Online orientation distribution function reconstruction in constant solid angle and its application to motion detection in HARDI

Abstract

The diffusion orientation distribution function (ODF) can be reconstructed from q-ball imaging (QBI) to map the complex intravoxel structure of water diffusion. As acquisition time is particularly large for high angular resolution diffusion imaging (HARDI), fast estimation algorithms have recently been proposed, as an on-line feedback on the reconstruction accuracy. Thus the acquisition could be stopped or continued on demand. We adapt these real-time algorithms to the mathematically correct definition of ODF in constant solid angle (CSA), and develop a motion detection algorithm upon this reconstruction. Results of improved fiber crossing detection by CSA ODF are shown, and motion detection was implemented and tested in vivo.

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apport ?
?
de recherche?
ISSN 0249-6399
ISRN INRIA/RR--7102--FR+ENG
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Online orientation distribution function
reconstruction in constant solid angle and its
application to motion detection in high angular
resolution diffusion imaging
Emmanuel Caruyer — Iman Aganj — Christophe Lenglet — Guillermo Sapiro — Rachid
Deriche
N° 7102
Novembre 2009
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Centre de recherche INRIA Sophia Antipolis – Méditerranée
2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex
Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65
Online orientation distribution function
reconstruction in constant solid angle and its
application to motion detection in high angular
resolution di?usion imaging
Emmanuel Caruyer∗, Iman Aganj†, Christophe Lenglet†‡,
Guillermo Sapiro†, Rachid Deriche∗
Thème : Thème BIO - Systèmes biologiques
Équipe-Projet Odyssée
Rapport de recherche n° 7102 ? Novembre 2009 ? 9 pages
Abstract: The di?usion orientation distribution function (ODF) can be recon-
structed from q-ball imaging (QBI) to map the complex intravoxel structure of
water di?usion. As acquisition time is particularly large for high angular resolu-
tion di?usion imaging (HARDI), fast estimation algorithms have recently been
proposed, as an on-line feedback on the reconstruction accuracy. Thus the ac-
quisition could be stopped or continued on demand. We adapt these real-time
algorithms to the mathematically correct de?nition of ODF in constant solid
angle (CSA), and develop a motion detection algorithm upon this reconstruc-
tion. Results of improved ?ber crossing detection by CSA ODF are shown, and
motion detection was implemented and tested in vivo.
Key-words: On-line model estimation, high angular resolution di?usion imag-
ing (hardi), q-ball imaging (qbi), orientation distribution function (odf), motion
estimation.
∗Équipe-Projet Odyssée, INRIA Sophia Antipolis Méditerranée
†Department of Electrical and Computer Engineering, University of Minnesota, Minneapo-
lis, MN, USA
‡Center for Magnetic Resonance Research, University of Minnesota, Minneapolis, MN,
USA
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Reconstruction en ligne de la fonction de
distribution d'orientation en angle solide constant
et application à la détection du mouvement en
imagerie de di?usion à haute résolution angulaire
Résumé : La fonction de distribution d'orientation de di?usion (ODF) peut
être reconstruite à partir d'imagerie Q-ball et permet alors de cartographier la
structure complexe de la connectivité cérébrale par tractographie à partir de
l'imagerie par résonance magnétique de di?usion (IRMd). Pour répondre ef-
?cacement aux problémes liés au temps d'acquisition des séquences d'images
IRMd, particulièrement long pour l'imagerie à haute résolution angulaire, des
algorithmes d'estimation et de reconstruction en temps réel de l'ODF ont été
récemment proposés. Dans ce rapport de recherche, nous adaptons notre précé-
dente contribution sur l'estimation temps réel de l'ODF au cas de l'ODF calculée
en angle solide constant et on montre comment tirer pro?t de notre formalisme
à base de ?ltrage de Kalman pour l'estimer en temps-réel et l'utiliser pour dé-
tecter d'éventuels mouvement opérés par un patient au cours de l'acquisition.
L'algorithme de détection de mouvement fait plein usage des résultats inter-
médiaires fournis par la reconstruction de l'ODF par ?ltrage de Kalman et
permet de décider en temps réel de la nécessité de continuer ou de stopper
l'acquisition en cours. Une expérience a été menée pour tester et valider cette
approche. Nous montrons sur nos résultats que les croisements de faisceaux de
?bres sont mieux détectés par l'ODF en angle solide constant, et que l'approche
proposée permet e?ectivement de bien détecter le mouvement opéré par un pa-
tient. Cette approche o?re de nouvelles perspectives dans l'amélioration des
protocoles d'acquisition d'IRM de Di?usion.
Mots-clés :
angulaire, imagerie q-ball, fonction de distribution d'orientation, estimation de
mouvement, ?ltrage de Kalman.
Estimation temps-réel, imagerie de di?usion à haute résolution
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Online CSA ODF reconstruction and motion detection in HARDI
3
1Introduction
Di?usion Magnetic Resonance Imaging (MRI) is a recent technological advance
which has rapidly been considered by the medical community as a help for diag-
nosis and preoperative planning. By measuring the local di?usion properties of
water molecules, it allows to infer underlying tissue structure and physiological
properties. White matter ?ber bundles are especially well suited to this analysis.
By mapping the Orientation Distribution Function (ODF), i.e. the angular in-
formation of the di?usion Probability Density Function (PDF), q-Ball Imaging
(QBI) [1] can resolve complex intravoxel structure. So as to speed up the re-
construction, analytical computation of the ODF has further been proposed [2],
allowing ODF reconstruction in a clinical context. Recently, the commonly used
ODF de?nition was corrected to match the de?nition of the marginal PDF of
di?usion in a Constant Solid Angle (CSA) [3, 4]. This moves the community to-
wards more accurate models, and more ?exible algorithms to reconstruct these
models.
An important limitation of the widespread adoption of di?usion MRI by
clinicians remains its sensitivity to patient motion. It is indeed delicate to com-
pensate patient motion through registration of the Di?usion Weighted Images
(DWI) prior to model estimation [5]. A real-time (on-line) reconstruction algo-
rithm has recently been proposed [6, 7], together with an acquisition sequence
designed to be optimal whenever the scan is interrupted. This is based on a
Kalman ?lter adaptive approach. In this work, we extend the framework pro-
posed in [7]: we adapt it to reconstruct the CSA ODF, and show how to use
this as a motion detection tool during acquisition.
In Section 2, we review the estimation of CSA ODF and present the subse-
quent changes in the Kalman ?lter implementation. We also propose a solution
to the motion detection based on the incremental estimation of the di?usion
signal. Section 3 presents experimental results of ?ber crossing detection from
synthetic and real data, comparing the original ODF to the CSA ODF, and
results on how motion is readily detected in vivo with our method. Section 4
concludes with a review of the contributions.
2Material and Methods
2.1Kalman ?ltering with ODF in constant solid angle
The ODF has originally been de?ned and estimated in QBI as [1]:
ODForig.(ˆ u) :=1
Z
?∞
0
P(rˆ u)dr ≈1
ZFRT{S(ˆ u)}
(1)
with P(rˆ u) the 3D probability function (PDF) of the di?usion of water molecules,
S(ˆ u) the di?usion signal in the direction ˆ u, FRT the Funk-Radon transform
[1, 8], and Z the normalization constant which needs to be computed. Z is re-
quired because, by lacking the Jacobian factor r2, the above de?nition of ODF
is not the actual marginal PDF of di?usion in a given direction in CSA (see
[3, 4]). It was shown in [3] that, by considering the Jacobian factor, the follow-
ing expression for the CSA ODF can be derived, better resolving multiple ?ber
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E. Caruyer, I. Aganj, C. Lenglet, G. Sapiro, R. Deriche
orientations:
ODFCSA(ˆ u):=
?∞
1
4π+
0
P(rˆ u)r2dr
≈
1
16π2FRT?∇2
bln(−lnE(ˆ u))?
where E(ˆ u) := S(ˆ u)/S0 with S0 the baseline image, and ∇2
Beltrami operator. The above dimensionless and intrinsically normalized for-
mula obviates the need for post-processing such as manual normalization or
arti?cial sharpening. The framework has been extended to multiple q-shells in
[9].
A robust implementation of the original ODF was proposed in [2, 10, 11],
which uses a least square error scheme to approximate the signal in the spherical
harmonic (SH) basis, and then computes the FRT analytically. This method
has also been exploited in [3] to calculate the CSA ODF, where ln(−lnE(ˆ u)),
instead of S(ˆ u), is approximated in the SH basis and the FRT and ∇2
are afterwards computed analytically. Additionally, [2] introduces a regulariza-
tion term proven to be useful especially when dealing with noisy data.
While this implementation scheme can be very e?cient for o?ine computa-
tion of the ODFs, the required pseudo-inversion may be computationally too ex-
pensive for real-time reconstruction of almost a million ODFs in a single HARDI
scan. Besides, in online ODF estimation, we do not expect intuitively that the
new measurement at each iteration bring such fundamental information as to
necessitate re-solving the entire system of equations. Kalman ?ltering [12], is
generally the ?rst and simplest answer in such cases where we desire to minimize
the computational cost of the real-time solution to a growing linear problem by
making optimum use of the solution at each step to update it on the ?y. A par-
ticular Kalman ?ltering framework has been proposed in [6] to incrementally
compute the original QBI ODFs taking into account the regularization factor in
[2]. Recently, the authors of [7] showed the sub-optimality of [6] and suggested
a new implementation of the method which does not compromise the optimality
of the Kalman ?lter.
In this work we use the same approach as in [7], yet this time to compute the
CSA ODFs [3] on-line. This is again done by SH approximation of the double
logarithm of the signal, instead of the signal itself:
bthe Laplace-
boperators
ln(−lnE(ˆ u)) ≈
R
?
j=1
ˆ cjYj(ˆ u),
where {Yj, j = 1, ..., R} is the l-th order SH basis for real and symmetric
functions on the unit sphere [2], R = (l+1)(l+2)/2 is the dimension of this basis,
and ˆ c = (ˆ cj)j=1, ..., Rthe corresponding vector of coe?cients. An incremental
computation of the coe?cients ˆ c lets us reconstruct the CSA ODF at each step,
in a similar way as done o?ine in [3]. The corresponding Kalman ?lter equations
are:















ˆ c0
˜P0
P0
gk
Pk
ˆ ck
=
=
=
=
=
=
E[c0]
E[(c − ˆ c0)(c − ˆ c0)T]
(˜P−1
0
+ λL)−1
Pk−1XT
(I − gkXk)Pk−1
ˆ ck−1+ gk(yk− Xkˆ ck−1)
k(XkPk−1XT
k+ σ2
k)−1
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where yk= ln(−lnE(ˆ uk)) is the double logarithm of the signal measured in the
direction ˆ uk, Xkis the k-th row of the SH matrix X, where Xk,j= Yj(ˆ uk). Pk
and gkare standard notations for the Kalman covariance matrix and gain, and
σ2
the parameters, vectors and matrices, we kindly refer the reader to [7].
Finally, the SH coe?cients of the CSA ODF are computed as:


where lj is the order associated to the j-th element of the modi?ed SH basis
(see [2, 3] for details). We emphasize that at each iteration, we estimate the
same ODF as we would obtain by o?-line processing.
kis the covariance of the signal measured at iteration k. For more details on
ˆ c?
j=



1
2√π
j = 1
−1
8π(−1)
lj
21×3×...×(lj+1)
2×4×...×(lj−2)ˆ cj
j > 1
2.2 Motion detection
The Kalman ?lter introduced in Section 2.1 reconstructs the spherical harmonics
coe?cients ˆ c to best ?t the signal ln(−ln(E)), for the ?2 norm. After a few
iterations, the estimate ˆ ck−1is stable enough to predict with good accuracy the
next signal outcome yk.
However, if the subject moves within the scanner, the di?usion weighted
images will no longer be registered to the baseline image, and pixels will not
match from one volume to another. An abnormal increase in the prediction error
of the di?usion signal during on-line reconstruction is a direct and expected
consequence of patient motion. We propose to use this quantity as an indicator
to warn the scanner operator to stop the scan in such a situation, as the next
acquisitions would hardly be useful for ODF reconstruction. For each iteration,
we calculate the prediction error ?k= yk−Xkˆ ck−1at each voxel, and compute
the average squared prediction error on the whole masked volume.
Next, we present results of simulated and in vivo ?ber crossing detection, as
well as motion detection.
3Experiments and results
3.1Results on arti?cial data
We simulated ?ber crossing by generating di?usion images from the sum of two
exponentials:
E(ˆ u) =1
2
where D1 is a diagonal matrix with diagonal entries (9,2,2) and D2 is D1
rotated about the y-axis, once by 60◦and another time by 90◦. Assuming an
apparent di?usion coe?cient (ADC) of 0.7×10−3mm2/s (the mean di?usivity in
brain parenchyma), these di?usion values correspond to a b-value of 4800 s/mm2.
The ODFs were reconstructed in the fourth order SH basis using 30 iterations.
The results are shown in Fig. 1 for CSA QBI, and also for original QBI followed
by Laplace-Beltrami sharpening (I − α∇2
α = 0.15 chosen to produce the optimal results. CSA QBI was shown in [3] to
?
e−ˆ uTD1ˆ u+ e−ˆ uTD2ˆ u?
(2)
boperator, see [13]), with parameter
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E. Caruyer, I. Aganj, C. Lenglet, G. Sapiro, R. Deriche
(a)
(b)
Figure 1: Incrementally reconstructed ODFs from synthetic data with ?ber crossings
of (a) 60◦and (b) 90◦, for iterations 1 to 30 (left to right), using (top) CSA QBI, and
(bottom) original QBI with Laplace-Beltrami sharpening.
Figure 2: Coronal slice: CSA ODF, after iteration 100 of the Kalman ?lter. The
ODFs are shown on the tensor Fractional Anisotropy map.
resolve the ?ber crossings starting at a smaller angle compared to the original
QBI. Accordingly, we can see in Fig. 1 that the 60◦crossing is better resolved
by the CSA ODF at most iterations, and the 90◦crossing is reconstructed
equally well by both methods. Additionally, further experiments demonstrated
that Kalman ?ltering in practice converges slightly faster when used with the
CSA ODF than the original ODF (rate of convergence was smaller by 0.001 for
crossing angles less than 50◦, and about the same for larger angles).
3.2In vivo motion detection
A sequence of 200 directions computed following the optimal sampling scheme
of [7] was implemented on a Siemens 3T scanner at the Center for Magnetic Res-
onance Research, University of Minnesota, to scan a healthy volunteer. During
the acquisition, the subject was asked to tilt his head after about 80 di?usion
weighted images were acquired. A pilot sequence without motion was also ac-
quired for comparison (see Fig. 2).
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Online CSA ODF reconstruction and motion detection in HARDI
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Figure 3: Axial slices of two baseline images: (left) before and (right) after the motion
occurred.
50100150
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Average Squared Prediction Error
200 dirs, Regular
200 dirs, With motion
Figure 4: Comparison of the average squared prediction error with and without mo-
tion.
The motion was quantitatively investigated (but not corrected) through rigid
registration of two baseline images acquired before and after the motion occured.
We used the Slicer3 rigid registration module, which estimated the transforma-
tion to be a rotation of about 20◦around the z-axis, combined with a small
translation. Axial views of both positions are shown in Fig. 3.
The Kalman ?lter of Section 2.1 was used to estimate the SH coe?cients of
the double logarithm of the signal, with a regularization parameter λ = 0.006.
The voxel-wise squared prediction error was calculated at each iteration of the
Kalman ?lter, and averaged over the relevant part of the imaging volume. This
region of interest was de?ned by a simple thresholding of the baseline image.
In Fig. 4, we compare the evolution of the reconstruction error during both
acquisition sequences. For the pilot acquisition, the ?t gets better as new mea-
surements are acquired, and the prediction error decreases, as expected. On the
contrary, there is a sudden increase in prediction error for the second experi-
ment, at a point corresponding exactly to the motion (iteration 80).
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E. Caruyer, I. Aganj, C. Lenglet, G. Sapiro, R. Deriche
4Conclusion
We have adapted and extended the on-line reconstruction algorithm to the
mathematically correct CSA ODF; our Kalman ?ltering solution is shown to
detect crossings in white matter along the acquisition and reconstruction pro-
cess beyond what was possible with regular ODF. We have also studied local
changes in di?usion signal induced by subject's motion, and have shown that
motion could be detected on-line when a coherent increase in prediction error
is reported over the imaging volume. We believe these improvements provide
the clinicians with a more accurate and more ?exible tool for QBI acquisition
and analysis, in particular for children or patients who have di?culty sustaining
prolonged scanning sessions.
The next challenging step towards these objectives could be the development
of a motion compensation solution to combine with the Kalman ?lter. Real-time
model estimation also opens the way for a wide variety of on-line post-processing
methods, such as scalar index computation or ?ber tractography, giving valuable
feedback to the operator.
Acknowledgements
This work was partly supported by the Computational Di?usion MRI (CD-
MRI) INRIA Associate Team program, NIH (P41 RR008079, P30NS057091,
R01 EB007813), ONR, NGA, NSF, DARPA, ARO. We would like to thank
Ryan Muetzel for helping us with the HARDI acquisition performed at CMRR,
University of Minnesota, Minneapolis.
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