The multivariate A/C/E model and the genetics of fiber architecture

We present a new algorithm to compute the voxel-wise genetic contribution to brain fiber microstructure using diffusion tensor imaging (DTI) in a dataset of 25 monozygotic (MZ) twins and 25 dizygotic (DZ) twin pairs (100 subjects total). First, the structural and DT scans were linearly co-registered. Structural MR scans were nonlinearly mapped via a 3D fluid transformation to a geometrically centered mean template, and the deformation fields were applied to the DTI volumes. After tensor re-orientation to realign them to the anatomy, we computed several scalar and multivariate DT-derived measures including the geodesic anisotropy (GA), the tensor eigenvalues and the full diffusion tensors. A covariance-weighted distance was measured between twins in the Log-Euclidean framework [2], and used as input to a maximum-likelihood based algorithm to compute the contributions from genetics (A), common environmental factors (C) and unique environmental ones (E) to fiber architecture. Quanititative genetic studies can take advantage of the full information in the diffusion tensor, using covariance weighted distances and statistics on the tensor manifold.

Multivariate variance-components analysis in DTI

Twin studies are a major research direction in imaging genetics, a new field, which combines algorithms from quantitative genetics and neuroimaging to assess genetic effects on the brain. In twin imaging studies, it is common to estimate the intraclass correlation (ICC), which measures the resemblance between twin pairs for a given phenotype. In this paper, we extend the commonly used Pearson correlation to a more appropriate definition, which uses restricted maximum likelihood methods (REML). We computed proportion of phenotypic variance due to additive (A) genetic factors, common (C) and unique (E) environmental factors using a new definition of the variance components in the diffusion tensor-valued signals. We applied our analysis to a dataset of Diffusion Tensor Images (DTI) from 25 identical and 25 fraternal twin pairs. Differences between the REML and Pearson estimators were plotted for different sample sizes, showing that the REML approach avoids severe biases when samples are smaller. Measures of genetic effects were computed for scalar and multivariate diffusion tensor derived measures including the geodesic anisotropy (tGA) and the full diffusion tensors (DT), revealing voxel-wise genetic contributions to brain fiber microstructure.

Automatic clustering and population analysis of white matter tracts using maximum density paths

We introduce a framework for population analysis of white matter tracts based on diffusion-weighted images of the brain. The framework enables extraction of fibers from high angular resolution diffusion images (HARDI); clustering of the fibers based partly on prior knowledge from an atlas; representation of the fiber bundles compactly using a path following points of highest density (maximum density path; MDP); and registration of these paths together using geodesic curve matching to find local correspondences across a population. We demonstrate our method on 4-Tesla HARDI scans from 565 young adults to compute localized statistics across 50 white matter tracts based on fractional anisotropy (FA). Experimental results show increased sensitivity in the determination of genetic influences on principal fiber tracts compared to the tract-based spatial statistics (TBSS) method. Our results show that the MDP representation reveals important parts of the white matter structure and considerably reduces the dimensionality over comparable fiber matching approaches.

How does angular resolution affect diffusion imaging measures?

A key question in diffusion imaging is how many diffusion-weighted images suffice to provide adequate signal-to-noise ratio (SNR) for studies of fiber integrity. Motion, physiological effects, and scan duration all affect the achievable SNR in real brain images, making theoretical studies and simulations only partially useful. We therefore scanned 50 healthy adults with 105-gradient high-angular resolution diffusion imaging (HARDI) at 4T. From gradient image subsets of varying size (6 ≤ N ≤ 94) that optimized a spherical angular distribution energy, we created SNR plots (versus gradient numbers) for seven common diffusion anisotropy indices: fractional and relative anisotropy (FA, RA), mean diffusivity (MD), volume ratio (VR), geodesic anisotropy (GA), its hyperbolic tangent (tGA), and generalized fractional anisotropy (GFA). SNR, defined in a region of interest in the corpus callosum, was near-maximal with 58, 66, and 62 gradients for MD, FA, and RA, respectively, and with about 55 gradients for GA and tGA. For VR and GFA, SNR increased rapidly with more gradients. SNR was optimized when the ratio of diffusion-sensitized to non-sensitized images was 9.13 for GA and tGA, 10.57 for FA, 9.17 for RA, and 26 for MD and VR. In orientation density functions modeling the HARDI signal as a continuous mixture of tensors, the diffusion profile reconstruction accuracy rose rapidly with additional gradients. These plots may help in making trade-off decisions when designing diffusion imaging protocols.

Working covariance model selection for generalized estimating equations

We investigate methods for data-based selection of working covariance models in the analysis of correlated data with generalized estimating equations. We study two selection criteria: Gaussian pseudolikelihood and a geodesic distance based on discrepancy between model-sensitive and model-robust regression parameter covariance estimators. The Gaussian pseudolikelihood is found in simulation to be reasonably sensitive for several response distributions and noncanonical mean-variance relations for longitudinal data. Application is also made to a clinical dataset. Assessment of adequacy of both correlation and variance models for longitudinal data should be routine in applications, and we describe open-source software supporting this practice.

An elementary approach to gap theorems

Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension >= 3. Suppose that the sectional curvature K satisfies -1-s(r) <= K <= -1, where r denotes distance to a fixed point in M. If lim(r ->infinity) e(2r) s(r) = 0, then (M, g) has to be isometric to H-n.The same proof also yields that if K satisfies -s(r) <= K <= 0 where lim(r ->infinity) r(2) s(r) = 0, then (M, g) is isometric to R-n, a result due to Greene and Wu.Our second result is a local one: Let (M, g) be any Riemannian manifold. For a E R, if K < a on a geodesic ball Bp (R) in M and K = a on partial derivative B-p (R), then K = a on B-p (R).

Beyond Gauss: Image-set matching on the Riemannian manifold of PDFs

State-of-the-art image-set matching techniques typically implicitly model each image-set with a Gaussian distribution. Here, we propose to go beyond these representations and model image-sets as probability distribution functions (PDFs) using kernel density estimators. To compare and match image-sets, we exploit Csiszar´ f-divergences, which bear strong connections to the geodesic distance defined on the space of PDFs, i.e., the statistical manifold. Furthermore, we introduce valid positive definite kernels on the statistical manifold, which let us make use of more powerful classification schemes to match image-sets. Finally, we introduce a supervised dimensionality reduction technique that learns a latent space where f-divergences reflect the class labels of the data. Our experiments on diverse problems, such as video-based face recognition and dynamic texture classification, evidence the benefits of our approach over the state-of-the-art image-set matching methods.

Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam

We explore an isoparametric interpolation of total quaternion for geometrically consistent, strain-objective and path-independent finite element solutions of the geometrically exact beam. This interpolation is a variant of the broader class known as slerp. The equivalence between the proposed interpolation and that of relative rotation is shown without any recourse to local bijection between quaternions and rotations. We show that, for a two-noded beam element, the use of relative rotation is not mandatory for attaining consistency cum objectivity and an appropriate interpolation of total rotation variables is sufficient. The interpolation of total quaternion, which is computationally more efficient than the one based on local rotations, converts nodal rotation vectors to quaternions and interpolates them in a manner consistent with the character of the rotation manifold. This interpolation, unlike the additive interpolation of total rotation, corresponds to a geodesic on the rotation manifold. For beam elements with more than two nodes, however, a consistent extension of the proposed quaternion interpolation is difficult. Alternatively, a quaternion-based procedure involving interpolation of relative rotations is proposed for such higher order elements. We also briefly discuss a strategy for the removal of possible singularity in the interpolation of quaternions, proposed in [I. Romero, The interpolation of rotations and its application to finite element models of geometrically exact rods, Comput. Mech. 34 (2004) 121–133]. The strain-objectivity and path-independence of solutions are justified theoretically and then demonstrated through numerical experiments. This study, being focused only on the interpolation of rotations, uses a standard finite element discretization, as adopted by Simo and Vu-Quoc [J.C. Simo, L. Vu-Quoc, A three-dimensional finite rod model part II: computational aspects, Comput. Methods Appl. Mech. Engrg. 58 (1986) 79–116]. The rotation update is achieved via quaternion multiplication followed by the extraction of the rotation vector. Nodal rotations are stored in terms of rotation vectors and no secondary storages are required.

Application of novel method for surface-ray tracing over hyperboloidal scatterers

A novel geodesic constant method has been developed for the hitherto unsolved problem of surface-ray tracing over a class of surface, namely the general hyperboloid of revolution (GHOR). All the ray-geometric parameters are obtained analytically in a one-parameter form. The ray parameters derived here for the first time can be readily used in the UTD formulation for computing the mutual coupling between the antennas located on the GHOR.

Surface-ray tracing on hybrid surfaces of revolution for UTD mutual coupling analysis

An angle invariance property based on Hertz's principle of particle dynamics is employed to facilitate the surface-ray tracing on nondevelopable hybrid quadric surfaces of revolution (h-QUASOR's). This property, when used in conjunction with a Geodesic Constant Method, yields analytical expressions for all the ray-parameters required in the UTD formulation. Differential geometrical considerations require that some of the ray-parameters (defined heuristically in the UTD for the canonical convex surfaces) be modified before the UTD can be applied to such hybrid surfaces. Mutual coupling results for finite-dimensional slots have been presented as an example on a satellite launch vehicle modeled by general paraboloid of revolution and right circular cylinder.

Surface-ray tracing solution of ellipsoid of revolution for UTD mutual coupling applications

An analytical surface-ray tracing has been carried out for the prolate ellipsoid of revolution using a novel geodesic constant method. This method yields closed form expressions for all the ray-geometric parameters required for the UTD mutual coupling calculations for the antennas located arbitrarily in three dimensions, on the ellipsoid of revolution.

A differential geometric method for kinematic analysis of two- and three-degree-of-freedom rigid body motions

In this paper, we present a novel differential geometric characterization of two- and three-degree-of-freedom rigid body kinematics, using a metric defined on dual vectors. The instantaneous angular and linear velocities of a rigid body are expressed as a dual velocity vector, and dual inner product is defined on this dual vector, resulting in a positive semi-definite and symmetric dual matrix. We show that the maximum and minimum magnitude of the dual velocity vector, for a unit speed motion, can be obtained as eigenvalues of this dual matrix. Furthermore, we show that the tip of the dual velocity vector lies on a dual ellipse for a two-degree-of-freedom motion and on a dual ellipsoid for a three-degree-of-freedom motion. In this manner, the velocity distribution of a rigid body can be studied algebraically in terms of the eigenvalues of a dual matrix or geometrically with the dual ellipse and ellipsoid. The second-order properties of the two- and three-degree-of-freedom motions of a rigid body are also obtained from the derivatives of the elements of the dual matrix. This results in a definition of the geodesic motion of a rigid body. The theoretical results are illustrated with the help of a spatial 2R and a parallel three-degree-of-freedom manipulator.

Geosphere laminations in free groups

We construct for free groups, which are codimension one analogues of geodesic laminations on surfaces. Other analogues that have been constructed by several authors are dimension-one instead of codimension-one. Our main result is that the space of such laminations is compact. This in turn is based on the result that crossing, in the sense of Scott-Swarup, is an open condition. Our construction is based on Hatcher's normal form for spheres in the model manifold.

Badly approximable numbers and vectors in Cantor-like sets

We show that a large class of Cantor-like sets of R-d, d >= 1, contains uncountably many badly approximable numbers, respectively badly approximable vectors, when d >= 2. An analogous result is also proved for subsets of R-d arising in the study of geodesic flows corresponding to (d+1)-dimensional manifolds of constant negative curvature and finite volume, generalizing the set of badly approximable numbers in R. Furthermore, we describe a condition on sets, which is fulfilled by a large class, ensuring a large intersection with these Cantor-like sets.

Rainbow connectivity: hardness and tractability

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph such that G is (strongly) rainbow connected. In this paper we study the rainbow connectivity problem and the strong rainbow connectivity problem from a computational point of view. Our main results can be summarised as below: 1) For every fixed k >= 3, it is NP-Complete to decide whether src(G) <= k even when the graph G is bipartite. 2) For every fixed odd k >= 3, it is NP-Complete to decide whether rc(G) <= k. This resolves one of the open problems posed by Chakraborty et al. (J. Comb. Opt., 2011) where they prove the hardness for the even case. 3) The following problem is fixed parameter tractable: Given a graph G, determine the maximum number of pairs of vertices that can be rainbow connected using two colors. 4) For a directed graph G, it is NP-Complete to decide whether rc(G) <= 2.