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A variational approach for the correction of field-inhomogeneities in EPI sequences

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A variational approach for the correction of ﬁeld-inhomogeneities in EPI sequences
Janine Olesch
a
, Lars Ruthotto
b
, Harald Kugel
c
, Bernd Fischer
a
and Carsten H. Wolters
ba
Institute of Mathematics, University of L¨ubeck, Germany;
b
Institute for Biomagnetism and Biosignalanalysis, University of M¨unster, Germany;
c
Department of Clinical Radiology, University of M¨unster, Germany
ABSTRACT
A wide range of medical applications in clinic and research exploit images acquired by fast magnetic resonanceimaging (MRI) sequences such as echo-planar imaging (EPI), e.g. functional MRI (fMRI) and diﬀusion tensorMRI (DT-MRI). Since the underlying assumption of homogeneous static ﬁelds fails to hold in practical applica-tions, images acquired by those sequences suﬀer from distortions in both geometry and intensity. In the presentpaper we propose a new variational image registration approach to correct those EPI distortions. To this endwe acquire two reference EPI images without diﬀusion sensitizing and with inverted phase encoding gradients inorder to calculate a rectiﬁed image. The idea is to apply a specialized registration scheme which on the one handmimics the elastic behavior of the underlying tissue and on the other hand compensates for the characteristicaldirection dependent image distortions. In addition the proposed scheme automatically corrects for intensity dis-tortions. This is done by evoking a problem dependent distance measure incorporated into a variational setting.Unlike existing approaches we adjust not only the image volumes but also the phase encoding direction aftercorrecting for patients head-movements between the acquisitions. Finally, we present ﬁrst successful results of the new algorithm for the registration of DT-MRI datasets.
Keywords:
Image registration, EPI distortion correction, MRI, DTI, fMRI
1. INTRODUCTION
EPI sequences that are sensitive to ﬁeld inhomogeneities are widely used in medical research and in clinicalapplications, e.g., in DT-MRI and fMRI. All those applications require undistorted images with respect toboth geometry and intensity. The high non-linearity of the distortions causes the registration of EPI to bea challenging task. However, a physical model for the occurring distortions exists which states that the ﬁeldinhomogeneity deforms the image along the phase-encoding and slice-selection direction.
1,2
The deformationcauses also a change in intensities given by its Jacobian. As Chang and Fitzpatrick
1
pointed out, the eﬀects of the inhomogeneity when inverting the phase-encoding gradient are reversed along that direction. Two diﬀerentclasses of correction approaches can be distinguished. The ﬁrst kind of approaches tries to measure the ﬁeldmap itself and then correct the acquired image.
3
However, since the static ﬁeld inhomogeneities are caused notonly by imperfections in the magnet but also by spatial varying susceptibility of the object being imaged
4
thoseﬁeld maps vary from subject to subject. On the other hand there are image-based approaches. The physicalmodel and the CF-methodology were developed and validated in.
1
Weiskopf et. al
5
applied those methodsto fMRI data and corrected them in real-time. Like in the srcinal proposal they consider each column of voxels individually which may result in a non-smooth deformation ﬁeld. Moreover the accuracy relies heavilyon the correct detection of edges.
5
To achieve smoothness, studies conducted by Skare and Andersson
2
and Taoet. al
4
modeled the deformation as a linear combination of B-splines or as a solution of a PDE. EvaluatingB-Spline basis functions however is computationally expensive. In this work we follow a variational approachusing a discretize-optimize strategy. Therefor our model tempts to combine the smoothness and regularity of the latter two studies
2,4
with the speed of the ﬁrst two.
1,5
On the one hand we avoid the speciﬁcation of basis
Further author information: (Send correspondence to the ﬁrst two authors who equally contributed)Janine Olesch: olesch@math.uni-luebeck.deLars Ruthotto: lars.ruthotto@uni-muenster.de
functions and allow the deformation to freely move voxels along the phase-encoding direction. This increasesthe transformation’s ﬂexibility and drastically reduces computation time. On the other hand its smoothness andone-to-one point correspondence is controlled by introducing an elastic regularization term. To further improverobustness against noise and increase speed a multi-level registration technique is introduced.
2. METHODS
In the following we present a variational approach to solve the problem of ﬁeld inhomogeneities in EPI sequencesfollowing the idea of the CF-method.
1
For two EPI-scans
M
1
,M
2
with inverted phase encoding gradients,geometric and intensity deformations due to ﬁeld inhomogeneities occur in both images along the phase encodingdirections
v
and its inverse
−
v
∈
R
3
. We restrict the algorithm to ﬁnd the optimal directional transformation
d
:
R
3
→
R
along
v
and
−
v
, respectively. In a usual registration setting, a distance measure compares a movingtemplate image to a ﬁxed reference image. To correct the ﬁeld inhomogeneities which lead to artifacts in oppositedirections
v
and
−
v
in both acquired images we register two moving images
M
1
and
M
2
. To realize this weintroduce˜
M
(
d
;
M,v
) :=
M
(
x
+
d
(
x
)
v
))(1+
<
∇
d
(
x
)
,v >
)where in the ﬁrst part, the image is evaluated after a voxel
x
∈
R
3
is moved by a
d
(
x
)
∈
R
along
v
. We like to pointout that we do not use B-Splines or any other parametric approach to derive the transformation. The secondpart covers the intensity-modulation of the voxel depending on the Jacobian of the transformation. The imagesare compared by the sum of squared diﬀerences measure (SSD) which is a reasonable choice since both imagesare taken on the same device. In case of head-movements which are likely to happen during the acquisitionprocess of
M
1
and
M
2
, the assumption of the two opposite directions
v
and
−
v
does not hold. In
2
a rigidcorrection was proposed to handle those artifacts simultaneously to the correction of the ﬁeld inhomogeneities.We propose here the following extension to their approach. In a ﬁrst step, in order to correct patients headmovements we register
M
1
to
M
2
rigidly and gain a transformation matrix
Q
∈
R
3
×
3
. The rotation of image
M
2
by the orthogonal matrix
Q
following this ﬁrst step also aﬀects the phase encoding direction in this image from
−
v
to
−
Qv
. Although the expected rotation should be minimal, even this eﬀect should not be neglected sinceotherwise corrections along the true phase encoding direction
−
Qv
will be impossible. Our resulting distancemeasure reads
D
(
d
;
v,Q
) = 12
Ω
˜
M
(
d
;
M
1
,v
)
−
˜
M
(
d
;
M
2
,
−
Qv
)
2
dx.
To ensure smooth deformation ﬁelds with respect to the elastic behavior of the underlying tissue we apply anelastic regularizer
6
to the resulting displacement ﬁeld which is given for our directional registration as
S
(
d
;
v
) = 12
Ω3
ℓ
=1
µ
∇
(
d
(
x
)
v
)
ℓ
2
+ (
µ
+
λ
)div
2
(
d
(
x
)
v
)) d
x.
Here
λ
and
µ
∈
R
+
are the so called Navier-Lam´econstants, which control the elastic behavior of the deformation.The regularizer only needs to be evaluated for
d
and
v
but not
−
Qv
since both resulting deformation ﬁelds haveidentical smoothness. Adding the distance measure
D
(
d
;
v,Q
) and the smoother
S
(
d
;
v
) we yield the variationalformulation of the registration setting. Find transformation parameters
d
:
R
3
→
R
which solve the optimizationproblemmin
d
J
(
d
;
v,Q
) =
D
(
d
;
v,Q
) +
α
S
(
d
;
v
)with a regularizing parameter
α
∈
R
. When the optimal deformation
d
is found we combine both correctedimages ˜
M
(
d
;
M
1
,v
) and ˜
M
(
d
;
M
2
,
−
Qv
) by averaging both to the ﬁnal result of the rectiﬁed image in order toimprove signal-to-noise ratio.
2
The workﬂow to solve the problem is to ﬁrst calculate the rigid pre-registrationbased on the SSD-distance measure to ﬁnd a suitable initial guess for the non-linear approach and to get therotational component which is needed to reorient
−
v
. To avoid local minima and to speed up the registrationprocess we apply a multilevel-strategy on the images. We ﬁrst calculate a solution an a low resolution level,then using this solution as a starting guess for the next ﬁner level and so on. Each level is attacked using adiscretize-optimize approach which means we discretize the functional
J
and apply an optimization strategy.
The discretization enables us to use the Gauss-Newton-method.
7
The search direction is calculated by applyingthe iterative Projected Conjugate Gradient (PCG) method
8
to solve the upcoming linear system. A strong Wolfeline search is applied to ﬁnd a suitable step-length for the search-direction.
3. RESULTS
We tested our algorithm on four EPI datasets with b=0 measured with the same acquisition protocol of healthysubjects on a Philips 3 Tesla scanner. The dimensions of the reconstructed matrix are 256
×
256
×
36 andthe intensities where scaled to a range between 0 and 255. The phase-encoding directions are initially parallelto the anterior-posterior direction. We performed the tests extending the FAIR package
9
for
Matlab
R2008aon a Windows XP 32-bit machine with an Intel Core 2 duo P8400 (2
×
2.26 GHz) with 3 GB of RAM. Weperformed a three step coarse-to-ﬁne multi-level optimization on grid-sizes of 32
×
32
×
5
,
64
×
64
×
9 and128
×
128
×
18. The results of all datasets are comparable to the one we show exemplarily in Figure 1. In theﬁrst step we rigidly corrected
M
1
and
M
2
for head-movement. The rigid correction resulted in a rotation of [
φ
x
,φ
y
,φ
z
] = [4
.
17
e
−
5
,
1
.
39
e
−
4
,
−
4
.
98
e
−
5] which is small as expected. The other datasets were also hardlyaﬀected by head-movement. Choosing
α
= 80
, λ
= 0
.
1
, µ
= 1
.
0 the algorithm was able to correct geometricdistortions of up to 1.6 cm without showing grid foldings, see Figure 1 (h),(i). One-to-one correspondences arereliable in the rectiﬁed image.The SSD was reduced by 71 % and the algorithm was even able to reduce the highly non-linear distortions inthe frontal area, where
M
2
was stretched in the fronto-central but heavily edged in the fronto-lateral areas(see1 (c)).The local minimum of our functional resulting in a distortion correction as shown was achieved after 270seconds. In further tests using diﬀerent tolerance levels we achieved much faster runtimes (
<
70 seconds) witha moderate loss of correction quality.
4. CONCLUSIONS
We have presented a new fast variational correction approach for EPI sequences based on image registrationtechniques following a multi-level strategy. The algorithm corrects distortions caused by the ﬁeld inhomogeneitywith respect to geometry and intensity and is easily extendable to additional tasks. We corrected for head-movement in both the image volume and the phase-encoding direction. The approach was tested on unweightedDTI images on a standard platform and even there an accurate and fast correction was achieved. Due to the useof a regularization term, a ﬂexible deformation model and a fast optimization strategy EPI deformations can becorrected within very short time. The ﬁrst very promising results highly encourage us to apply the algorithm onadditional distortion-problems.
ACKNOWLEDGMENTS
The authors thank Jan Modersitzki for making the FAIR (Flexible Algorithms for Image Registration) codeavailable. This research was supported by the German Research Foundation (DFG), projects WO1425/1-1 andJU445/5-1. The authors would like to thank F. Gigengack for his help when starting the project.
REFERENCES
[1] Chang, H. and Fitzpatrick, J. M., “A Technique for Accurate Magnetic Resonance Imaging in the Presenceof Field Inhomogeneities,”
IEEE Transactions on Medical Imaging
11
(3), 319 – 329 (1992).[2] Skare, S. and Andersson, J. L. R., “Correction of MR Image Distortions Induced by Metallic Objects Usinga 3D Cubic B-Spline Basis Set: Application to Stereotactic Surgical Planning,”
Magnetic Resonance in Medicine
54
, 169–181 (2005).[3] Wu, M., Chang, L.-C., Walker, L., Lemaitre, H., Barnett, A. S., Marenco, S., and Pierpaoli, C., “Comparisonof EPI Distortion Correction Methods in Diﬀusion Tensor MRI Using a Novel Framework,” in [
Medical Image Computing and Computer-Assisted Intervention – MICCAI 2008
], (2008).[4] Tao, R., Fletcher, P. T., Gerber, S., and Whitaker, R. T., “A Variational Image-Based Approach to the Cor-rection of Susceptibility Artifacts in the Alignment of Diﬀusion Weighted and Structural MRI,”
Information Processing in Medical Imaging
(2009).
[5] Weiskopf, N., Klose, U., Birmbaumer, N., and Mathiak, K., “Single-shot compensation of image distortionand BOLD contrast optimization using multi-echo EPI for real-time fMRI,”
NeuroImage
24
, 1068–1079(2005).[6] Modersitzki, J., [
Numerical Methods for Image Registration
], Oxford University Press (2004).[7] Nocedal, J. and Wright, S. J., [
Numerical Optimization
], Springer, Berlin (2000).[8] Golub, G. H. and Van Loan, C. F., [
Matrix Computations
], The Johns Hopkins University Press (1996).[9] Modersitzki, J., [
FAIR: Flexible Algorithms for Image Registration
], Siam (to appear 2009).
−200−1000100200
(a) initial
M
1
(b) corrected
M
1
(c) initial diﬀerence
−200−1000100200
(d) initial, reoriented
M
2
(e) corrected
M
2
(f) ﬁnal diﬀerence(g) rectiﬁed image (h) grid of
M
1
transformation (i) grid of
M
2
transformation
Figure 1. Axial slice of the results of the 3 step multi-level optimization of initial
M
1
(a) and initial but rigidly reoriented
M
2
(d). The diﬀerence of both images is presented in (c). The corrected images and the diﬀerence between them areshown in (b), (e) and (f), resp. (g) shows the resulting averaged image and the deformation grids are shown in (h) and(i).

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