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Characterization of anisotropy in high angular resolution diffusionweighted MRI
 Magn. Reson. Med
, 2002
"... The methods of group theory are applied to the problem of characterizing the diffusion measured in high angular resolution MR experiments. This leads to a natural representation of the local diffusion in terms of spherical harmonics. In this representation, it is shown that isotropic diffusion, anis ..."
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Cited by 160 (0 self)
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The methods of group theory are applied to the problem of characterizing the diffusion measured in high angular resolution MR experiments. This leads to a natural representation of the local diffusion in terms of spherical harmonics. In this representation, it is shown that isotropic diffusion, anisotropic diffusion from a single fiber, and anisotropic diffusion from multiple fiber directions fall into distinct and separable channels. This decomposition can be determined for any voxel without any prior information by a spherical harmonic transform, and for special cases the magnitude and orientation of the local diffusion may be determined. Moreover, nondiffusion–related asymmetries produced by experimental artifacts fall into channels distinct from the fiber channels, thereby allowing their separation and a subsequent reduction in noise from the reconstructed fibers. In the case of a single fiber, the method
Qball imaging
 Magnetic Resonance in Medicine
, 2004
"... Magnetic resonance diffusion tensor imaging (DTI) provides a powerful tool for mapping neural histoarchitecture in vivo. However, DTI can only resolve a single fiber orientation within each imaging voxel due to the constraints of the tensor model. For example, DTI cannot resolve fibers crossing, be ..."
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Cited by 153 (0 self)
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Magnetic resonance diffusion tensor imaging (DTI) provides a powerful tool for mapping neural histoarchitecture in vivo. However, DTI can only resolve a single fiber orientation within each imaging voxel due to the constraints of the tensor model. For example, DTI cannot resolve fibers crossing, bending, or twisting within an individual voxel. Intravoxel fiber crossing can be resolved using qspace diffusion imaging, but qspace imaging requires large pulsed field gradients and timeintensive sampling. It is also possible to resolve intravoxel fiber crossing using mixture model decomposition of the high angular resolution diffusion imaging (HARDI) signal, but mixture modeling requires a model of the underlying diffusion process. Recently, it has been shown that the HARDI signal can be reconstructed modelindependently using a spherical tomographic inversion called the Funk–Radon transform, also known as the spherical Radon transform. The resulting imaging method, termed qball imaging, can resolve multiple intravoxel fiber orientations and does not require any assumptions on the diffusion process such as Gaussianity or multiGaussianity. The present paper reviews the theory of qball imaging and describes a simple linear matrix formulation for the qball reconstruction based on spherical radial basis function interpolation. Open aspects of the qball reconstruction algorithm are
Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging
 Magn. Reson. Med
, 2003
"... A new method for mapping diffusivity profiles in tissue is presented. The BlochTorrey equation is modified to include a diffusion term with an arbitrary rank Cartesian tensor. This equation is solved to give the expression for the generalized StejskalTanner formula quantifying diffusive attenuatio ..."
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Cited by 103 (5 self)
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A new method for mapping diffusivity profiles in tissue is presented. The BlochTorrey equation is modified to include a diffusion term with an arbitrary rank Cartesian tensor. This equation is solved to give the expression for the generalized StejskalTanner formula quantifying diffusive attenuation in complicated geometries. This makes it possible to calculate the components of higherrank tensors without using the computationallydifficult spherical harmonic transform. General theoretical relations between the diffusion tensor (DT) components measured by traditional (rank2) DT imaging (DTI) and 3D distribution of diffusivities, as measured by high angular resolution diffusion imaging (HARDI) methods, are derived. Also, the spherical tensor components from HARDI are related to the rank2 DT. The relationships between higher and lowerrank Cartesian DTs are also presented. The inadequacy of the traditional
Mapping human wholebrain structural networks with diffusion MRI.
 PLoS One
, 2007
"... Understanding the largescale structural network formed by neurons is a major challenge in system neuroscience. A detailed connectivity map covering the entire brain would therefore be of great value. Based on diffusion MRI, we propose an efficient methodology to generate large, comprehensive and i ..."
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Cited by 47 (2 self)
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Understanding the largescale structural network formed by neurons is a major challenge in system neuroscience. A detailed connectivity map covering the entire brain would therefore be of great value. Based on diffusion MRI, we propose an efficient methodology to generate large, comprehensive and individual white matter connectional datasets of the living or dead, human or animal brain. This noninvasive tool enables us to study the basic and potentially complex network properties of the entire brain. For two human subjects we find that their individual brain networks have an exponential node degree distribution and that their global organization is in the form of a small world. Citation: Hagmann P, Kurant M, Gigandet X, Thiran P, Wedeen VJ, et al (2007) Mapping Human WholeBrain Structural Networks with Diffusion MRI. PLoS ONE 2(7): e597.
MR connectomics: principles and challenges.
 J. Neurosci. Methods
, 2010
"... a b s t r a c t MR connectomics is an emerging framework in neuroscience that combines diffusion MRI and whole brain tractography methodologies with the analytical tools of network science. In the present work we review the current methods enabling structural connectivity mapping with MRI and show ..."
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Cited by 33 (5 self)
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a b s t r a c t MR connectomics is an emerging framework in neuroscience that combines diffusion MRI and whole brain tractography methodologies with the analytical tools of network science. In the present work we review the current methods enabling structural connectivity mapping with MRI and show how such data can be used to infer new information of both brain structure and function. We also list the technical challenges that should be addressed in the future to achieve highresolution maps of structural connectivity. From the resulting tremendous amount of data that is going to be accumulated soon, we discuss what new challenges must be tackled in terms of methods for advanced network analysis and visualization, as well data organization and distribution. This new framework is well suited to investigate key questions on brain complexity and we try to foresee what fields will most benefit from these approaches.
Generalized scalar measures for diffusion MRI using trace, variance and entropy
 Magn. Reson. Med. 53:866–876. Biophysical Journal 94(7) 2809–2818 Özarslan et al
, 2005
"... This paper details the derivation of rotationally invariant scalar measures from higherrank diffusion tensors (DTs) and functions defined on a unit sphere. This was accomplished with the use of an expression that generalizes the evaluation of the trace operator to tensors of arbitrary rank, and eve ..."
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Cited by 31 (6 self)
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This paper details the derivation of rotationally invariant scalar measures from higherrank diffusion tensors (DTs) and functions defined on a unit sphere. This was accomplished with the use of an expression that generalizes the evaluation of the trace operator to tensors of arbitrary rank, and even to functions whose domains are the unit sphere. It is shown that the mean diffusivity is invariant to the selection of tensor rank for the model used. However, this rank invariance does not apply to the anisotropy measures. Therefore, a variancebased, general anisotropy measure is proposed. Also an information theoretical parametrization of anisotropy is introduced that is frequently more consistent with the meaning attributed to anisotropy. We accomplished this by associating anisotropy with the amount of orientational information present in the data, regardless of the imaging technique used. Using a simplified model of fibrous tissue, we simulated anisotropy values with varying orientational complexity and tensor models. Simulations suggested that a lowerrank tensor model may produce artificially low anisotropy values in voxels with complex structure. This was confirmed with a spinecho experiment performed on an excised
Superresolution in MRI: application to human white matter fiber tract visualization by diffusion tensor imaging. Magn Reson Med 2001;45:29– 35
"... A superresolution algorithm was applied to spatially shifted, singleshot, diffusionweighted brain images to generate a new image with increased spatial resolution. Detailed twodimensional white matter fiber tract maps of the human brain resulting from application of the technique are shown. The m ..."
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Cited by 30 (0 self)
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A superresolution algorithm was applied to spatially shifted, singleshot, diffusionweighted brain images to generate a new image with increased spatial resolution. Detailed twodimensional white matter fiber tract maps of the human brain resulting from application of the technique are shown. The method provides a new means for improving the resolution in cases where kspace segmentation is difficult to implement. Diffusionweighted imaging and diffusion tensor imaging in vivo stand to benefit in particular because the necessity of obtaining highresolution scans is matched by the difficulty in obtaining them. Magn Reson Med
P.: AxCaliber: a method for measuring axon diameter distribution from diffusion MRI. Magnetic Resonance in Medicine 59(6
, 2008
"... Myeloarchitecture refers to different histological features, e.g., axonal density, myelin basic protein content, oligodendrocyte count and the axon diameter distribution, with which one can distinguish different white matter bundles. These parameters have been measured by histological methods but co ..."
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Cited by 30 (1 self)
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Myeloarchitecture refers to different histological features, e.g., axonal density, myelin basic protein content, oligodendrocyte count and the axon diameter distribution, with which one can distinguish different white matter bundles. These parameters have been measured by histological methods but could not be quantified by conventional magnetic resonance imaging (MRI). The composite hindered and restricted model of diffusion (CHARMED) MRI framework can be used to measure at least two of these histological features: the axonal density (i.e., the volume
Relationships between diffusion tensor and qspace MRI
 MRM
"... Fundamental relationships between diffusion tensor (DT) and 3D qspace MRI are derived which establish conditions when these two complementary MR methods are equivalent. It is shown that the displacement distribution measured by qspace MRI in both the large displacement (i.e., large r) and the long ..."
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Cited by 26 (1 self)
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Fundamental relationships between diffusion tensor (DT) and 3D qspace MRI are derived which establish conditions when these two complementary MR methods are equivalent. It is shown that the displacement distribution measured by qspace MRI in both the large displacement (i.e., large r) and the longwavelength (i.e., small q) limits is the same 3D Gaussian displacement distribution assumed in DTMRI. In these limiting cases, qspace MR yields a dispersion tensor that is identical to the effective DT, D, measured in DTMRI. An experiment is then proposed to measure D using qspace methods. These findings establish that the effective DT, measured in DTMRI, characterizes molecule motions on a coarse lengthscale. Finally, the feasibility of and requirements for performing 3D qspace MRI on a clinical scanner are considered. Magn Reson Med 47:
Maximum likelihood estimation of signal amplitude and noise variance from MR data. Magnetic Resonance in Medicine
, 2004
"... In magnetic resonance imaging, the raw data, which are acquired in spatial frequency space, are intrinsically complex valued and corrupted by Gaussian distributed noise. After applying an inverse Fourier transform the data remain complex valued and Gaussian distributed. If the signal amplitude is t ..."
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Cited by 22 (0 self)
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In magnetic resonance imaging, the raw data, which are acquired in spatial frequency space, are intrinsically complex valued and corrupted by Gaussian distributed noise. After applying an inverse Fourier transform the data remain complex valued and Gaussian distributed. If the signal amplitude is to be estimated, one has two options. It can be estimated directly from the complex valued data set, or one can first perform a magnitude operation on this data set, which changes the distribution of the data from Gaussian to Rician, and estimate the signal amplitude from the thus obtained magnitude image. Similarly, the noise variance can be estimated from both the complex and magnitude data sets. This paper addresses the question whether it is better to use complex valued data or magnitude data for the estimation of these parameters using the Maximum Likelihood method. As a performance criterion, the meansquared error (MSE) is used. 1