Joint T1 and brain fiber diffeomorphic registration using the demons

International audienceImage registration is undoubtedly one of the most active areas of research in medical imaging. Within inter-individual comparison, registration should align images as well as cortical and external structures such as sulcal lines and fibers. While using image-based registration[1], neural fibers appear uniformly white giving no information to the registration. Tensor-based registration was recently proposed to improve white-matter alignment[2,3], however misregistration may also persist in regions where the tensor field appears uniform[4]. We propose an hybrid approach by extending the Diffeomorphic Demons(D)[5] registration to incorporate geometric constrains. Combining the deformation field induce by the image and the geometry, we define a mathematically sound framework to jointly register images and geometric descriptors such as fibers or sulcal lines

A Riemannian Framework for Tensor Computing

Positive definite symmetric matrices (so-called tensors in this article) are nowadays a common source of geometric information. In this paper, we propose to provide 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 manifold of constant curvature 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 schemes and Gaussian filtering 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 Laplacian 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 that are particularly simple and efficient to solve

Fast and Simple Computations on Tensors with Log-Euclidean Metrics.

Computations on tensors, i.e. symmetric positive definite real matrices in medical imaging, appear in many contexts. In medical imaging, these computations have become common with the use of DT-MRI. The classical Euclidean framework for tensor computing has many defects, which has recently led to the use of Riemannian metrics as an alternative. So far, only affine-invariant metrics had been proposed, which have excellent theoretical properites but lead to complex algorithms with a high computational cost. In this article, we present a new familly of metrics, called Log-Euclidean. These metrics have the same excellent theoretical properties as affine-invariant metrics and yield very similar results in practice. But they lead to much more simple computations, with a much lighter computational cost, very close to the cost of the classical Euclidean framework. Indeed, Riemannian computations become Euclidean computations in the logarithmic domain with Log-Euclidean metrics. We present in this article the complete theory for these metrics, and show experimental results for multilinear interpolation, dense extrapolation of tensors and anisotropic diffusion of tensor fields

Evaluating Brain Anatomical Correlations via Canonical Correlation Analysis of Sulcal Lines

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Joint Estimation and Smoothing of Clinical DT-MRI with a Log-Euclidean Metric

Diffusion tensor MRI is an imaging modality that is gaining importance in clinical applications. However, in a clinical environment, data has to be acquired rapidly at the detriment of the image quality. We propose a new variational framework that specifically targets low quality DT-MRI. The hypothesis of an additive Gaussian noise on the images leads us to estimate the tensor field directly on the image intensities. To further reduce the influence of the noise, we optimally exploit the spatial correlation by adding to the estimation an anisotropic regularization term. This criterion is easily optimized thanks to the use of the recently introduced Log-Euclidean metrics. Results on real clinical data show promising improvements of fiber tracking in the brain and we present the first successful attempt, up to our knowledge, to reconstruct the spinal cord

Quantitative evaluation of 10 tractography algorithms on a realistic diffusion MR phantom.

International audienceAs it provides the only method for mapping white matter fibers in vivo, diffusion MRI tractography is gaining importance in clinical and neuroscience research. However, despite the increasing availability of different diffusion models and tractography algorithms, it remains unclear how to select the optimal fiber reconstruction method, given certain imaging parameters. Consequently, it is of utmost importance to have a quantitative comparison of these models and algorithms and a deeper understanding of the corresponding strengths and weaknesses. In this work, we use a common dataset with known ground truth and a reproducible methodology to quantitatively evaluate the performance of various diffusion models and tractography algorithms. To examine a wide range of methods, the dataset, but not the ground truth, was released to the public for evaluation in a contest, the "Fiber Cup". 10 fiber reconstruction methods were evaluated. The results provide evidence that: 1. For high SNR datasets, diffusion models such as (fiber) orientation distribution functions correctly model the underlying fiber distribution and can be used in conjunction with streamline tractography, and 2. For medium or low SNR datasets, a prior on the spatial smoothness of either the diffusion model or the fibers is recommended for correct modelling of the fiber distribution and proper tractography results. The phantom dataset, the ground truth fibers, the evaluation methodology and the results obtained so far will remain publicly available on: http://www.lnao.fr/spip.php?rubrique79 to serve as a comparison basis for existing or new tractography methods. New results can be submitted to [email protected] and updates will be published on the webpage

Traitement riemannien des tenseurs pour l'IRM de diffusion et l'anatomie algorithmique du cerveau.

Symmetric, positive-definite matrices, or tensors, are nowadays a common geometrical tool for image processing and analysis. The recent emergence of diffusion tensor MRI (DTI) and computational anatomy (CA) brought importance of tensors out to the medical community. However, working with those is difficult: the positive-definite constraint must be satisfied at any cost, which cannot be ensured in general with standard matrix operations. In this work, we propose two alternatives to the standard Euclidean calculus on tensors. Instead of seeing the tensor space as a vector space, we consider it as a manifold, i.e., a smooth curved space. Thanks to the Riemannian geometry, we are able to ``unfold'' this space, and to generalize any operation on tensors with astonishing simple implementations. In a second step, we review the applications of such frameworks in the context of clinical DTI and brain CA. In DTI, we show that very noisy data, typical of clinical acquisitions, can be optimally exploited and eventually produce a meaningful and clinically relevant fiber reconstruction. In brain CA, we show that, by considering simple brain anatomical landmarks - the sulcal lines - we are able to precisely measure the inter-individual variability of the cortex. Finally, we develop a new framework to study the anatomical correlations between brain regions, and present results of so far unknown relationships between symmetric sulcal positions, and between a-priori unrelated sulci, which raises new fundamental questions about the origin of such statistical dependencies.Les matrices symétriques et définies positives, ou tenseurs, sont aujourd'hui fréquemment utilisées en traitement et analyse des images. Leur importance a été mise à jour avec l'apparition récente de l'IRM du tenseur de diffusion (ITD) et de l'anatomie algorithmique (AA). Cependant, il est difficile de travailler avec : la contrainte de positivité doit être satisfaite à tout prix, ce qui n'est pas garanti avec les opérations matricielles standard. Dans ce travail, nous proposons deux alternatives au calcul euclidien sur les tenseurs. Au lieu de voir l'espace des tenseurs comme un espace vectoriel, nous le considérons comme une variété, i.e., un espace courbe et lisse. Grâce à la géométrie riemannienne, il est alors possible de " déplier " cet espace et de généraliser aux tenseurs toute opération avec des implémentations étonnamment simples. Dans un deuxième temps, nous passons en revue les applications de tels cadres de calcul en ITD clinique et en AA du cerveau. En ITD, nous montrons qu'il est possible de traiter de manière optimale des données très bruitées typiques d'acquisitions cliniques, et de produire des reconstructions de fibres plausibles. En AA du cerveau, nous montrons qu'en considérant des repères anatomiques simples - les lignes sulcales - il est possible de mesurer précisément la variabilité interindividuelle du cortex. Finalement, nous développons un cadre nouveau pour étudier les corrélations anatomiques entre régions du cerveau, et présentons des résultats jusqu'à maintenant inconnus de dépendances entre sillons symétriques, et entre sillons à priori non reliés, soulevant ainsi de nouvelles questions sur l'origine de telles dépendances statistiques

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