Geodesic regression on orientation distribution functions with its application to an aging study.

In this paper, we treat orientation distribution functions (ODFs) derived from high angular resolution diffusion imaging (HARDI) as elements of a Riemannian manifold and present a method for geodesic regression on this manifold. In order to find the optimal regression model, we pose this as a least-squares problem involving the sum-of-squared geodesic distances between observed ODFs and their model fitted data. We derive the appropriate gradient terms and employ gradient descent to find the minimizer of this least-squares optimization problem. In addition, we show how to perform statistical testing for determining the significance of the relationship between the manifold-valued regressors and the real-valued regressands. Experiments on both synthetic and real human data are presented. In particular, we examine aging effects on HARDI via geodesic regression of ODFs in normal adults aged 22 years old and above.

Evolution of hippocampal shapes across the human lifespan.

Aberrant hippocampal morphology plays an important role in the pathophysiology of aging. Volumetric analysis of the hippocampus has been performed in aging studies; however, the shape morphometry--which is potentially more informative in terms of related cognition--has yet to be examined. In this paper, we employed an advanced brain mapping technique, large deformation diffeomorphic metric mapping (LDDMM), and a dimensionality reduction approach, locally linear diffeomorphic metric embedding (LLDME), to explore age-related changes in hippocampal shape as delineated from magnetic resonance (MR) images of 302 healthy adults aged from 18 to 94 years. Compared with the hippocampal volumes, the hippocampal shapes clearly showed the nonlinear trajectory of biological aging across the human lifespan, where the variation of hippocampal shapes by age was characterized by a cubic polynomial. By integrating of LDDMM and LLDME, we were also able to illustrate the average hippocampal shapes in each individual decade. In addition, LDDMM and LLDME facilitated the identification of 63 years as a threshold beyond which hippocampal morphological changes were accelerated. Adults over 63 years of age showed the inward-deformation bilaterally in the head of the hippocampi and the left subiculum regardless of hippocampal volume reduction when compared to adults younger than 63. Hence, we demonstrated that the shape of anatomical structures added another dimension of structural morphological quantification beyond the volume in understanding aging.

Diffeomorphic metric mapping of high angular resolution diffusion imaging based on Riemannian structure of orientation distribution functions.

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 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.

Approximations of the diffeomorphic metric and their applications in shape learning.

In neuroimaging studies based on anatomical shapes, it is well-known that the dimensionality of the shape information is much higher than the number of subjects available. A major challenge in shape analysis is to develop a dimensionality reduction approach that is able to efficiently characterize anatomical variations in a low-dimensional space. For this, there is a need to characterize shape variations among individuals for N given subjects. Therefore, one would need to calculate (2(N)) mappings between any two shapes and obtain their distance matrix. In this paper, we propose a method that reduces the computational burden to N mappings. This is made possible by making use of the first- and second-order approximations of the metric distance between two brain structural shapes in a diffeomorphic metric space. We directly derive these approximations based on the so-called conservation law of momentum, i.e., the diffeomorphic transformation acting on anatomical shapes along the geodesic is completely determined by its velocity at the origin of a fixed template. This allows for estimating morphological variation of two shapes through the first- and second-order approximations of the initial velocity in the tangent space of the diffeomorphisms at the template. We also introduce an alternative representation of these approximations through the initial momentum, i.e., a linear transformation of the initial velocity, and provide a simple computational algorithm for the matrix of the diffeomorphic metric. We employ this algorithm to compute the distance matrix of hippocampal shapes among an aging population used in a dimensionality reduction analysis, namely, ISOMAP. Our results demonstrate that the first- and second-order approximations are sufficient to characterize shape variations when compared to the diffeomorphic metric constructed through (2(N)) mappings in ISOMAP analysis.

Large deformation diffeomorphic metric mapping of orientation distribution functions.

We propose a novel large deformation diffeomorphic registration algorithm to align high angular resolution diffusion images (HARDI) characterized by Orientation Distribution Functions (ODF). 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. We first extend ODFs traditionally defined in a unit sphere to a generalized ODF defined in R3. This makes it easy for an affine transformation as well as a diffeomorphic group action to be applied on the ODF. We then construct a Riemannian space of the generalized ODFs and incorporate its Riemannian metric for the similarity of ODFs 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 generalized ODFs, and present its numerical implementation. Both synthetic and real brain HARDI data are used to illustrate the performance of our registration algorithm.

A nonparametric Riemannian framework for processing high angular resolution diffusion images and its applications to ODF-based morphometry.

High angular resolution diffusion imaging (HARDI) has become an important technique for imaging complex oriented structures in the brain and other anatomical tissues. This has motivated the recent development of several methods for computing the orientation probability density function (PDF) at each voxel. However, much less work has been done on developing techniques for filtering, interpolation, averaging and principal geodesic analysis of orientation PDF fields. In this paper, we present a Riemannian framework for performing such operations. The proposed framework does not require that the orientation PDFs be represented by any fixed parameterization, such as a mixture of von Mises-Fisher distributions or a spherical harmonic expansion. Instead, we use a nonparametric representation of the orientation PDF. We exploit the fact that under the square-root re-parameterization, the space of orientation PDFs forms a Riemannian manifold: the positive orthant of the unit Hilbert sphere. We show that various orientation PDF processing operations, such as filtering, interpolation, averaging and principal geodesic analysis, may be posed as optimization problems on the Hilbert sphere, and can be solved using Riemannian gradient descent. We illustrate these concepts with numerous experiments on synthetic, phantom and real datasets. We show their application to studying left/right brain asymmetries.

Locally Linear Diffeomorphic Metric Embedding (LLDME) for surface-based anatomical shape modeling.

This paper presents the algorithm, Locally Linear Diffeomorphic Metric Embedding (LLDME), for constructing efficient and compact representations of surface-based brain shapes whose variations are characterized using Large Deformation Diffeomorphic Metric Mapping (LDDMM). Our hypothesis is that the shape variations in the infinite-dimensional diffeomorphic metric space can be captured by a low-dimensional space. To do so, traditional Locally Linear Embedding (LLE) that reconstructs a data point from its neighbors in Euclidean space is extended to LLDME that requires interpolating a shape from its neighbors in the infinite-dimensional diffeomorphic metric space. This is made possible through the conservation law of momentum derived from LDDMM. It indicates that initial momentum, a linear transformation of the initial velocity of diffeomorphic flows, at a fixed template shape determines the geodesic connecting the template to a subject's shape in the diffeomorphic metric space and becomes the shape signature of an individual subject. This leads to the compact linear representation of the nonlinear diffeomorphisms in terms of the initial momentum. Since the initial momentum is in a linear space, a shape can be approximated by a linear combination of its neighbors in the diffeomorphic metric space. In addition, we provide efficient computations for the metric distance between two shapes through the first order approximation of the geodesic using the initial momentum as well as for the reconstruction of a shape given its low-dimensional Euclidean coordinates using the geodesic shooting with the initial momentum as the initial condition. Experiments are performed on the hippocampal shapes of 302 normal subjects across the whole life span (18-94years). Compared with Principal Component Analysis and ISOMAP, LLDME provides the most compact and efficient representation of the age-related hippocampal shapes. Even though the hippocampal volumes among young adults are as variable as those in older adults, LLDME disentangles the hippocampal local shape variation from the hippocampal size and thus reveals the nonlinear relationship of the hippocampal morphometry with age.

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

As 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 fibercup09@gmail.com and updates will be published on the webpage.

Estimating orientation distribution functions with probability density constraints and spatial regularity.

High angular resolution diffusion imaging (HARDI) has become an important magnetic resonance technique for in vivo imaging. Current techniques for estimating the diffusion orientation distribution function (ODF), i.e., the probability density function of water diffusion along any direction, do not enforce the estimated ODF to be nonnegative or to sum up to one. Very often this leads to an estimated ODF which is not a proper probability density function. In addition, current methods do not enforce any spatial regularity of the data. In this paper, we propose an estimation method that naturally constrains the estimated ODF to be a proper probability density function and regularizes this estimate using spatial information. By making use of the spherical harmonic representation, we pose the ODF estimation problem as a convex optimization problem and propose a coordinate descent method that converges to the minimizer of the proposed cost function. We illustrate our approach with experiments on synthetic and real data.