Abstract:
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 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 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.

Abstract:
Many diffusion MRI researchers, including the Human Connectome Project (HCP), acquire data using multishell (e.g., WU-Minn consortium) and diffusion spectrum imaging (DSI) schemes (e.g., USC-Harvard consortium). However, these data sets are not readily accessible to high angular resolution diffusion imaging (HARDI) analysis methods that are popular in connectomics analysis. Here we introduce a scheme conversion approach that transforms multishell and DSI data into their corresponding HARDI representations, thereby empowering HARDI-based analytical methods to make use of data acquired using non-HARDI approaches. This method was evaluated on both phantom and in-vivo human data sets by acquiring multishell, DSI, and HARDI data simultaneously, and comparing the converted HARDI, from non-HARDI methods, with the original HARDI data. Analysis on the phantom shows that the converted HARDI from DSI and multishell data strongly predicts the original HARDI (correlation coefficient > 0.9). Our in-vivo study shows that the converted HARDI can be reconstructed by constrained spherical deconvolution, and the fiber orientation distributions are consistent with those from the original HARDI. We further illustrate that our scheme conversion method can be applied to HCP data, and the converted HARDI do not appear to sacrifice angular resolution. Thus this novel approach can benefit all HARDI-based analysis approaches, allowing greater analytical accessibility to non-HARDI data, including data from the HCP.

Abstract:
Objective Up to now, fiber tractography in the clinical routine is mostly based on diffusion tensor imaging (DTI). However, there are known drawbacks in the resolution of crossing or kissing fibers and in the vicinity of a tumor or edema. These restrictions can be overcome by tractography based on High Angular Resolution Diffusion Imaging (HARDI) which in turn requires larger numbers of gradients resulting in longer acquisition times. Using compressed sensing (CS) techniques, HARDI signals can be obtained by using less non-collinear diffusion gradients, thus enabling the use of HARDI-based fiber tractography in the clinical routine. Methods Eight patients with gliomas in the temporal lobe, in proximity to the optic radiation (OR), underwent 3T MRI including a diffusion-weighted dataset with 30 gradient directions. Fiber tractography of the OR using a deterministic streamline algorithm based on DTI was compared to tractography based on reconstructed diffusion signals using HARDI+CS. Results HARDI+CS based tractography displayed the OR more conclusively compared to the DTI-based results in all eight cases. In particular, the potential of HARDI+CS-based tractography was observed for cases of high grade gliomas with significant peritumoral edema, larger tumor size or closer proximity of tumor and reconstructed fiber tract. Conclusions Overcoming the problem of long acquisition times, HARDI+CS seems to be a promising basis for fiber tractography of the OR in regions of disturbed diffusion, areas of high interest in glioma surgery.

Abstract:
In this paper, we propose a novel large deformation diffeomorphic registration algorithm to align high angular resolution diffusion images (HARDI) characterized by orientation distribution functions (ODFs). Our proposed algorithm seeks an optimal diffeomorphism of large deformation between two ODF fields in a spatial volume domain and at the same time, locally reorients an ODF in a manner such that it remains consistent with the surrounding anatomical structure. To this end, we first review the Riemannian manifold of ODFs. We then define the reorientation of an ODF when an affine transformation is applied and subsequently, define the diffeomorphic group action to be applied on the ODF based on this reorientation. We incorporate the Riemannian metric of ODFs for quantifying the similarity of two HARDI images into a variational problem defined under the large deformation diffeomorphic metric mapping (LDDMM) framework. We finally derive the gradient of the cost function in both Riemannian spaces of diffeomorphisms and the ODFs, and present its numerical implementation. Both synthetic and real brain HARDI data are used to illustrate the performance of our registration algorithm.

Abstract:
We present a Bayesian probabilistic model to estimate the brain white matter atlas from high angular resolution diffusion imaging (HARDI) data. This model incorporates a shape prior of the white matter anatomy and the likelihood of individual observed HARDI datasets. We first assume that the atlas is generated from a known hyperatlas through a flow of diffeomorphisms and its shape prior can be constructed based on the framework of large deformation diffeomorphic metric mapping (LDDMM). LDDMM characterizes a nonlinear diffeomorphic shape space in a linear space of initial momentum uniquely determining diffeomorphic geodesic flows from the hyperatlas. Therefore, the shape prior of the HARDI atlas can be modeled using a centered Gaussian random field (GRF) model of the initial momentum. In order to construct the likelihood of observed HARDI datasets, it is necessary to study the diffeomorphic transformation of individual observations relative to the atlas and the probabilistic distribution of orientation distribution functions (ODFs). To this end, we construct the likelihood related to the transformation using the same construction as discussed for the shape prior of the atlas. The probabilistic distribution of ODFs is then constructed based on the ODF Riemannian manifold. We assume that the observed ODFs are generated by an exponential map of random tangent vectors at the deformed atlas ODF. Hence, the likelihood of the ODFs can be modeled using a GRF of their tangent vectors in the ODF Riemannian manifold. We solve for the maximum a posteriori using the Expectation-Maximization algorithm and derive the corresponding update equations. Finally, we illustrate the HARDI atlas constructed based on a Chinese aging cohort of 94 adults and compare it with that generated by averaging the coefficients of spherical harmonics of the ODF across subjects.

Abstract:
This work investigates the possibilities of applying high-angular-resolution-diffusion-imaging- (HARDI-) based methods in a clinical setting by investigating the performance of non-Gaussian diffusion probability density function (PDF) estimation for a range of b-values and diffusion gradient direction tables. It does so at realistic SNR levels achievable in limited time on a high-performance 3T system for the whole human brain in vivo. We use both computational simulations and in vivo brain scans to quantify the angular resolution of two selected reconstruction methods: Q-ball imaging and the diffusion orientation transform. We propose a new analytical solution to the ODF derived from the DOT. Both techniques are analytical decomposition approaches that require identical acquisition and modest postprocessing times and, given the proposed modifications of the DOT, can be analyzed in a similar fashion. We find that an optimal HARDI protocol given a stringent time constraint (<10？min) combines a moderate b-value (around 2000？s/mm2) with a relatively low number of acquired directions (>48). Our findings generalize to other methods and additional improvements in MR acquisition techniques. 1. Introduction Diffusion-weighted magnetic resonance imaging (DW-MRI) is a clinical medical imaging technique that provides a unique view on the structure of brain white matter in vivo. Right from its early stages, in the early 1990s, DW-MRI was perceived to have immediate value for the evaluation of neuropathologies such as acute ischemic stroke. Since then, numerous advents in diffusion imaging technology have greatly augmented image quality unraveling new clinical applications. Moreover, the debut of diffusion tensor imaging (DTI) and fiber tractography enabled a completely new, noninvasive view on white matter fibre bundles connecting gray matter neural populations, of increasing importance for cognitive neuroimaging applications. With DTI and fiber tractography, the understanding of several neurological and psychiatric disorders, such as schizophrenia, traumas, stroke, and edemas, has been increased, and they have also been applied clinically to aid presurgical planning before intracranial mass resections. In DW-MRI, white matter fiber bundles are probed indirectly by measuring the directional specificity (anisotropy) of local water diffusion. During a small time interval (the “so-called” effective diffusion time ), at each location of the tissue, the diffusion causes a displacement of water molecules. Postprocessing of diffusion-weighted images is fundamentally aimed

Abstract:
We identify close pairs of galaxies from 278 deg^2 of Sloan Digital Sky Survey commissioning imaging data. The pairs are drawn from a sample of 330,041 galaxies with 18 < r^* < 20. We determine the angular correlation function of galaxy pairs, and find it to be stronger than the correlation function of single galaxies by a factor of 2.9 +/- 0.4. The two correlation functions have the same logarithmic slope of 0.77. We invert Limber's equation to estimate the three-dimensional correlation functions; we find clustering lengths of r_0= 4.2 +/- 0.4 h^{-1} Mpc for galaxies and 7.8 +/- 0.7 h^{-1} Mpc for galaxy pairs. These results agree well with the global richness dependence of the correlation functions of galaxy systems.

Abstract:
The connectivity and structural integrity of the white matter of the brain is nowadays known to be implicated into a wide range of brain-related disorders. However, it was not before the advent of diffusion Magnetic Resonance Imaging (dMRI) that researches have been able to examine the properties of white matter in vivo. Presently, among a range of various methods of dMRI, high angular resolution diffusion imaging (HARDI) is known to excel in its ability to provide reliable information about the local orientations of neural fasciculi (aka fibre tracts). Moreover, as opposed to the more traditional diffusion tensor imaging (DTI), HARDI is capable of distinguishing the orientations of multiple fibres passing through a given spatial voxel. Unfortunately, the ability of HARDI to discriminate between neural fibres that cross each other at acute angles is always limited, which is the main reason behind the development of numerous post-processing tools, aiming at the improvement of the directional resolution of HARDI. Among such tools is spherical deconvolution (SD). Due to its ill-posed nature, however, SD standardly relies on a number of a priori assumptions which are to render its results unique and stable. In this paper, we propose a different approach to the problem of SD in HARDI, which accounts for the spatial continuity of neural fibres as well as the presence of isotropic diffusion. Subsequently, we demonstrate how the proposed solution can be used to successfully overcome the effect of partial voluming, while preserving the spatial coherency of cerebral diffusion at moderate-to-severe noise levels. In a series of both in silico and in vivo experiments, the performance of the proposed method is compared with that of several available alternatives, with the comparative results clearly supporting the viability and usefulness of our approach.

Abstract:
A new mechanism of pattern formation different from the Turing and oscillatory instabilities in the reaction–diffusion systems was found. It is closely connected with the resonance phenomenon that appears in the models when Jacobi's matrix of the kinetic part is equivalent to Jordan cell and diffusion coefficients are cited. Some results of numerical calculations of the blood coagulation model are discussed. The pattern formation regimes that can be treated as the results from the resonance phenomenon were observed.

Abstract:
We discuss the diffusion phenomenon in the parabolic and hyperbolic regimes. New effects related to the finite velocity of the diffusion process are predicted, that can partially explain the strange behavior associated to adsorption phenomenon. For sake of simplicity, the analysis is performed by considering a sample in the shape of a slab limited by two perfectly blocking surfaces, in such a manner that the problem is one-dimensional in the space. Two cases are investigated. In the former, the initial distribution of the diffusing particles is assumed of gaussian type, centered around the symmetry surface in the middle of the sample. In the latter, the initial distribution is localized close to the limiting surfaces. In both cases, we show that the evolution toward to the equilibrium distribution is not monotonic. In particular, close to the limiting surfaces the bulk density of diffusing particles present maxima and minima related to the finite velocity of the diffusion process connected to the second order time derivative in the partial differential equation describing the evolution of the bulk density in the sample.