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Rachid Deriche - One of the best experts on this subject based on the ideXlab platform.
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A Computational Framework For Generating Rotation Invariant Features And Its Application In Diffusion MRI
Medical Image Analysis, 2020Co-Authors: Mauro Zucchelli, Samuel Deslauriers-gauthier, Rachid DericheAbstract:In this work, we present a novel computational framework for analytically generating a complete set of algebraically independent Rotation Invariant Features (RIF) given the Laplace-series expansion of a Spherical Function. Our computational framework provides a closed-form solution for these new invariants, which are the natural expansion of the well known Spherical mean, power-spectrum and bispectrum invariants. We highlight the maximal number of algebraically independent invariants which can be obtained from a truncated Spherical Harmonic (SH) representation of a Spherical Function and show that most of these new invariants can be linked to statistical and geometrical measures of Spherical Functions, such as the mean, the variance and the volume of the Spherical signal. Moreover, we demonstrate their application to dMRI signal modeling including the Apparent Diffusion Coefficient (ADC), the diffusion signal and the fiber Orientation Distribution Function (fODF). In addition, using both synthetic and real data, we test the ability of our invariants to estimate brain tissue microstructure in healthy subjects and show that our framework provides more flexibility and open up new opportunities for innovative development in the domain of microstructure recovery from diffusion MRI.
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A Closed-Form Solution of Rotation Invariant Spherical Harmonic Features in Diffusion MRI
2018Co-Authors: Mauro Zucchelli, Samuel Deslauriers-gauthier, Rachid DericheAbstract:Rotation invariant features are an indispensable tool for characterizing diffusion Magnetic Resonance Imaging (MRI) and in particular for brain tissue microstructure estimation. In this work, we propose a new mathematical framework for efficiently calculating a complete set of such invariants from any Spherical Function. Specifically, our method is based on the Spherical harmonics series expansion of a given Function of any order and can be applied directly to the resulting coefficients by performing a simple integral operation analytically. This enable us to derive a general closed-form equation for the invariants. We test our invariants on the diffusion MRI fiber orientation distribution Function obtained from the diffusion signal both in-vivo and in synthetic data. Results show how it is possible to use these invariants for characterizing the white matter using a small but complete set of features.
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Choosing Tractography Parameters to Improve Connectivity Mapping
2014Co-Authors: Girard Gabriel, Rachid Deriche, Whittingstall Kevin, Maxime DescoteauxAbstract:Diffusion-weighted imaging (DWI) is often used as a starting point for in vivo white matter (WM) connectivity to reconstruct potential WM pathways between brain areas. Tractography algorithms have many parameters which can influence reconstruction and connectivity. Various choices of parameters have been proposed. But how does one choose the best set of parameters? In this study, we varied three critical parameters while monitoring connectivity score using the Tractometer evaluation system on the International Symposium on Biomedical Imaging (ISBI) Challenge synthetic dataset. The three parameters were: * θ: The maximum deviation angle between two consecutive tractography steps. This addresses the hypothesis of smoothness of the WM pathways. * τ: The Spherical Function (SF) threshold. This aims at removing noisy propagation directions during the tractography process. * τ init : The initial SF threshold. This aims at removing initial noise at the seeds and to start tractography in a good tangent direction to the WM bundle.
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A polynomial approach for extracting the extrema of a Spherical Function and its application in diffusion MRI
Medical Image Analysis, 2013Co-Authors: Aurobrata Ghosh, Elias Tsigaridas, Bernard Mourrain, Rachid DericheAbstract:Antipodally symmetric Spherical Functions play a pivotal role in diffusion MRI in representing sub-voxel-resolution microstructural information of the underlying tissue. This information is described by the geometry of the Spherical Function. In this paper we propose a method to automatically compute all the extrema of a Spherical Function. We then classify the extrema as maxima, minima and saddle-points to identify the maxima. We take advantage of the fact that a Spherical Function can be described equivalently in the Spherical harmonic (SH) basis, in the symmetric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. We extract the extrema of the Spherical Function by computing the stationary points of its constrained HP representation. Instead of using traditional optimization approaches, which are inherently local and require exhaustive search or re-initializations to locate multiple extrema, we use a novel polynomial system solver which analytically brackets all the extrema and refines them numerically, thus missing none and achieving high precision. To illustrate our approach we consider the Orientation Distribution Function (ODF). In diffusion MRI the ODF is a Spherical Function which represents a state-of-the-art reconstruction algorithm whose maxima are aligned with the dominant fiber bundles. It is, therefore, vital to correctly compute these maxima to detect the fiber bundle directions. To demonstrate the potential of the proposed polynomial approach we compute the extrema of the ODF to extract all its maxima. This polynomial approach is, however, not dependent on the ODF and the framework presented in this paper can be applied to any Spherical Function described in either the SH basis, ST basis or the HP basis.
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A Polynomial Based Approach to Extract Fiber Directions from the ODF and its Experimental Validation
2009Co-Authors: Aurobrata Ghosh, Maxime Descoteaux, Elias Tsigaridas, Rachid DericheAbstract:In Diffusion MRI, Spherical Functions are commonly employed to represent the diffusion information. The ODF is an intuitive Spherical Function since its maxima are aligned with the dominant fiber directions. Therefore, it is important to correctly determine these maximal directions, as they are the key to tracing fiber tracts. A tractography algorithm will suffer from cumulative error when the maximal directions are incorrectly estimated locally. The goal of this work is to present a polynomial based approach for estimating the maximal directions correctly. The paper will also present a measure of the correctness of the estimation. This approach will be tested on synthetic, phantom, and real data, and will be compared to an existing discrete “mesh-search” approach [2]. It will be shown how this approach naturally overcomes the inherent shortcomings of the discrete search. Finally, although, the approach is demonstrated on the ODF, it can be equally applied to any Spherical Function.
Maxime Descoteaux - One of the best experts on this subject based on the ideXlab platform.
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Choosing Tractography Parameters to Improve Connectivity Mapping
2014Co-Authors: Girard Gabriel, Rachid Deriche, Whittingstall Kevin, Maxime DescoteauxAbstract:Diffusion-weighted imaging (DWI) is often used as a starting point for in vivo white matter (WM) connectivity to reconstruct potential WM pathways between brain areas. Tractography algorithms have many parameters which can influence reconstruction and connectivity. Various choices of parameters have been proposed. But how does one choose the best set of parameters? In this study, we varied three critical parameters while monitoring connectivity score using the Tractometer evaluation system on the International Symposium on Biomedical Imaging (ISBI) Challenge synthetic dataset. The three parameters were: * θ: The maximum deviation angle between two consecutive tractography steps. This addresses the hypothesis of smoothness of the WM pathways. * τ: The Spherical Function (SF) threshold. This aims at removing noisy propagation directions during the tractography process. * τ init : The initial SF threshold. This aims at removing initial noise at the seeds and to start tractography in a good tangent direction to the WM bundle.
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A Polynomial Based Approach to Extract Fiber Directions from the ODF and its Experimental Validation
2009Co-Authors: Aurobrata Ghosh, Maxime Descoteaux, Elias Tsigaridas, Rachid DericheAbstract:In Diffusion MRI, Spherical Functions are commonly employed to represent the diffusion information. The ODF is an intuitive Spherical Function since its maxima are aligned with the dominant fiber directions. Therefore, it is important to correctly determine these maximal directions, as they are the key to tracing fiber tracts. A tractography algorithm will suffer from cumulative error when the maximal directions are incorrectly estimated locally. The goal of this work is to present a polynomial based approach for estimating the maximal directions correctly. The paper will also present a measure of the correctness of the estimation. This approach will be tested on synthetic, phantom, and real data, and will be compared to an existing discrete “mesh-search” approach [2]. It will be shown how this approach naturally overcomes the inherent shortcomings of the discrete search. Finally, although, the approach is demonstrated on the ODF, it can be equally applied to any Spherical Function.
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a polynomial based approach to extract the maxima of an antipodally symmetric Spherical Function and its application to extract fiber directions from the orientation distribution Function in diffusion mri
Medical Image Computing and Computer-Assisted Intervention, 2008Co-Authors: Aurorata Ghosh, Maxime Descoteaux, Elias Tsigaridas, Bernard Mourrain, Pierre Comon, Rachid DericheAbstract:In this paper we extract the geometric characteristics from an antipodally symmetric Spherical Function (ASSF), which can be de- scribed equivalently in the Spherical harmonic (SH) basis, in the symmet- ric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. All three bases span the same vector space and are bijective when the rank of the SH series equals the order of the ST and equals the degree of the HP. We show, therefore, how it is possible to extract the maxima and minima of an ASSF by computing the stationary points of a constrained HP. In Diffusion MRI, the Orientation Distribution Function (ODF), repre- sents a state of the art reconstruction method whose maxima are aligned with the dominant fiber bundles. It is, therefore, important to be able to correctly estimate these maxima to detect the fiber directions. The ODF is an ASSF. To illustrate the potential of our method, we take up the example of the ODF, and extract its maxima to detect the fiber directions. Thanks to our method we are able to extract the maxima without limiting our search to a discrete set of values on the sphere, but by searching the maxima of a continuous Function. Our method is also general, not dependent on the ODF, and the framework we present can be applied to any ASSF described in one of the three bases.
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High Angular Resolution Diffusion MRI: from Local Estimation to Segmentation and Tractography
2008Co-Authors: Maxime DescoteauxAbstract:At the current resolution of diffusion-weighted (DW) magnetic resonance imaging (MRI), research groups agree that there are between one third to two thirds of imaging voxels in the human brain white matter that contain fiber crossing bundles. This thesis tackles the important problem of recovering crossing fiber bundles from DWMRI measurements. The main goal is to overcome the limitations of diffusion tensor imaging (DTI). It is well-known that imaging voxels where there are multiple fiber crossings produce a non-Gaussian DW signal. This is precisely where DTI is limited due to the intrinsic Gaussian assumption of the technique. Hence, this thesis is dedicated to the development of local reconstruction methods, segmentation and tractography algorithms able to infer multiple fiber crossing from DW-MRI data. To do so, high angular resolution diffusion imaging (HARDI) is used to measure DW images along several directions. Q-ball imaging (QBI) is a recent such HARDI technique that reconstructs the diffusion orientation distribution Function (ODF), a Spherical Function that has its maxima aligned with the underlying fiber directions at every voxel. QBI and the diffusion ODF will play a central role in this thesis. There are many original contributions in this thesis. First, we propose a robust estimation of the HARDI signal using a closed-form regularization algorithm based on the Spherical harmonics. Then, we estimate the apparent coefficient coefficient (ADC) to study HARDI anisotropy measures and to discriminate voxels with underlying isotropic, single fiber andmultiple fiber distributions. Next, we develop a linear, robust and analytical QBI solution using the Spherical harmonic basis, which is used in a new statistical region-based active contour algorithmto segment important white matter fiber bundles. In addition, we develop a new Spherical deconvolution sharpening method that transforms the diffusion q-ball ODF into a fiber ODF. Finally, we propose a new deterministic tractography algorithm and a new probabilistic tractography algorithm exploiting the full distribution of the fiber ODF. Overall, we show local reconstruction, segmentation and tracking results on complex fiber regions with known fiber crossing on simulated HARDI data, on a biological phantom and on multiple human brain datasets. Most current DTI based methods neglect these complex fibers, which might lead to wrong interpretations of the brain anatomy and Functioning.
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Diffusion Maps Clustering for Magnetic Resonance Q-Ball Imaging Segmentation
International Journal of Biomedical Imaging, 2008Co-Authors: Demian Wassermann, Maxime Descoteaux, Rachid DericheAbstract:White matter fiber clustering aims to get insight about anatomical structures in order to generate atlases, perform clear visualizations, and compute statistics across subjects, all important and current neuroimaging problems. In this work, we present a diffusion maps clustering method applied to diffusion MRI in order to segment complex white matter fiber bundles. It is well known that diffusion tensor imaging (DTI) is restricted in complex fiber regions with crossings and this is why recent high-angular resolution diffusion imaging (HARDI) such as Q-Ball imaging (QBI) has been introduced to overcome these limitations. QBI reconstructs the diffusion orientation distribution Function (ODF), a Spherical Function that has its maxima agreeing with the underlying fiber populations. In this paper, we use a Spherical harmonic ODF representation as input to the diffusion maps clustering method.We first show the advantage of using diffusion maps clustering over classical methods such as N-Cuts and Laplacian eigenmaps. In particular, our ODF diffusion maps requires a smaller number of hypothesis from the input data, reduces the number of artifacts in the segmentation, and automatically exhibits the number of clusters segmenting the Q-Ball image by using an adaptive scalespace parameter.We also show that our ODF diffusion maps clustering can reproduce published results using the diffusion tensor (DT) clustering with N-Cuts on simple synthetic images without crossings. On more complex data with crossings, we show that our ODF-based method succeeds to separate fiber bundles and crossing regions whereas the DT-based methods generate artifacts and exhibit wrong number of clusters. Finally, we show results on a real-brain dataset where we segment well-known fiber bundles.
Juan Tirao - One of the best experts on this subject based on the ideXlab platform.
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Spherical FunctionS: THE SPHERES VS. THE PROJECTIVE SPACES
arXiv: Representation Theory, 2012Co-Authors: Juan Tirao, Ignacio ZurriánAbstract:In this paper we establish a close relationship between the Spherical Functions of the n-dimensional sphere S n ≃ SO(n + 1)/SO(n) and the Spherical Functions of the n-dimensional real projective space P n (R) ≃ SO(n+ 1)/O(n). In fact, for n odd a Function on SO(n+ 1) is an irreducible Spherical Function of some type � ∈ ˆ SO(n) if and only if it is an irreducible Spherical Function of some type ∈ ˆ O(n). When n is even this is also true for certain types, and in the other cases we exhibit a clear correspondence between the irreducible Spherical Functions of both pairs (SO(n + 1),SO(n)) and (SO(n + 1),O(n)). Summarizing, to find all Spherical Functions of one pair is equivalent to do so for the other pair.
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MATRIX VALUED Spherical FunctionS ASSOCIATED TO THE THREE DIMENSIONAL HYPERBOLIC SPACE
International Journal of Mathematics, 2002Co-Authors: F. A. Grünbaum, Inés Pacharoni, Juan TiraoAbstract:The main purpose of this paper is to compute all irreducible Spherical Functions on of arbitrary type , where K = SU(2). This is accomplished by associating to a Spherical Function Φ on G a matrix valued Function H on the three dimensional hyperbolic space ℍ = G/K. The entries of H are solutions of two coupled systems of ordinary differential equations. By an appropriate twisting involving Hahn polynomials we uncouple one of the systems and express the entries of H in terms of Gauss' Functions 2F1. Just as in the compact instance treated in [7], there is a useful role for a special class of generalized hypergeometric Functions p+1 Fp.
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Matrix Valued Spherical Functions Associated to the Complex Projective Plane
Journal of Functional Analysis, 2002Co-Authors: F. A. Grünbaum, Inés Pacharoni, Juan TiraoAbstract:Abstract The main purpose of this paper is to compute all irreducible Spherical Functions on G=SU(3) of arbitrary type δ∈K, where K=S(U(2)×U(1))≃U(2). This is accomplished by associating to a Spherical Function Φ on G a matrix valued Function H on the complex projective plane P2( C )=G/K. It is well known that there is a fruitful connection between the hypergeometric Function of Euler and Gauss and the Spherical Functions of trivial type associated to a rank one symmetric pair (G, K). But the relation of Spherical Functions of types of dimension bigger than one with classical analysis has not been worked out even in the case of an example of a rank one pair. The entries of H can be described by solutions of two systems of ordinary differential equations. There is no ready-made approach to such a pair of systems or even to a single system of this kind. In our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitly in terms of a special class of generalized hypergeometric Functions p+1Fp.
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Matrix Valued Spherical Functions Associated to the Complex Projective Plane
arXiv: Representation Theory, 2001Co-Authors: F. A. Grünbaum, Inés Pacharoni, Juan TiraoAbstract:The main purpose of this paper is to compute all irreducible Spherical Functions on $G=\SU(3)$ of arbitrary type $\delta\in \hat K$, where $K={\mathrm{S}}(\mathrm{U}(2)\times\mathrm{U}(1))\simeq\mathrm{U}(2)$. This is accomplished by associating to a Spherical Function $\Phi$ on $G$ a matrix valued Function $H$ on the complex projective plane $P_2(\mathbb{C})=G/K$. It is well known that there is a fruitful connection between the hypergeometric Function of Euler and Gauss and the Spherical Functions of trivial type associated to a rank one symmetric pair $(G,K)$. But the relation of Spherical Functions of types of dimension bigger than one with classical analysis, has not been worked out even in the case of an example of a rank one pair. The entries of $H$ are solutions of two systems of ordinary differential equations. There is no ready made approach to such a pair of systems, or even to a single system of this kind. In our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitely in terms of a special class of generalized hypergeometric Functions ${}_{p+1}F_p$.
Aurobrata Ghosh - One of the best experts on this subject based on the ideXlab platform.
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A polynomial approach for extracting the extrema of a Spherical Function and its application in diffusion MRI
Medical Image Analysis, 2013Co-Authors: Aurobrata Ghosh, Elias Tsigaridas, Bernard Mourrain, Rachid DericheAbstract:Antipodally symmetric Spherical Functions play a pivotal role in diffusion MRI in representing sub-voxel-resolution microstructural information of the underlying tissue. This information is described by the geometry of the Spherical Function. In this paper we propose a method to automatically compute all the extrema of a Spherical Function. We then classify the extrema as maxima, minima and saddle-points to identify the maxima. We take advantage of the fact that a Spherical Function can be described equivalently in the Spherical harmonic (SH) basis, in the symmetric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. We extract the extrema of the Spherical Function by computing the stationary points of its constrained HP representation. Instead of using traditional optimization approaches, which are inherently local and require exhaustive search or re-initializations to locate multiple extrema, we use a novel polynomial system solver which analytically brackets all the extrema and refines them numerically, thus missing none and achieving high precision. To illustrate our approach we consider the Orientation Distribution Function (ODF). In diffusion MRI the ODF is a Spherical Function which represents a state-of-the-art reconstruction algorithm whose maxima are aligned with the dominant fiber bundles. It is, therefore, vital to correctly compute these maxima to detect the fiber bundle directions. To demonstrate the potential of the proposed polynomial approach we compute the extrema of the ODF to extract all its maxima. This polynomial approach is, however, not dependent on the ODF and the framework presented in this paper can be applied to any Spherical Function described in either the SH basis, ST basis or the HP basis.
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A Polynomial Based Approach to Extract Fiber Directions from the ODF and its Experimental Validation
2009Co-Authors: Aurobrata Ghosh, Maxime Descoteaux, Elias Tsigaridas, Rachid DericheAbstract:In Diffusion MRI, Spherical Functions are commonly employed to represent the diffusion information. The ODF is an intuitive Spherical Function since its maxima are aligned with the dominant fiber directions. Therefore, it is important to correctly determine these maximal directions, as they are the key to tracing fiber tracts. A tractography algorithm will suffer from cumulative error when the maximal directions are incorrectly estimated locally. The goal of this work is to present a polynomial based approach for estimating the maximal directions correctly. The paper will also present a measure of the correctness of the estimation. This approach will be tested on synthetic, phantom, and real data, and will be compared to an existing discrete “mesh-search” approach [2]. It will be shown how this approach naturally overcomes the inherent shortcomings of the discrete search. Finally, although, the approach is demonstrated on the ODF, it can be equally applied to any Spherical Function.
Atsushi Matsuo - One of the best experts on this subject based on the ideXlab platform.
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Integrable connections related to zonal Spherical Functions
Inventiones Mathematicae, 1992Co-Authors: Atsushi MatsuoAbstract:We define a system of differential equations of first order for a Function valued in the group algebra of the Weyl group associated with an arbitrary root system. This is equivalent to the system of differential equations given by Heckman and Opdam which is a deformation of the system satisfied by the zonal Spherical Function of the Riemannian symmetric spaceG/K of non-compact type. When the root system is A n -type, our equation is related to the Knizhnik-Zamolodchikov equation in conformal field theory.