorporating

an inverse consistency constraint as suggested by the recent ex-

tension of the diffeomorphic Demons algorithm [41] should not

be difficult.

An interesting observation from the synthetic warp experi-

ment in Section V-F is that the best registration occurs when the

kernelsizeissuchthattheharmonicenergyisabout0.14,which

is close to the average harmonic energy of the synthetic warps.

In practice, no ground truth deformation is available, making

selection of the optimal kernel size difficult. Furthermore, we

believe the amount of deformation required is dependent on the

application of interest; the appropriate kernel size is likely to

vary with applications.

Itmayalsobethecasethatdifferentanatomicalregionsmight

require different optimal warp smoothness. While using a dif-

ferent kernel at each spatial location is possible, this would re-

duce the efficiency of the demons algorithm. More importantly,

itbecomesunclearwhetherstep2ofthedemonsalgorithm(spa-

tial smoothing) is justified. To get around such a situation, one

could instead shift the burden to step 1 of the demons algo-

rithm. In this paper, thevariability matrix

tity matrix. Allowing for a nonconstant diagonal matrix

effectively result in spatially varying warp smoothness, since

smaller values of the th diagonal entry place greater emphasis

on matching the

th tensor of the fixed image to the moving

image. Estimating

and an optimal registration regularization

tradeoff is an active area of research [4], [19], [33], [40], [44],

[45] that we do not deal with in this paper.

TheexactFSdifferentialisusefulevenwithadifferentmodel

of deformation or dissimilarity metric from the ones we employ

in this paper. Mutual Information (MI) has been proposed as a

criterion to register diffusion images [16], [28], [39]. Because

MI can handle nonlinear change in intensities across images, it

can potentially handle diffusion image registration without any

reorientation. In fact, [39] suggests that MI without reorienta-

and are both zeros, so the

is set to be theiden-

will

tion results in better registration than MI with iterative reorien-

tation with either FS or PPD. Future work could involve testing

their observation when reorientation is properly taken into ac-

countusingtheanalyticaldifferentialwepresentedinthispaper.

VII. CONCLUSION

In this work, we derive the exact differential of the FS

reorientation. We propose a fast diffeomorphic DT image

registration algorithm DT-REFinD using the exact FS differ-

ential. We show that the use of the exact gradient achieves

better tensor alignment than the approximate gradient which

ignores reorientation, over an entire spectrum of harmonic

energies. The improvements persist even if we use an error

metric different from the objective function we optimize and

if we use PPD reorientation for applying the final deforma-

tion. We also show that the exact gradient method recovered

randomly generated warps significantly better than the approx-

imate gradient method—1.56 mm versus 2.34 mm error on

average. DT-REFinD has been incorporated into the freely

available MedINRIA software, which can be downloaded at

http://www-sop.inria.fr/asclepios/software/MedINRIA.

APPENDIX A

FS DIFFERENTIAL

In [20], the differential of the matrix

derived, where

thecontextof[20],

of labeled points and

rigid body motion.

componentoftherigidmotion usingtheleast-squaresestimate

. Finding the differential

andtherefore allows the error analysis of the estimate

the measurements andare noisy.

From

and

by ignoring second order terms [20]. Defining

, we have

is

andandare matrices. In

containsthemeasuredcoordinatesofaset

contains their measured positions after

andcan be used to estimate therotation

in terms of

when

, we get

(38)

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YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL1927

From (38),

the form

is a skew symmetric matrix, and therefore takes

(39)

We define

Then, the major result of [20] can be expressed as follows:

and.

(40)

where

product operator.

Recall that we are interested in

. By setting

, we obtain

denotes the th column ofanddenotes the cross

, where

and

and. Therefore

(41)

Since

, by multiplying (41) by, we obtain

(42)

By setting,

expression for

and, we finally arrive at the

(43)

(44)

where we have used the fact that

.

APPENDIX B

ROTATION DERIVATIVES

For completeness, we now derive the expressions for

, whereare the neighboring voxels

of voxel

are the neighbors of voxel

tionsrespectively.

the displacement field in the

Forconvenience,

. Recall that

in the

are

,

,anddirec-

thecomponents

and

of

directions.

we denote

. Using the chain rule, we have

(45)

(46)

(47)

The second and third equalities come from evaluating

, which are mostly zeros. No-

tice that

and

voxels. Similarly, we have

are evaluated at two different

(48)

ACKNOWLEDGMENT

The authors would like to thank D. Ducreux, M.D., Ph.D.,

Bicêtre Hospital, Paris, for the DTI data and the reviewers for

their manyhelpful suggestions. B. T.T. Yeo would like to thank

C.-F. Westin, G. Kindlmann, and M. Sabuncu for useful discus-

sions and feedback.

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