Standard particle filtering technique have previously been applied to the problem of fiber tracking by Brun et al. (2002) and Bjornemo et al. (2002). However, these previous attempts have not utilised the full power of the technique, and as a result the fiber paths were tracked in a goal directed way. In this paper we provide an advanced technique by presenting a fast and novel probabilistic method for white matter fiber tracking in diffusion weighted MRI (DWI), which takes advantage of the weighting and resampling mechanism of particle filtering. We formulate fiber tracking using a nonlinear state space model which captures both smoothness regularity of the fibers and the uncertainties in the local fiber orientations due to noise and partial volume effects. Global fiber tracking is then posed as a problem of particle filtering. To model the posterior distribution, we classify voxels of the white matter as either prolate or oblate tensors. We then construct the orientation distributions for prolate and oblate tensors separately. Finally, the importance density function for particle filtering is modeled using the von Mises-Fisher distribution on a unit sphere. Fast and efficient sampling is achieved using Ulrich-Wood’s simulation algorithm. Given a seed point, the method is able to rapidly locate the globally optimal fiber and also provides a probability map for potential connections. The proposed method is validated and compared to alternative methods both on synthetic data and real-world brain MRI datasets.

In this article, we review recent mathematical models and computational methods for the processing of diffusion Magnetic Resonance Images, including state-of-the-art reconstruction of diffusion models, cerebral white matter connectivity analysis, and segmentation techniques. We focus on Diffusion Tensor Images (DTI) and Q-Ball Images (QBI).

Abstract. A mapping of unit vectors onto a 5D hypersphere is used to model and partition ODFs from HARDI data. This mapping has a number of useful and interesting properties and we make a link to interpretation of the second order spherical harmonic decompositions of HARDI data. The paper presents the working theory and experiments of using a von Mises-Fisher mixture model for directional samples. The MLE of the second moment of the HvMF pdf can also be related to fractional anisotropy. We perform error analysis of the estimation scheme in single and multi-fibre regions and then show how a penalised-likelihood model selection method can be employed to differentiate single and multiple fibre regions. 1

Abstract. We introduce a fibre tract segmentation algorithm based on the geometric coherence of fibre orientations as indicated by a streamline flow model. The inference of local flow approximations motivates a pairwise consistency measure between fibre ODF maxima. We use this measure in a recursive algorithm to cluster consistent ODF maxima, leading to the segmentation of white matter pathways. The method requires minimal seeding compared to streamline tractography-based methods, and allows multiple tracts to pass through the same voxels. We illustrate the approach with a segmentation of the corpus callosum and one of the cortico-spinal tract, with each example seeded at a single voxel. 1 Introduction and Related Work The development of algorithms for automatic white matter fibre tract segmentation in diffusion MRI is of significant interest in the community. A number of methods approach this problem by clustering individual diffusion tensors (DTs) or diffusion orientation distribution functions (ODFs), relying on Euclidean or

Abstract. In this article, 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 phantom, 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. 1

Abstract. We denoise HARDI (High Angular Resolution Diffusion Imaging) data arising in medical imaging. Diffusion imaging is a relatively new and powerful method to measure the 3D profile of water diffusion at each point. This can be used to reconstruct fiber directions and pathways in the living brain, providing detailed maps of fiber integrity and connectivity. HARDI is a powerful new extension of diffusion imaging, which goes beyond the diffusion tensor imaging (DTI) model: mathematically, intensity data is given at every voxel and at any direction on the sphere. However, HARDI data is usually highly contaminated with noise, depending on the b-value which is a tuning parameter preselected to collect the data. Larger b-values help to collect more accurate information in terms of measuring diffusivity, but more noise is generated by many factors as well. So large b-values are preferred, if we can satisfactorily reduce the noise without losing the data structure. We propose a variational method to denoise HARDI data by denoising the spherical

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We address the problem of segmenting high angular resolution diffusion images of the brain into cerebral regions corresponding to distinct white matter fiber bundles. We cast this problem as a manifold clustering problem in which distinct fiber bundles correspond to different submanifolds of the space of orientation distribution functions (ODFs). Our approach integrates tools from sparse representation theory into a graph theoretic segmentation framework. By exploiting the Riemannian properties of the space of ODFs, we learn a sparse representation for the ODF at each voxel and infer the segmentation by applying spectral clustering to a similarity matrix built from these representations. We evaluate the performance of our method via experiments on synthetic, phantom and real data. Index Terms — image segmentation, compressed sensing, graph theory, harmonic analysis, diffusion magnetic resonance imaging. 1.

Abstract. Compared with Diffusion Tensor Imaging (DTI), High Angular Resolution Imaging (HARDI) can better explore the complex microstructure of white matter. Orientation Distribution Function (ODF) is used to describe the probability of the fiber direction. Fisher information metric has been constructed for probability density family in Information Geometry theory and it has been successfully applied for tensor computing in DTI. In this paper, we present a state of the art Riemannian framework for ODF computing based on Information Geometry and sparse representation of orthonormal bases. In this Riemannian framework, the exponential map, logarithmic map and geodesic have closed forms. And the weighted Frechet mean exists uniquely on this manifold. We also propose a novel scalar measurement, named Geometric Anisotropy (GA), which is the Riemannian geodesic distance between the ODF and the isotropic ODF. The Renyi entropy H 1 of the ODF can be computed from the GA. Moreover, we present 2 an Affine-Euclidean framework and a Log-Euclidean framework so that we can work in an Euclidean space. As an application, Lagrange interpolation on ODF field is proposed based on weighted Frechet mean. We validate our methods on synthetic and real data experiments. Compared with existing Riemannian frameworks on ODF, our framework is model-free. The estimation of the parameters, i.e. Riemannian coordinates, is robust and linear. Moreover it should be noted that our theoretical results can be used for any probability density function (PDF) under an orthonormal basis representation. 1

By measuring the anisotropic self-diffusion rates of water, Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) provides a unique noninvasive probe of fibrous tissue. In particular, it has been explored widely for imaging nerve fiber tracts in the human brain. Geometric features provide a quick visual overview of the complex datasets that arise from DW-MRI. At the same time, they build a bridge towards quantitative analysis, by extracting explicit representations of structures in the data that are relevant to specific research questions. Therefore, features in DW-MRI data are an active research topic not only within scientific visualization, but have received considerable interest from the medical image analysis, neuroimaging, and computer vision communities. It is the goal of this paper to survey contributions from all these fields, concentrating on streamline clustering, edge detection and segmentation, topological methods, and extraction of anisotropy creases. We point out interrelations between these topics and make suggestions for future research.