Team Odyssée

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Section: New Results

Brain anatomical imaging using Diffusion MRI

Non Rigid Registration of Diffusion Tensor Images

Participants : Rachid Deriche, Olivier Faugeras, Christophe Lenglet [ Siemens Corporate Research Princeton, NJ USA ] , Théo Papadopoulo.

We propose a novel variational framework for the dense non-rigid registration of Diffusion Tensor Images (DTI). Our approach relies on the differential geometrical properties of the Riemannian manifold of multivariate normal distributions endowed with the metric derived from the Fisher information matrix. The availability of closed form expressions for the geodesics and the Christoffel symbols allows us to define statistical quantities and to perform the parallel transport of tangent vectors in this space. We propose a matching energy that aims to minimize the difference in the local statistical content (means and covariance matrices) of two DT images through a gradient descent procedure. The result of the algorithm is a dense vector field that can be used to wrap the source image into the target image. This article is essentially a mathematical study of the registration problem. Some numerical experiments are provided as a proof of concept.

This work has been submitted to the SIAM Journal of Applied Mathematics and has appeared as a technical report [Oops!] .

Regularized, Fast, and Robust Analytical Q-Ball Imaging

Keywords : diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI), Q-Ball imaging (QBI), fiber tractography, orientation distribution function (ODF), regularization, Funk Radon transform, spherical harmonic.

Participants : Rachid Deriche, Maxime Descoteaux, Elaine Angelino [ Odyssée/Division of Engineering and Applied Sciences, Harvard University, Boston, Massachusetts, USA ] , Shaun Fitzgibbons [ Odyssée/Division of Engineering and Applied Sciences, Harvard University, Boston, Massachusetts, USA ] .

This work was partially supported by the CRSNG Canada graduate scholarship, FQRNT-INRIA and INRIA (International internships program)

In this work, we propose a regularized, fast, and robust analytical solution for the Q-ball imaging (QBI) reconstruction of the orientation distribution function (ODF) together with its detailed validation and a discussion on its benefits over the state-of-the-art. Our analytical solution is achieved by modeling the raw high angular resolution diffusion imaging signal with a spherical harmonic basis that incorporates a regularization term based on the Laplace-Beltrami operator defined on the unit sphere. This leads to an elegant mathematical simplification of the Funk-Radon transform which approximates the ODF. We prove a new corollary of the Funk-Hecke theorem to obtain this simplification. Then, we show that the Laplace-Beltrami regularization is theoretically and practically better than Tikhonov regularization. At the cost of slightly reducing angular resolution, the Laplace-Beltrami regularization reduces ODF estimation errors and improves fiber detection while reducing angular error in the ODF maxima detected. Finally, a careful quantitative validation is performed against ground truth from synthetic data and against real data from a biological phantom and a human brain dataset. We show that our technique is also able to recover known fiber crossings in the human brain and provides the practical advantage of being up to 15 times faster than original numerical QBI method.

This work has been published in [Oops!] .

Diffusion MRI : From 2nd order to 4th Order Diffusion Tensor

Keywords : High Order Diffusion tensor imaging (HO-DTI), 4th order DTI (HARDI).

Participants : Rachid Deriche, Aurobratha Ghosh.

This work was partially supported by the ARC Diffusion MRI

To detect fiber crossings, today the HARDI acquisition approach has produced a plethora of new techniques and mathematical tools such as radial basis functions, Spherical Harmonics (SH), Higher Order Tensors (HOT), etc. The mathematical properties of these high order tools need to be better understood to be fully exploited. In particular it seems appropriate to explore HOT while leveraging the extensive framework already established for classical DTI. In this work, we have started to explore HOT and in particular the space of 4th order diffusion tensors.

A major limitation of the 2nd order DTI, is its incapacity to discriminate multiple fibers crossing in the same voxel. In the tensor framework this can be overcome by accomodating HOTs to the diffusivity function. However in these spaces of higher dimensions, it gets harder to enforce the physical constraint of positive diffusion in the inverse problem of estimation. In particular, the space of 4th order tensors, which already makes it possible to detect at least three separate fibers, needs to be better understood. In this work, we started to explore the space of 4th order diffusion tensors, and rewrite them in matrix form, to be able to extend the Riemannian framework of S+(3) to S+(6). We also started to explore the different symmetries of a 4th order tensor, which make the problem of estimation non-unique.

ODF sharpening improves clinically feasible Q-ball imaging reconstructions

Keywords : Q-Ball imaging (QBI), sharpening, orientation distribution function (ODF), spherical harmonic (SH), Funk Radon Transform (FRT).

Participants : Rachid Deriche, Maxime Descoteaux.

Recent High Angular Resolution Diffusion Imaging (HARDI) acquisitions use low b-values (b = 1000 s/mm2) and small number of gradient encoding directions, N, (less than 100) to describe local non-Gaussian diffusion process in clinically feasible acquisitions. One such technique is Q-Ball Imaging (QBI) , which reconstructs the diffusion orientation distribution function (ODF) of water molecules in a biological tissue. However, at low b-values and small N and because of the intrinsic Bessel function smoothing in the Funk-Radon Transform used to reconstruct the ODF , the ODF profiles are quite smooth and ODF maxima (giving the underlying fiber orientation) are difficult to find and sometimes missed when compared to ODF reconstructed from research-oriented HARDI acquisitions with higher b-values (b greater than 3000 s/mm2) and large N (N greater than 100). In this work [Oops!] , we define a general sharpening operation that can be used with any HARDI reconstruction method and in particular, we show that if the sharpening is applied on the ODF, it considerably improves fiber detection and increases angular resolution of QBI. This work has been presented and published in [Oops!] .

Tensor transform improves clinically feasible Q-ball imaging reconstructions

Keywords : Q-Ball imaging (QBI), sharpening, orientation distribution function (ODF), spherical harmonic (SH), Funk Radon Transform (FRT).

Participants : Rachid Deriche, Maxime Descoteaux.

In this work   [Oops!] , we focus on the development of a well-defined deconvolution method that transforms the diffusion tensor (DT) into a sharpened version, the fiber tensor. We show how to transform the diffusion tensors into so-called fiber tensors and we demonstrate that this tensor transform is a natural pre-processing task when one is interested in fiber tracking. It also leads to a dramatic improvement of the tractography results obtained by front propagation techniques on the full diffusion tensor. We compare and validate sharpening and tracking results on synthetic data and on known fiber bundles in the human brain.

This work has been presented and published in [Oops!] .

Disambiguation of Complex Subvoxel Fibre Configurations in High Angular Resolution Fibre Tractography

Keywords : High angular resolution diffusion imaging (HARDI), Q-ball imaging (QBI), orientation distribution function (ODF), 3D Curve inference, Crossing, Fanning, Branching, tractography.

Participants : Rachid Deriche, Maxime Descoteaux, Peter Savadjiev [ School of Computer Science, McGill University, Montreal Canada ] , Jennifer S. W. Campbell [ School of Computer Science/McConnell Brain Imaging Center, McGill University, Montreal Canada ] , G. Bruce Pike [ McConnell Brain Imaging Center, McGill University, Montreal Canada ] , Kaleem Siddiqi [ School of Computer Science, McGill University, Montreal Canada ] .

This work was partially supported by the CRSNG Canada graduate scholarship and FQRNT-INRIA

In this work   [Oops!] , we apply the 3D curve inference algorithm, described in Savadjiev et al (Medical Image Analysis 10(5):799-813,2006), in a labeling scheme that disambiguates ODFs in the presence of complex subvoxel fibre configurations that result from (1) single fibres (possibly with subvoxel curvature), (2) subvoxel fanning and (3) crossing configurations. The labeling will allow these cases to be treated properly in tractography and thus reduce the occurrence of false positive and false negative connections. For example, the identification of fibre fanning should help to reconstruct the entire cortical-spinal tract as it fans out to the motor cortex, a situation that currently confounds algorithms using only the ODF maxima. We provide labeling results and the recovery of fanning polarity on synthetic data as well as in-vivo human brain data, and show preliminary results that demonstrate a substantial improvement in the performance of a fibre tracking algorithm.

This work has been presented and published in [Oops!] .

Crossings of Callosal Fibers

Keywords : Probabilistic fiber tractography, Q-Ball imaging (QBI), diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI) crossings, callosal Fibers.

Participants : Rachid Deriche, Maxime Descoteaux, Alfred Anwander [ Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany ] .

This work was partially supported by the PAI Procope.

The corpus callosum (CC) is involved in the inter-hemispheric interaction of cortical regions and the exact reconstruction of fibers connecting the cerebral hemispheres is of major interest. In this contribution   [Oops!] , we investigate and show how the reconstruction of transcallosal fiber connections intersecting with the corona radiata and the superior longitudinal fasciculus can be improved with a local model of crossing fibers using diffusion weighted imaging and Q-Ball tractography. Current DTI based methods are shown to produces incomplete fiber reconstructions in the CC and neglect all lateral fibers to prefrontal areas and reconstructs only fibers to the medial/dorsal cortex. Q-Ball tractography additional finds strong interhemispheric connectivity of the inferior and middle frontal gyrus and the ventral premotor cortex. These additional lateral fibers influence the cartography of the transcallosal fibers, and might lead to new insights to inter-hemispheric cognitive networks.

This work has been presented and published in [Oops!] .

Multidirectional Q-Ball Tracking

Participants : Rachid Deriche, Maxime Descoteaux.

This work was partially supported by FQRNT/INRIA.

We present a new tracking algorithm based on the full multidirectional information of the diffusion orientation distribution function (ODF) estimated from Q-Ball Imaging (QBI). From the ODF, we extract all available maxima and then extend streamline (STR) tracking to allow for splitting in multiple directions (SPLIT-STR). Our new algorithm SPLIT-STR overcomes important limitations of classical diffusion tensor streamline tracking in regions of low anisotropy and regions of fiber crossings. Not only can the tracking propagate through fiber crossings but it can also deal with fibers fanning and branching. SPLIT-STR algorithm is efficient and validated on synthetic data, on a biological phantom and compared against probabilistic tensor tracking on a human brain dataset with known crossing fibers.

This work has been presented and published in [Oops!] .

Deterministic and Probabilistic Q-Ball Tractography: from Diffusion to Sharp Fiber Distributions

Keywords : Probabilistic and deterministic fiber tractography, Q-Ball imaging (QBI), diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI) crossings, callosal Fibers.

Participants : Rachid Deriche, Maxime Descoteaux, Alfred Anwander [ Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany ] .

This work was partially supported by the PAI Procope.

In this work   [Oops!] , we propose a deterministic and a probabilistic extension of classical diffusion tensor imaging (DTI) tractography algorithms based on a sharp fiber orientation distribution function (ODF) reconstruction from Q-Ball Imaging (QBI). An important contribution of the paper is the integration of some of the latest state-of-the-art high angular resolution diffusion imaging (HARDI) data processing methods to obtain accurate and convincing results of complex fiber bundles with crossing, fanning and branching configurations. First, we develop a new deconvolution sharpening transformation from diffusion ODF (dODF) to fiber ODF (fODF). We show that this sharpening transformation improves angular resolution and fiber detection of QBI and thus greatly improves tractography results. The angular resolution of QBI is in fact improved by approximately 20 and the fODF is shown to behave very similarly of the fiber orientation density (FOD) estimated from the spherical deconvolution method of Tournier et al. Another major contribution of the paper is the extensive comparison study on human brain datasets of our new deterministic and probabilistic tracking algorithms. As an application, we show how the reconstruction of transcallosal fiber connections intersecting with the corona radiata and the superior longitudinal fasciculus can be improved with the fODF in a group of 8 subjects. Current DTI based methods neglect these fibers, which might lead to wrong interpretations of the brain functions.

This work has been published in [Oops!] .

Avoiding Artifacts in Spectral white matter fiber clustering and embedding

Keywords : Q-Ball Imaging, orientation distribution function (ODF), Laplacian Eigenmaps, Clustering, Spectral Embedding.

Participants : Rachid Deriche, Demian Wassermann.

This work was partially supported by the ARC Diffusion MRI

Tractography applied to the tensor field in diffusion tensor imaging (DTI) results in sets of streamlines which can be associated with major fiber tracts. If fibers are reconstructed and visualized individually through the complete white matter, the display gets easily cluttered making it difficult to get insight in the data. The goeal of this work was to recover fibers tracts from white matter from DTI or HARDI imaging and embed and cluster them in order to perform statistical analysis and improve readability. In this work   [Oops!] , we show that spectral embedding clustering techniques can provide a fast, non-linear way of performing this process and we present an approach based in Diffusion Maps that minimizes the dependance in uniform clustering avoiding artifacs by performing a previous normalization step.

Segmentation of Q-Ball Images Using Statistical Surface Evolution

Keywords : Q-Ball imaging (QBI), segmentation, orientation distribution function (ODF), diffusion tensor imaging (DTI), spherical harmonic (SH).

Participants : Rachid Deriche, Maxime Descoteaux.

This work was partially supported by the CRSNG Canada graduate scholarship, FQRNT-INRIA and PAI Procope

In this work   [Oops!] , we develop a new method to segment Q-Ball imaging (QBI) data. We first estimate the orientation distribution function (ODF) using a fast and robust spherical harmonic (SH) method. Then, we use a region-based statistical surface evolution on this image of ODFs to efficiently find coherent white matter fiber bundles. We show that our method is appropriate to propagate through regions of fiber crossings and we show that our results outperform state-of-the-art diffusion tensor (DT) imaging segmentation methods, inherently limited by the DT model. Results obtained on synthetic data, on a biological phan- tom, on real datasets and on all 13 subjects of a public QBI database show that our method is reproducible, automatic and brings a strong added value to diffusion MRI segmentation.

This work has been presented and published in [Oops!] . More details can be found in the INRIA Research Report   [Oops!] .

Diffusion Maps Segmentation of Magnetic Resonance Q-Ball Imaging

Keywords : Q-Ball Imaging, orientation distribution function (ODF), N-Cuts Segmentation, Laplacian Eigenmaps, Clustering, Spectral Embedding.

Participants : Rachid Deriche, Maxime Descoteaux, Demian Wassermann.

This work was partially supported by the ARC Diffusion MRI

In this work   [Oops!] , 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 has Q-Ball Imaging (QBI) have been introduced to overcome these limitations. QBI reconstructs the diffusion orientation distribution function (ODF), a spherical function that has its maximum(a) agreeing with the underlying fiber population. In this paper, we use the ODF representation in a small set of spherical harmonic coefficients 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 adaptative scale-space 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 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 successfully segment the fiber bundles.

This work has been presented and published in [Oops!] .

Validation and Comparison of Analytical Q-Ball Imaging Methods

Keywords : Diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI), q-ball imaging (QBI), orientation distribution function (ODF), regularization.

Participants : Rachid Deriche, Maxime Descoteaux, Peter Savadjiev [ School of Computer Science, McGill University, Montreal Canada ] , Jennifer S. W. Campbell [ School of Computer Science/McConnell Brain Imaging Center, McGill University, Montreal Canada ] , G. Bruce Pike [ McConnell Brain Imaging Center, McGill University, Montreal Canada ] , Kaleem Siddiqi [ School of Computer Science, McGill University, Montreal Canada ] .

This work was partially supported by the CRSNG Canada graduate scholarship and FQRNT-INRIA

Q-ball imaging (QBI), introduced by D. Tuch, reconstructs the diffusion orientation distribution function (ODF) of the underlying fiber population of a biological tissue. An analytical solution for QBI was recently proposed by several independent groups, using a spherical harmonic (SH) representation of the input signal. The methods differ primarily in the way SH are estimated. In this work [Oops!] , we validate these methods and compare them against Tuch's numerical QBI on synthetic data, on a biological phantom and on a human brain dataset. We show that analytical QBI results in a speed-up factor of 15 over Tuch's QBI, while providing results that are in strong agreement. We also show that at the cost of slightly reducing angular resolution, QBI with Laplace-Beltrami regularization provides the strongest robustness to noise and the most accurate detection of fiber crossings.

This work has been presented and published in [Oops!] .


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