scispace - formally typeset

Proceedings ArticleDOI

Computation of the mid-sagittal plane in diffusion tensor MR brain images

08 Mar 2007-Vol. 6512, Iss: 31, pp 522-530

TL;DR: This plane is estimated as the one that best superposes the two hemispheres of the brain by reflection symmetry via the automated minimisation of a correlation-type global criterion over the tensor image via the NEWUOA algorithm in a multiresolution framework.

AbstractWe propose a method for the automated computation of the mid-sagittal plane of the brain in diffusion tensor MR images. We estimate this plane as the one that best superposes the two hemispheres of the brain by reflection symmetry. This is done via the automated minimisation of a correlation-type global criterion over the tensor image. The minimisation is performed using the NEWUOA algorithm in a multiresolution framework. We validate our algorithm on synthetic diffusion tensor MR images. We quantitatively compare this computed plane with similar planes obtained from scalar diffusion images (such as FA and ADC maps) and from the B0 image (that is, without diffusion sensitisation). Finally, we show some results on real diffusion tensor MR images.

Topics: Diffusion MRI (58%), Tractography (58%)

Summary (2 min read)

1. INTRODUCTION

  • The human brain displays an approximately bilateral symmetry with respect to the mid-sagittal plane (MSP).
  • After correction of the tilt, the remaining normal or abnormal asymmetries can be assessed qualitatively or quantitatively, which can be relevant for many pathologies.
  • A more general approach is to view the mid-sagittal plane as the one that best separates the brain into two similar parts.
  • 5–12 Diffusion Tensor Magnetic Resonance Imaging (DT-MRI or DTI for short) complements conventional, anatomical, MR imaging in that it adds specific structural information about the brain.
  • Such an algorithm has the same potential applications as the methods developed for anatomical and functional images.

2.1. Data acquisition

  • A single-shot echo-planar sequence combined with coil sensitivity encoding was used to acquire the diffusion-weighted MR images with a spin-echo Stejskal-Tanner sequence (TE/TR(ms) = 56/7991, flip angle = 90◦).
  • For each subject, an image without diffusion sensitisation was acquired (b value = 0 sec/mm2), together with diffusion-weighted MR images (b value = 800 sec/mm2) with diffusion-sensitising gradients applied in 16 different directions.

2.2. Pre-processing

  • The pre-processing stage includes the following steps: Correction of distortions.
  • Diffusion-weighted images acquired with echo-planar sequences are significantly distorted, mainly due to the eddy currents induced by the large diffusion gradients.
  • To deal with this artefact, each diffusion-weighted image was registered to the image without diffusion sensitisation.
  • The alignment is based on the maximisation of a similarity measure on image intensities (the local correlation) and a transformation model based on the imaging physics.
  • 24 Estimation of the diffusion tensor by multiple linear regression on a voxel-by-voxel basis.25 Diagonalisation of the diffusion tensor by classical linear algebra (computation of the eigenvalues and eigenvectors).

2.4. Formulation of the problem

  • If the brain was perfectly symmetrical, there would exist a symmetry plane P superposing each voxel v of the grid with its anatomical counterpart.
  • In real life, the brain is only grossly symmetrical, and such an ideal symmetry plane does not exist.
  • This average distance would be null for a perfect symmetry plane.

2.5.2. Interpolation scheme

  • Thus, an interpolation is needed to estimate the tensor value I(v′) at this non-grid point.
  • A natural choice would be to interpolate in log-Euclidean or affine-invariant frameworks, which would preserve the constraint of symmetric positive definiteness.
  • 26–29 This unfortunately leads to a high computational cost and a prohibitive computation time for the overall algorithm.
  • The authors choice is to simply use the nearest neighbor interpolation instead, which also guarantees that the interpolated value is a SPD matrix.

2.5.4. Optimisation

  • The authors choose the algorithm NEWUOA (NEW Unconstrained Optimisation Algorithm) to solve the optimisation problem.
  • Powell,30 has been shown to be significantly more accurate, more robust and faster than other deterministic, derivative-free algorithms such as Nelder-Mead’s downhill simplex or Powell’s direction set algorithms for intensity-based rigid-body image registration.
  • A multiresolution (simple decimation) approach is implemented to increase the robustness, accuracy and speed of the algorithm.
  • A subsampled image is built by taking only one voxel out of fx (resp. fy, fz) in the x- (resp. y-, z-) direction in the original image.

2.5.5. Realignment of the plane

  • Once the MSP is estimated, it is realigned at the centre of the image grid for improved image display.
  • Such a realignment is performed by applying the rigid transformation R to both the anatomical and the tensor information.
  • The log-Euclidean interpolation28 is used for computation of the realigned DTI.

3. VALIDATION AND RESULTS

  • The authors use the measures δ and to evaluate the capture range (i.e. the maximal tilt that it is able to correct) and the accuracy (i.e. the closeness between the estimated plane and the ground truth plane) of the algorithm, as shown in Fig.
  • To investigate this question, the authors have modified the algorithm to make it suitable for scalar images.
  • It appears that the latter is significantly more accurate than the three other ones (p < 5.10−4).
  • The computation of the MSP in these three scalar images is much faster and seems as robust as the computation of the MSP in the DTI.

4. CONCLUSION

  • The authors have proposed a method for the automated computation of the mid-sagittal plane of the brain in diffusion tensor MR images.
  • Moreover, the authors introduced a new optimisation algorithm to deal with the high computational burden of DTI data.
  • The most straightforward modifications could be made in the choice of the interpolation scheme and of the distance function.
  • The authors will also evaluate the robustness and accuracy properties of the algorithm in presence of image artefacts (noise, intensity inhomogeneities) and brain normal and pathological asymmetries.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

HAL Id: inserm-00140454
https://www.hal.inserm.fr/inserm-00140454
Submitted on 3 May 2011
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Computation of the mid-sagittal plane in diusion
tensor MR brain images
Sylvain Prima, Nicolas Wiest-Daesslé
To cite this version:
Sylvain Prima, Nicolas Wiest-Daesslé. Computation of the mid-sagittal plane in diusion tensor
MR brain images. Medical Imaging 2007: Image Processing, Feb 2007, San Diego, United States.
pp.65121I, �10.1117/12.709467�. �inserm-00140454�

Computation of the mid-sagittal plane
in diffusion tensor MR brain images
Sylvain Prima
a,b,c,d
and Nicolas Wiest-Daessl´e
a,b,c,d
a
INRIA, VisAGeS U746 Unit/Project, IRISA, Campus de Beaulieu, F-35042 Rennes, France
b
University of Rennes I, IRISA, Campus de Beaulieu, F-35042 Rennes, France
c
CNRS, IRISA UMR 6074, Campus de Beaulieu, F-35042 Rennes, France
d
INSERM, VisAGeS U746 Unit/Project, IRISA, Campus de Beaulieu, F-35042 Rennes, France
ABSTRACT
We propose a method for the automated computation of the mid-sagittal plane of the brain in diffusion tensor
MR images. We estimate this plane as the one that best superposes the two hemispheres of the brain by reflection
symmetry. This is done via the automated minimisation of a correlation-type global criterion over the tensor
image. The minimisation is performed using the NEWUOA algorithm in a multiresolution framework. We
validate our algorithm on synthetic diffusion tensor MR images. We quantitatively compare this computed plane
with similar planes obtained from scalar diffusion images (such as FA and ADC maps) and from the B0 image
(that is, without diffusion sensitisation). Finally, we show some results on real diffusion tensor MR images.
Keywords: Registration, optimisation methods, diffusion tensor imaging, brain symmetry, mid-sagittal plane.
1. INTRODUCTION
The human brain displays an approximately bilateral symmetry with respect to the mid-sagittal plane (MSP).
This gross symmetry is often partially hidden in medical images, when the head is scanned in a tilted position.
In this situation, homologous bilateral structures (ventricles, grey nuclei, etc.) do not appear on the same 2D
axial or coronal slices, which can lead to misinterpretation of the images. Having a coherent display of medical
images with respect to the cerebral MSP can be useful for many applications.
First, it allows to remove the inter-hemispheric differences only due to the head tilt. After correction of the
tilt, the remaining normal or abnormal asymmetries can be assessed qualitatively or quantitatively, which can
be relevant for many pathologies. Second, the MSP itself is a crucial landmark for further analyses, such as
morphometry of the corpus callosum, delineation of the AC-PC line, or global quantification of brain symmetries
and asymmetries. Third, knowing the position of this plane is useful in registration tasks, where such an a priori
knowledge about the images to be matched can improve the robustness, accuracy and computation speed of the
methods.
Many algorithms have been developed for the automated computation of the cerebral mid-sagittal plane in
medical images. Two different approaches have been mainly used. Some works define the mid-sagittal plane
as the plane that best matches the inter-hemispheric fissure.
1–4
Such techniques are only applicable when this
fissure and its characteristic features are clearly visible on images, limiting their application to MR and CT
images. A more general approach is to view the mid-sagittal plane as the one that best separates the brain into
two similar parts. Many methods have been developed based on this idea, dealing with either anatomical (MR
and CT) or functional images (PET, SPECT).
5–12
Diffusion Tensor Magnetic Resonance Imaging (DT-MRI or DTI for short) complements conventional, anatom-
ical, MR imaging in that it adds specific structural information about the brain. By measuring diffusion properties
of water molecules in the brain, it allows the indirect visualisation of the microscopic organisation of tissues,
and more especially the white matter. There is an increasing interest in how DTI conveys brain symmetry and
asymmetry.
13–20
Further author information: (Send correspondence to Sylvain Prima)
Sylvain Prima: E-mail: sprima@irisa.fr, Telephone: 33 2 99 84 73 59
Medical Imaging 2007: Image Processing, edited by Josien P. W. Pluim, Joseph M. Reinhardt,
Proc. of SPIE Vol. 6512, 65121I, (2007) · 1605-7422/07/$18 · doi: 10.1117/12.709467
Proc. of SPIE Vol. 6512 65121I-1

In this paper, we propose a method for the automated computation of the mid-sagittal plane of the brain
in DTI. We estimate this plane as the one that best superposes the two hemispheres of the brain by reflection
symmetry. This is done via the automated minimisation of a correlation-type global criterion over the image
data, i.e. diffusion tensors. The minimisation is performed using the NEWUOA algorithm in a multiresolution
framework.
Such an algorithm has the same potential applications as the methods developed for anatomical and functional
images. More specifically, it has a direct application on DTI visualisation, which is often a critical task due to the
complex, non-scalar, nature of the data under study. For example, the classical red-green-blue colormap applied
to the principal direction of diffusion for easier display of DTI (red for left-right, green for antero-posterior and
blue for top-bottom)
21
is of poor interest if the mid-sagittal plane (and thus the left-right direction) is not aligned
with the image grid.
We detail our method in Section 2. We make a quantitative evaluation of its robustness and accuracy
properties, and present some results on real data in Section 3. We conclude and give some perspectives in
Section 4.
2. METHOD
2.1. Data acquisition
Images were obtained on a whole-body Philips 3T MR scanner. A single-shot echo-planar sequence combined
with coil sensitivity encoding (SENSE) was used to acquire the diffusion-weighted MR images with a spin-echo
Stejskal-Tanner sequence (TE/TR(ms) = 56/7991, flip angle = 90
).
22
For each subject, an image without
diffusion sensitisation was acquired (b value = 0 sec/mm
2
), together with diffusion-weighted MR images (b value
= 800 sec/mm
2
) with diffusion-sensitising gradients applied in 16 different directions. The characteristics of the
diffusion-weighted MR images are: matrix size = 256 × 256, pixel size (mm) = 0.875 × 0.875, FOV (mm) =
224 × 224 × 120, number of slices = 60, slice thickness (mm) = 2.
2.2. Pre-processing
The pre-processing stage includes the following steps:
Correction of distortions. Diffusion-weighted images acquired with echo-planar sequences are signifi-
cantly distorted, mainly due to the eddy currents induced by the large diffusion gradients. To deal with
this artefact, each diffusion-weighted image was registered to the image without diffusion sensitisation. The
alignment is based on the maximisation of a similarity measure on image intensities (the local correlation)
and a transformation model based on the imaging physics.
23
Computation of a brain mask as the intersection of individual brain masks computed on each diffusion-
weighted MR image by histogram analysis.
24
Estimation of the diffusion tensor by multiple linear regression on a voxel-by-voxel basis.
25
Diagonalisation of the diffusion tensor by classical linear algebra (computation of the eigenvalues and
eigenvectors).
Computation of different quantitative MR diffusion parameters and related 3D maps: Appar-
ent Diffusion Coefficient (ADC), Fractional Anisotropy (FA), Relative Anisotropy (RA), direction of the
eigenvector with the largest eigenvalue.
Proc. of SPIE Vol. 6512 65121I-2

2.3. Computation of the mid-sagittal plane
2.3.1. Some notations
I is a diffusion tensor image, defined on a regular grid G
For the sake of simplicity and without loss of generality we assume that G is isotropic with voxel size 1
mm
3
v is a voxel on the grid G, and its value I(v) is a tensor (symmetric positive definite (SPD) 3x3 matrix)
a(b) is the value of the function a for argument b, whereas ab is the product of matrices a and b;inthe
following, v is considered as either the argument of a function or as a column vector, depending on the
context
S
P
is the reflection symmetry with respect to the plane P
M
T
is the transpose of matrix M
x, y and z are the left-right, antero-posterior and top-bottom axes, respectively
2.4. Formulation of the problem
If the brain was perfectly symmetrical, there would exist a symmetry plane P superposing each voxel v of the
grid with its anatomical counterpart. If we note v
= S
P
(v), it means that the tensors I(v)andI(v
) would be
mirror-symmetrical with respect to P , which writes:
I(v)=S
P
(I(v
))
In real life, the brain is only grossly symmetrical, and such an ideal symmetry plane does not exist. Instead,
we define the approximate symmetry (or mid-sagittal) plane P as the plane minimising the average distance
between the tensor I(v) and the reflection of I(v
) with respect to P , where v
= S
P
(v) is the voxel homologous
to v in the contralateral hemisphere. This average distance would be null for a perfect symmetry plane. The
optimisation problem can then be written as:
˜
P = arg min
P
1
card(G
P
)
v G
P
d (I(v),S
P
(I(v
))) , where:
v
= S
P
(v)
G
P
is the overlapping area between the two sets of voxels to compare: G
P
= G S
P
(G)
d(., .) is a metric on tensors
2.5. Implementation details
2.5.1. Parameterisation of P
Any plane in IR
3
can be characterised by a normal unit vector n and its distance d to the origin. The unit vector
n is characterised by its angles α and β in a spherical coordinate system, where:
α is the latitude angle (the equator is in the xy-plane)
β is the longitude angle (the prime meridian is in the xz-plane)
Proc. of SPIE Vol. 6512 65121I-3

The coordinates of n are:
n
T
=
cos(β)cos(α)sin(β)cos(α)sin(α)
If we note H = Id 2nn
T
(Id is the 3x3 identity matrix) and t =2dn, then it can be easily shown that the
reflection of the voxel v and of the tensor I(v) with respect to P are equal to, respectively:
S
P
(v)=Hv + t and S
P
(I(v)) = HI(v)H
T
= HI(v)H,asH = H
T
2.5.2. Interpolation scheme
For an arbitrary plane P , the voxel v
= S
P
(v) does not necessarily coincide with a grid point of G.Thus,an
interpolation is needed to estimate the tensor value I(v
) at this non-grid point. A natural choice would be to
interpolate in log-Euclidean or affine-invariant frameworks, which would preserve the constraint of symmetric
positive definiteness.
26–29
This unfortunately leads to a high computational cost and a prohibitive computation
time for the overall algorithm. Our choice is to simply use the nearest neighbor interpolation instead, which also
guarantees that the interpolated value is a SPD matrix.
2.5.3. Distance function
The distance function d is simply based on the Frobenius norm. For two tensors C and D, d is defined as:
d(C, D)=
3
i=1
3
j=1
(C
ij
D
ij
)
2
2.5.4. Optimisation
We choose the algorithm NEWUOA (NEW Unconstrained Optimisation Algorithm) to solve the optimisation
problem. This algorithm, originally proposed by M.J.D. Powell,
30
has been shown to be significantly more
accurate, more robust and faster than other deterministic, derivative-free algorithms such as Nelder-Mead’s
downhill simplex or Powell’s direction set algorithms for intensity-based rigid-body image registration.
31
The
three parameters of interest are α, β and d (that define the plane P and thus the associated reflection symmetry
S
P
). The algorithm is initialised with a plane at the centre of the image grid, orthogonal to the left-right axis.
A multiresolution (simple decimation) approach is implemented to increase the robustness, accuracy and
speed of the algorithm. A subsampled image is built by taking only one voxel out of f
x
(resp. f
y
, f
z
)inthe
x-(resp. y-, z-) direction in the original image. A first solution is found at this low resolution, and is used to
initialise the algorithm at a higher resolution, and so on. In this coarse-to-fine scheme, typically two or three
levels can be used, depending on the resolution of the original image.
2.5.5. Realignment of the plane
Once the MSP is estimated, it is realigned at the centre of the image grid for improved image display.
7
Such
a realignment is performed by applying the rigid transformation R to both the anatomical and the tensor
information.
32
The rigid transformation R is computed as:
R =(S
K
S
P
)
1/2
where S
P
is the reflection symmetry with respect to the estimated mid-sagittal plane and S
K
is the reflection
symmetry with respect to K, the plane in the middle of the image grid. Let r be the rotation part of R.Each
tensor I(v) of the image I is rotated following: R(I(v)) = rI(v)r
T
. The log-Euclidean interpolation
28
is used for
computation of the realigned DTI. The overall computation time for a two-level multiresolution scheme is about
13 minutes for a typical DT image (matrix size 256 × 256 and 60 slices) on a standard PC (OS Linux), with an
Intel Xeon 4 CPU at 2.8GHz, 2 GBytes of RAM.
Proc. of SPIE Vol. 6512 65121I-4

Figures (4)
Citations
More filters

Proceedings ArticleDOI
12 Apr 2007
TL;DR: A new optimisation algorithm is introduced (called NEWUOA) to address the above registration problems, and its robustness and accuracy properties are demonstrated.
Abstract: A number of problems frequently encountered in brain image analysis can be conveniently solved within a registration framework, such as alignment of mono- or multi-sequence magnetic resonance images (MRI) for single or multiple subjects, computation of the cerebral mid-sagittal plane in anatomical or diffusion-tensor MRI, correction of acquisition distortions in diffusion-weighted MRI, etc. A widely used approach for registration tasks consists of maximising a similarity criterion between the intensities of the images to be matched. In this context, efficient optimisation methods are needed to obtain good performances. In this paper, we introduce a new optimisation algorithm (called NEWUOA) to address the above registration problems, and we demonstrate its robustness and accuracy properties

16 citations


Journal ArticleDOI
TL;DR: This paper presents a fast and robust MSP extraction method based on 3D scale-invariant feature transform (SIFT), which can match multiple pairs of 3D SIFT features in parallel and solve the optimal MSP on-the-fly.
Abstract: Midsagittal plane (MSP) extraction from 3D brain images is considered as a promising technique for human brain symmetry analysis In this paper, we present a fast and robust MSP extraction method based on 3D scale-invariant feature transform (SIFT) Unlike the existing brain MSP extraction methods, which mainly rely on the gray similarity, 3D edge registration or parameterized surface matching to determine the fissure plane, our proposed method is based on distinctive 3D SIFT features, in which the fissure plane is determined by parallel 3D SIFT matching and iterative least-median of squares plane regression By considering the relative scales, orientations and flipped descriptors between two 3D SIFT features, we propose a novel metric to measure the symmetry magnitude for 3D SIFT features By clustering and indexing the extracted SIFT features using a k-dimensional tree (KD-tree) implemented on graphics processing units, we can match multiple pairs of 3D SIFT features in parallel and solve the optimal MSP on-the-fly The proposed method is evaluated by synthetic and in vivo datasets, of normal and pathological cases, and validated by comparisons with the state-of-the-art methods Experimental results demonstrated that our method has achieved a real-time performance with better accuracy yielding an average yaw angle error below 091° and an average roll angle error no more than 089°

12 citations


Journal ArticleDOI
TL;DR: This article presents a fast and robust symmetry detection method for automatically extracting symmetry axis (fissure line) from a brain image based on a set of scale‐invariant feature transform (SIFT) features, where the symmetry axis is determined by parallel matching and voting of distinctive features within the brain image.
Abstract: Symmetry analysis for brain images has been considered as a promising technique for automatically extracting the pathological brain slices in conventional scanning. In this article, we present a fast and robust symmetry detection method for automatically extracting symmetry axis (fissure line) from a brain image. Unlike the existing brain symmetry detection methods which mainly rely on the intensity or edges to determine the symmetry axis, our proposed method is based on a set of scale-invariant feature transform (SIFT) features, where the symmetry axis is determined by parallel matching and voting of distinctive features within the brain image. By clustering and indexing the extracted SIFT features using a GPU KD-tree, we can match multiple pairs of features in parallel based on a novel symmetric similarity metric, which combines the relative scales, orientations, and flipped descriptors to measure the magnitude of symmetry between each pair of features. Finally, the dominant symmetry axis presented in the brain image is determined using a parallel voting algorithm by accumulating the pair-wise symmetry score in a Hough space. Our method was evaluated on both synthetic and in vivo datasets, including both normal and pathological cases. Comparisons with state-of-the-art methods were also conducted to validate the proposed method. Experimental results demonstrated that our method achieves a real-time performance and with a higher accuracy than previous methods, yielding an average polar angle error within 0.69° and an average radius error within 0.71 mm. © 2013 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 23, 314–326, 2013

6 citations


06 Sep 2008
TL;DR: The fully automated approach is superior to manual or semi-automated DT-MRI analyses regarding the robustness of the results (reproducibility and accuracy) and combining different preprocessing methods (3 estimation methods and 3 distortion correction methods) has little impact on such results.
Abstract: Diffusion tensor MRI (DT-MRI) allows the in vivo assessment of the abnormalities of white matter in multiple sclerosis (MS). DT-MRI is complementary to conventional MRI sequences where such abnormalities are often not visible. Most studies have shown differences of mean diffusivity (MD) and fractional anisotropy (FA) between patients and controls in MS lesions (MSL) and normal appearing white matter (NAWM) based on histogram analyses. However, the majority of these studies are based on histogram analysis, i.e. local information of DT-MRI is lost, and moreover a number of those studies were not conclusive, partly explained by methodological issues, because these tensor indices vary within the brain, which is likely to make such global, histogram-based analyses, fail. Here we propose a new framework to compare these indices between MSL and NAWM and between two populations (patients and controls). First, MSL are manually delineated in MS patients. The mid-sagittal plane is then automatically computed, allowing to define a contralateral region of interest (ROI) in NAWM for each MSL. This allows the local comparison of DTI indices in anatomically similar regions in each MS patient. Second, each MS patient is linearly registered to each control subject, and the same left-right comparison between MSL and contralateral NAWM is then performed in controls. The results (ANOVA with multiple comparisons procedure) show that 1) FA values are lower in MSL than in contralateral NAWM in MS patients (p < 0.05) but not in controls, 2) FA values are lower in MS patients (MSL and contralateral NAWM) compared to controls (p < 0.05), 3) MD values are not different between MSL/contralateral NAWM in MS patients and controls. We also show that combining different preprocessing methods (3 estimation methods and 3 distortion correction methods) has little impact on such results. Nevertheless, our fully automated approach is superior to manual or semi-automated DT-MRI analyses regarding the robustness of the results (reproducibility and accuracy).

2 citations


Cites methods from "Computation of the mid-sagittal pla..."

  • ...The contralateral ROIs are automatically computed using the mid-sagittal plane as a reference [26]....

    [...]


Proceedings ArticleDOI
12 Mar 2018
TL;DR: This work proposes a plane computation based on the structure of interest, directly on Diffusion Tensor Images (DTI), through the DTI-based divergence map, and explores the high organization of the fibers in the Corpus Callosum to establish a reference system that can be used to perform 2D CC studies.
Abstract: The Corpus Callosum (CC) is the largest white matter structure in the brain and subject of many relevant studies. In order to properly analyze this structure in 2D studies, the midsagittal plane (MSP) determination of the CC is required. Usually, this computation is done on structural MR images and transformed to diffusion space when necessary. Furthermore, most existing methods take into account the whole brain structure instead of only the object of study. Differently, our work proposes a plane computation based on the structure of interest, directly on Diffusion Tensor Images (DTI), through the DTI-based divergence map.1 Since our plane is computed in the diffusion domain, the method explores the high organization of the fibers in the CC to establish a reference system that can be used to perform 2D CC studies, while most existing MSP computation algorithms are based on structural characteristics of the brain, such as shape symmetry and inter-hemispheric fissure location. Experiments showed that the proposed method is reliable regarding repeatability and parameters choices. Results also indicate that the callosal fiber convergence plane (CFCP) found by our method is similar to MSP in most subjects. Nevertheless, when the CC is not well aligned with the brain intercommissural fissure, CFCP and MSP presented significant differences.

1 citations


References
More filters

Journal ArticleDOI

31,977 citations


"Computation of the mid-sagittal pla..." refers methods in this paper

  • ...• Computation of a brain mask as the intersection of individual brain masks computed on each diffusionweighted MR image by histogram analysis.(24) • Estimation of the diffusion tensor by multiple linear regression on a voxel-by-voxel basis....

    [...]


Journal ArticleDOI
TL;DR: This paper proposes to endow the tensor space with an affine-invariant Riemannian metric and demonstrates that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries, the geodesic between two tensors and the mean of a set of tensors are uniquely defined.
Abstract: Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc. We have previously shown that the Riemannian metric provides a powerful framework for generalizing statistics to manifolds. In this paper, we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. For instance, most interpolation and Gaussian filtering schemes can be tackled efficiently through a weighted mean computation. Linear and anisotropic diffusion schemes can be adapted to our Riemannian framework, through partial differential evolution equations, provided that the metric of the tensor space is taken into account. For that purpose, we provide intrinsic numerical schemes to compute the gradient and Laplace-Beltrami operators. Finally, to enforce the fidelity to the data (either sparsely distributed tensors or complete tensors fields) we propose least-squares criteria based on our invariant Riemannian distance which are particularly simple and efficient to solve.

1,409 citations


Journal ArticleDOI
Abstract: This article treats the theoretical underpinnings of diffusion-tensor magnetic resonance imaging (DT-MRI), as well as experimental design and data analysis issues. We review the mathematical model underlying DT-MRI, discuss the quantitative parameters that are derived from the measured effective diffusion tensor, and describe artifacts that arise in typical DT-MRI acquisitions. We also discuss difficulties in identifying appropriate models to describe water diffusion in heterogeneous tissues, as well as in interpreting experimental data obtained in such issues. Finally, we describe new statistical methods that have been developed to analyse DT-MRI data, and their potential uses in clinical and multi-site studies.

1,338 citations


Additional excerpts

  • ...• Estimation of the diffusion tensor by multiple linear regression on a voxel-by-voxel basis.(25) • Diagonalisation of the diffusion tensor by classical linear algebra (computation of the eigenvalues and eigenvectors)....

    [...]


Journal ArticleDOI
TL;DR: A new family of Riemannian metrics called Log‐Euclidean is proposed, based on a novel vector space structure for tensors, which can be converted into Euclidean ones once tensors have been transformed into their matrix logarithms.
Abstract: Diffusion tensor imaging (DT-MRI or DTI) is an emerging imaging modality whose importance has been growing considerably. However, the processing of this type of data (i.e., symmetric positive-definite matrices), called "tensors" here, has proved difficult in recent years. Usual Euclidean operations on matrices suffer from many defects on tensors, which have led to the use of many ad hoc methods. Recently, affine-invariant Riemannian metrics have been proposed as a rigorous and general framework in which these defects are corrected. These metrics have excellent theoretical properties and provide powerful processing tools, but also lead in practice to complex and slow algorithms. To remedy this limitation, a new family of Riemannian metrics called Log-Euclidean is proposed in this article. They also have excellent theoretical properties and yield similar results in practice, but with much simpler and faster computations. This new approach is based on a novel vector space structure for tensors. In this framework, Riemannian computations can be converted into Euclidean ones once tensors have been transformed into their matrix logarithms. Theoretical aspects are presented and the Euclidean, affine-invariant, and Log-Euclidean frameworks are compared experimentally. The comparison is carried out on interpolation and regularization tasks on synthetic and clinical 3D DTI data.

1,050 citations


Journal ArticleDOI
TL;DR: Directionally encoded color (DEC) maps of the human brain obtained using empirical or heuristic schemes clearly show the main association, projection, and commissural white matter pathways.
Abstract: This paper investigates the use of color to represent the directional information contained in the diffusion tensor. Ideally, one wants to take into account both the properties of human color vision and of the given display hardware to produce a representation in which differences in the orientation of anisotropic structures are proportional to the perceived differences in color. It is argued here that such a goal cannot be achieved in general and therefore, empirical or heuristic schemes, which avoid some of the common artifacts of previously proposed approaches, are implemented. Directionally encoded color (DEC) maps of the human brain obtained using these schemes clearly show the main association, projection, and commissural white matter pathways. In the brainstem, motor and sensory pathways are easily identified and can be differentiated from the transverse pontine fibers and the cerebellar peduncles. DEC maps obtained from diffusion tensor imaging data provide a simple and effective way to visualize fiber direction, useful for investigating the structural anatomy of different organs. Magn Reson Med 42:526‐540, 1999. r 1999 Wiley-Liss, Inc.

863 citations


Frequently Asked Questions (2)
Q1. What contributions have the authors mentioned in the paper "Computation of the mid-sagittal plane in diffusion tensor mr brain images" ?

The authors propose a method for the automated computation of the mid-sagittal plane of the brain in diffusion tensor MR images. Finally, the authors show some results on real diffusion tensor MR images. 

In this paper, the authors have proposed a method for the automated computation of the mid-sagittal plane of the brain in diffusion tensor MR images. In the future, the authors plan to experiment other implementation choices and compare them with the present method.