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

Noise reduction in magnitude diffusion-weighted images using spatial similarity and diffusion redundancy

PURPOSE

Diffusion-weighted imaging (DWI) has significant value in clinical application, which however suffers from a serious low signal-to-noise ratio (SNR) problem, especially at high spatial resolution and/or high diffusion sensitivity factor.

METHODS

Here, we propose a denoising method for magnitude DWI. The method consists of two modules: pre-denoising and post-filtering, the former mines the diffusion redundancy by local kernel principal component analysis, and the latter fully mines the non-local self-similarity using patch-based non-local mean.

RESULTS

Validated by simulation and in vivo datasets, the experiment results show that the proposed method is capable of improving the SNR of the whole brain, thus enhancing the performance for diffusion metrics estimation, crossing fiber discrimination, and human brain fiber tractography tracking compared with the different three state-of-the-art comparison methods. More importantly, the proposed method consistently exhibits superior performance to comparison methods when used for denoising diffusion data acquired with sensitivity encoding (SENSE).

CONCLUSION

The proposed denoising method is expected to show significant practicability in acquiring high-quality whole-brain diffusion data, which is crucial for many neuroscience studies.

A Riemannian framework for incorporating white matter bundle prior in orientation distribution function based tractography algorithms

Diffusion magnetic resonance imaging (dMRI) tractography is a powerful approach to study brain structural connectivity. However, its reliability in a clinical context is still highly debated. Recent studies have shown that most classical algorithms achieve to recover the majority of existing true bundles. However, the generated tractograms contain many invalid bundles. This is due to the crossing fibers and bottleneck problems which increase the number of false positive fibers. In this work, we proposed to overpass this limitation with a novel method to guide the algorithms in those challenging regions with prior knowledge of the anatomy. We developed a method to create a combination of anatomical prior applicable to any orientation distribution function (ODF)-based tractography algorithms. The proposed method captures the tract orientation distribution (TOD) from an atlas of segmented fiber bundles and incorporates it during the tracking process, using a Riemannian framework. We tested the prior incorporation method on two ODF-based state-of-the-art algorithms, iFOD2 and Trekker PTT, on the diffusion-simulated connectivity (DiSCo) dataset and on the Human Connectome Project (HCP) data. We also compared our method with two bundles priors generated by the bundle specific tractography (BST) method. We showed that our method improves the overall spatial coverage and connectivity of a tractogram on the two datasets, especially in crossing fiber regions. Moreover, the fiber reconstruction may be improved on clinical data, informed by prior extracted on high quality data, and therefore could help in the study of brain anatomy and function.

Neural orientation distribution fields for estimation and uncertainty quantification in diffusion MRI

Inferring brain connectivity and structure in-vivo requires accurate estimation of the orientation distribution function (ODF), which encodes key local tissue properties. However, estimating the ODF from diffusion MRI (dMRI) signals is a challenging inverse problem due to obstacles such as significant noise, high-dimensional parameter spaces, and sparse angular measurements. In this paper, we address these challenges by proposing a novel deep-learning based methodology for continuous estimation and uncertainty quantification of the spatially varying ODF field. We use a neural field (NF) to parameterize a random series representation of the latent ODFs, implicitly modeling the often ignored but valuable spatial correlation structures in the data, and thereby improving efficiency in sparse and noisy regimes. An analytic approximation to the posterior predictive distribution is derived which can be used to quantify the uncertainty in the ODF estimate at any spatial location, avoiding the need for expensive resampling-based approaches that are typically employed for this purpose. We present empirical evaluations on both synthetic and real in-vivo diffusion data, demonstrating the advantages of our method over existing approaches.

Flow-based Geometric Interpolation of Fiber Orientation Distribution Functions

The fiber orientation distribution function (FOD) is an advanced model for high angular resolution diffusion MRI representing complex fiber geometry. However, the complicated mathematical structures of the FOD function pose challenges for FOD image processing tasks such as interpolation, which plays a critical role in the propagation of fiber tracts in tractography. In FOD-based tractography, linear interpolation is commonly used for numerical efficiency, but it is prone to generate false artificial information, leading to anatomically incorrect fiber tracts. To overcome this difficulty, we propose a flowbased and geometrically consistent interpolation framework that considers peak-wise rotations of FODs within the neighborhood of each location. Our method decomposes a FOD function into multiple components and uses a smooth vector field to model the flows of each peak in its neighborhood. To generate the interpolated result along the flow of each vector field, we develop a closed-form and efficient method to rotate FOD peaks in neighboring voxels and realize geometrically consistent interpolation of FOD components. By combining the interpolation results from each peak, we obtain the final interpolation of FODs. Experimental results on Human Connectome Project (HCP) data demonstrate that our method produces anatomically more meaningful FOD interpolations and significantly enhances tractography performance.

Horizontal Flows and Manifold Stochastics in Geometric Deep Learning

We introduce two constructions in geometric deep learning for 1) transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy; and 2) allowing efficient evaluation of manifold convolution layers by sampling manifold valued random variables that center around a weighted diffusion mean. Both methods are inspired by stochastics on manifolds and geometric statistics, and provide examples of how stochastic methods - here horizontal frame bundle flows and non-linear bridge sampling schemes, can be used in geometric deep learning. We outline the theoretical foundation of the two methods, discuss their relation to Euclidean deep networks and existing methodology in geometric deep learning, and establish important properties of the proposed constructions.

ManifoldNet: A Deep Neural Network for Manifold-Valued Data With Applications

Geometric deep learning is a relatively nascent field that has attracted significant attention in the past few years. This is partly due to the availability of data acquired from non-euclidean domains or features extracted from euclidean-space data that reside on smooth manifolds. For instance, pose data commonly encountered in computer vision reside in Lie groups, while covariance matrices that are ubiquitous in many fields and diffusion tensors encountered in medical imaging domain reside on the manifold of symmetric positive definite matrices. Much of this data is naturally represented as a grid of manifold-valued data. In this paper we present a novel theoretical framework for developing deep neural networks to cope with these grids of manifold-valued data inputs. We also present a novel architecture to realize this theory and call it the ManifoldNet. Analogous to vector spaces where convolutions are equivalent to computing weighted sums, manifold-valued data 'convolutions' can be defined using the weighted Fréchet Mean ([Formula: see text]). (This requires endowing the manifold with a Riemannian structure if it did not already come with one.) The hidden layers of ManifoldNet compute [Formula: see text]s of their inputs, where the weights are to be learnt. This means the data remain manifold-valued as they propagate through the hidden layers. To reduce computational complexity, we present a provably convergent recursive algorithm for computing the [Formula: see text]. Further, we prove that on non-constant sectional curvature manifolds, each [Formula: see text] layer is a contraction mapping and provide constructive evidence for its non-collapsibility when stacked in layers. This captures the two fundamental properties of deep network layers. Analogous to the equivariance of convolution in euclidean space to translations, we prove that the [Formula: see text] is equivariant to the action of the group of isometries admitted by the Riemannian manifold on which the data reside. To showcase the performance of ManifoldNet, we present several experiments using both computer vision and medical imaging data sets.

Rigid motion invariant statistical shape modeling based on discrete fundamental forms data from the osteoarthritis initiative and the Alzheimer' disease neuroimaging initiative

We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free. By analyzing metric distortion and curvature of shapes as elements of Lie groups in a consistent Riemannian setting, we construct a framework that reliably handles large deformations. Due to the explicit character of Lie group operations, our non-Euclidean method is very efficient allowing for fast and numerically robust processing. This facilitates Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power. Additionally, as planar configurations form a submanifold in shape space, our representation allows for effective estimation of quasi-isometric surfaces flattenings. We evaluate the performance of our model w.r.t. shape-based classification of hippocampus and femur malformations due to Alzheimer's disease and osteoarthritis, respectively. In particular, we outperform state-of-the-art classifiers based on geometric deep learning as well as statistical shape modeling especially in presence of sparse training data. To provide insight into the model's ability of capturing biological shape variability, we carry out an analysis of specificity and generalization ability.

TractLearn: A geodesic learning framework for quantitative analysis of brain bundles

Deep learning-based convolutional neural networks have recently proved their efficiency in providing fast segmentation of major brain fascicles structures, based on diffusion-weighted imaging. The quantitative analysis of brain fascicles then relies on metrics either coming from the tractography process itself or from each voxel along the bundle. Statistical detection of abnormal voxels in the context of disease usually relies on univariate and multivariate statistics models, such as the General Linear Model (GLM). Yet in the case of high-dimensional low sample size data, the GLM often implies high standard deviation range in controls due to anatomical variability, despite the commonly used smoothing process. This can lead to difficulties to detect subtle quantitative alterations from a brain bundle at the voxel scale. Here we introduce TractLearn, a unified framework for brain fascicles quantitative analyses by using geodesic learning as a data-driven learning task. TractLearn allows a mapping between the image high-dimensional domain and the reduced latent space of brain fascicles using a Riemannian approach. We illustrate the robustness of this method on a healthy population with test-retest acquisition of multi-shell diffusion MRI data, demonstrating that it is possible to separately study the global effect due to different MRI sessions from the effect of local bundle alterations. We have then tested the efficiency of our algorithm on a sample of 5 age-matched subjects referred with mild traumatic brain injury. Our contributions are to propose: 1/ A manifold approach to capture controls variability as standard reference instead of an atlas approach based on a Euclidean mean. 2/ A tool to detect global variation of voxels' quantitative values, which accounts for voxels' interactions in a structure rather than analyzing voxels independently. 3/ A ready-to-plug algorithm to highlight nonlinear variation of diffusion MRI metrics. With this regard, TractLearn is a ready-to-use algorithm for precision medicine.

Interpolation and Averaging of Diffusion MRI Multi-Compartment Models

Multi-compartment models (MCM) are increasingly used to characterize the brain white matter microstructure from diffusion-weighted imaging (DWI). Their use in clinical studies is however limited by the inability to resample an MCM image towards a common reference frame, or to construct atlases from such brain microstructure models. We propose to solve this problem by first identifying that these two tasks amount to the same problem. We propose to tackle it by viewing it as a simplification problem, solved thanks to spectral clustering and the definition of semi-metrics between several usual compartments encountered in the MCM literature. This generic framework is evaluated for two models: the multi-tensor model where individual fibers are modeled as individual tensors and the diffusion direction imaging (DDI) model that differentiates intra- and extra-axonal components of each fiber. Results on simulated data, simulated transformations and real data show the ability of our method to well interpolate MCM images of these types. We finally present as an application an MCM template of normal controls constructed using our approach.

Denoise magnitude diffusion magnetic resonance images via variance-stabilizing transformation and optimal singular-value manipulation

Although shown to have a great utility for a wide range of neuroscientific and clinical applications, diffusion-weighted magnetic resonance imaging (dMRI) faces a major challenge of low signal-to-noise ratio (SNR), especially when pushing the spatial resolution for improved delineation of brain's fine structure or increasing the diffusion weighting for increased angular contrast or both. Here, we introduce a comprehensive denoising framework for denoising magnitude dMRI. The framework synergistically combines the variance stabilizing transform (VST) with optimal singular value manipulation. The purpose of VST is to transform the Rician data to Gaussian-like data so that an asymptotically optimal singular value manipulation strategy tailored for Gaussian data can be used. The output of the framework is the estimated underlying diffusion signal for each voxel in the image domain. The usefulness of the proposed framework for denoising magnitude dMRI is demonstrated using both simulation and real-data experiments. Our results show that the proposed denoising framework can significantly improve SNR across the entire brain, leading to substantially enhanced performances for estimating diffusion tensor related indices and for resolving crossing fibers when compared to another competing method. More encouragingly, the proposed method when used to denoise a single average of 7 ​Tesla Human Connectome Project-style diffusion acquisition provided comparable performances relative to those achievable with ten averages for resolving multiple fiber populations across the brain. As such, the proposed denoising method is expected to have a great utility for high-quality, high-resolution whole-brain dMRI, desirable for many neuroscientific and clinical applications.

Estimating uncertainty in MRF-based image segmentation: A perfect-MCMC approach

Typical methods for image segmentation, or labeling, formulate and solve an optimization problem to produce a single optimal solution. For applications in clinical decision support relying on automated medical image segmentation, it is also desirable for methods to inform about (i) the uncertainty in label assignments or object boundaries or (ii) alternate close-to-optimal solutions. However, typical methods fail to do so. To estimate uncertainty, while some Bayesian methods rely on simplified prior models and approximate variational inference schemes, others rely on sampling segmentations from the associated posterior model using (i) traditional Markov chain Monte Carlo (MCMC) methods based on Gibbs sampling or (ii) approximate perturbation models. However, in such typical approaches, in practice, the resulting inference or generated sample set are approximations that deviate significantly from those indicated by the true posterior. To estimate uncertainty, we propose the modern paradigm of perfect MCMC sampling to sample multi-label segmentations from generic Bayesian Markov random field (MRF) models, in finite time for exact inference. Furthermore, for exact sampling in generic Bayesian MRFs, we extend the theory underlying Fill's algorithm to generic MRF models by proposing a novel bounding-chain algorithm. On several classic problems in medical image analysis, and several modeling and inference schemes, results on simulated data and clinical brain magnetic resonance images show that our uncertainty estimates gain accuracy over several state-of-the-art inference methods.

Advances in computational and statistical diffusion MRI

Computational methods are crucial for the analysis of diffusion magnetic resonance imaging (MRI) of the brain. Computational diffusion MRI can provide rich information at many size scales, including local microstructure measures such as diffusion anisotropies or apparent axon diameters, whole-brain connectivity information that describes the brain's wiring diagram and population-based studies in health and disease. Many of the diffusion MRI analyses performed today were not possible five, ten or twenty years ago, due to the requirements for large amounts of computer memory or processor time. In addition, mathematical frameworks had to be developed or adapted from other fields to create new ways to analyze diffusion MRI data. The purpose of this review is to highlight recent computational and statistical advances in diffusion MRI and to put these advances into context by comparison with the more traditional computational methods that are in popular clinical and scientific use. We aim to provide a high-level overview of interest to diffusion MRI researchers, with a more in-depth treatment to illustrate selected computational advances.

Riemannian Nonlinear Mixed Effects Models: Analyzing Longitudinal Deformations in Neuroimaging

Statistical machine learning models that operate on manifold-valued data are being extensively studied in vision, motivated by applications in activity recognition, feature tracking and medical imaging. While non-parametric methods have been relatively well studied in the literature, efficient formulations for parametric models (which may offer benefits in small sample size regimes) have only emerged recently. So far, manifold-valued regression models (such as geodesic regression) are restricted to the analysis of cross-sectional data, i.e., the so-called "fixed effects" in statistics. But in most "longitudinal analysis" (e.g., when a participant provides multiple measurements, over time) the application of fixed effects models is problematic. In an effort to answer this need, this paper generalizes non-linear mixed effects model to the regime where the response variable is manifold-valued, i.e., f : Rd → ℳ. We derive the underlying model and estimation schemes and demonstrate the immediate benefits such a model can provide - both for group level and individual level analysis - on longitudinal brain imaging data. The direct consequence of our results is that longitudinal analysis of manifold-valued measurements (especially, the symmetric positive definite manifold) can be conducted in a computationally tractable manner.

SR-HARDI: Spatially Regularizing High Angular Resolution Diffusion Imaging

High angular resolution diffusion imaging (HARDI) has recently been of great interest in mapping the orientation of intra-voxel crossing fibers, and such orientation information allows one to infer the connectivity patterns prevalent among different brain regions and possible changes in such connectivity over time for various neurodegenerative and neuropsychiatric diseases. The aim of this paper is to propose a penalized multi-scale adaptive regression model (PMARM) framework to spatially and adaptively infer the orientation distribution function (ODF) of water diffusion in regions with complex fiber configurations. In PMARM, we reformulate the HARDI imaging reconstruction as a weighted regularized least-squares regression (WRLSR) problem. Similarity and distance weights are introduced to account for spatial smoothness of HARDI, while preserving the unknown discontinuities (e.g., edges between white matter and grey matter) of HARDI. The L₁ penalty function is introduced to ensure the sparse solutions of ODFs, while a scaled L₁ weighted estimator is calculated to correct the bias introduced by the L₁ penalty at each voxel. In PMARM, we integrate the multiscale adaptive regression models (Li et al., 2011), the propagation-separation method (Polzehl and Spokoiny, 2000), and Lasso (least absolute shrinkage and selection operator) (Tibshirani, 1996) to adaptively estimate ODFs across voxels. Experimental results indicate that PMARM can reduce the angle detection errors on fiber crossing area and provide more accurate reconstruction than standard voxel-wise methods.

Combined Tensor Fitting and TV Regularization in Diffusion Tensor Imaging Based on a Riemannian Manifold Approach

In this paper, we consider combined TV denoising and diffusion tensor fitting in DTI using the affine-invariant Riemannian metric on the space of diffusion tensors. Instead of first fitting the diffusion tensors, and then denoising them, we define a suitable TV type energy functional which incorporates the measured DWIs (using an inverse problem setup) and which measures the nearness of neighboring tensors in the manifold. To approach this functional, we propose generalized forward- backward splitting algorithms which combine an explicit and several implicit steps performed on a decomposition of the functional. We validate the performance of the derived algorithms on synthetic and real DTI data. In particular, we work on real 3D data. To our knowledge, the present paper describes the first approach to TV regularization in a combined manifold and inverse problem setup.

Kernel regression estimation of fiber orientation mixtures in diffusion MRI

We present and evaluate a method for kernel regression estimation of fiber orientations and associated volume fractions for diffusion MR tractography and population-based atlas construction in clinical imaging studies of brain white matter. This is a model-based image processing technique in which representative fiber models are estimated from collections of component fiber models in model-valued image data. This extends prior work in nonparametric image processing and multi-compartment processing to provide computational tools for image interpolation, smoothing, and fusion with fiber orientation mixtures. In contrast to related work on multi-compartment processing, this approach is based on directional measures of divergence and includes data-adaptive extensions for model selection and bilateral filtering. This is useful for reconstructing complex anatomical features in clinical datasets analyzed with the ball-and-sticks model, and our framework's data-adaptive extensions are potentially useful for general multi-compartment image processing. We experimentally evaluate our approach with both synthetic data from computational phantoms and in vivo clinical data from human subjects. With synthetic data experiments, we evaluate performance based on errors in fiber orientation, volume fraction, compartment count, and tractography-based connectivity. With in vivo data experiments, we first show improved scan-rescan reproducibility and reliability of quantitative fiber bundle metrics, including mean length, volume, streamline count, and mean volume fraction. We then demonstrate the creation of a multi-fiber tractography atlas from a population of 80 human subjects. In comparison to single tensor atlasing, our multi-fiber atlas shows more complete features of known fiber bundles and includes reconstructions of the lateral projections of the corpus callosum and complex fronto-parietal connections of the superior longitudinal fasciculus I, II, and III.

Diffusion MRI abnormalities detection with orientation distribution functions: a multiple sclerosis longitudinal study

We propose a new algorithm for the voxelwise analysis of orientation distribution functions between one image and a group of reference images. It relies on a generic framework for the comparison of diffusion probabilities on the sphere, sampled from the underlying models. We demonstrate that this method, combined to dimensionality reduction through a principal component analysis, allows for more robust detection of lesions on simulated data when compared to classical tensor-based analysis. We then demonstrate the efficiency of this pipeline on the longitudinal comparison of multiple sclerosis patients at an early stage of the disease: right after their first clinically isolated syndrome (CIS) and three months later. We demonstrate the predictive value of ODF-based scores for the early detection of lesions that will appear or heal.

Adaptive smoothing of multi-shell diffusion weighted magnetic resonance data by msPOAS

We present a novel multi-shell position-orientation adaptive smoothing (msPOAS) method for diffusion weighted magnetic resonance data. Smoothing in voxel and diffusion gradient space is embedded in an iterative adaptive multiscale approach. The adaptive character avoids blurring of the inherent structures and preserves discontinuities. The simultaneous treatment of all q-shells improves the stability compared to single-shell approaches such as the original POAS method. The msPOAS implementation simplifies and speeds up calculations, compared to POAS, facilitating its practical application. Simulations and heuristics support the face validity of the technique and its rigorousness. The characteristics of msPOAS were evaluated on single and multi-shell diffusion data of the human brain. Significant reduction in noise while preserving the fine structure was demonstrated for diffusion weighted images, standard DTI analysis and advanced diffusion models such as NODDI. MsPOAS effectively improves the poor signal-to-noise ratio in highly diffusion weighted multi-shell diffusion data, which is required by recent advanced diffusion micro-structure models. We demonstrate the superiority of the new method compared to other advanced denoising methods.

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Method for structure preserving smoothing multi-shell dMRI data

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Does not rely on any dMRI diffusion model

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Outperforms naive single-shell POAS and other approaches

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Feasible for real data application

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Implemented within a freely available package dti for the R Language

Segmentation of high angular resolution diffusion MRI using sparse riemannian manifold clustering

We address the problem of segmenting high angular resolution diffusion imaging (HARDI) data into multiple regions (or fiber tracts) with distinct diffusion properties. We use the orientation distribution function (ODF) to model diffusion and cast the ODF segmentation problem as a clustering problem in the space of ODFs. Our approach integrates tools from sparse representation theory and Riemannian geometry into a graph theoretic segmentation framework. By exploiting the Riemannian properties of the space of ODFs, we learn a sparse representation for each ODF and infer the segmentation by applying spectral clustering to a similarity matrix built from these representations. In cases where regions with similar (resp. distinct) diffusion properties belong to different (resp. same) fiber tracts, we obtain the segmentation by incorporating spatial and user-specified pairwise relationships into the formulation. Experiments on synthetic data evaluate the sensitivity of our method to image noise and to the concentration parameters, and show its superior performance compared to alternative methods when analyzing complex fiber configurations. Experiments on phantom and real data demonstrate the accuracy of the proposed method in segmenting simulated fibers and white matter fiber tracts of clinical importance.

Canonical Correlation Analysis on Riemannian Manifolds and Its Applications

Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multi-variate random variables and has found a multitude of applications in computer vision, medical imaging and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations and probability distributions, all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is rather sub-optimal. But perhaps more importantly, since the algorithms do not respect the underlying geometry of the data space, it is hard to provide statistical guarantees (if any) on the results. Using the space of SPD matrices as a concrete example, this paper gives a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful statistical relationships on the product Riemannian manifold. As a proof of principle, we present results on an Alzheimer's disease (AD) study where the analysis task involves identifying correlations across diffusion tensor images (DTI) and Cauchy deformation tensor fields derived from T1-weighted magnetic resonance (MR) images.

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.

Natural selection. V. How to read the fundamental equations of evolutionary change in terms of information theory

The equations of evolutionary change by natural selection are commonly expressed in statistical terms. Fisher's fundamental theorem emphasizes the variance in fitness. Quantitative genetics expresses selection with covariances and regressions. Population genetic equations depend on genetic variances. How can we read those statistical expressions with respect to the meaning of natural selection? One possibility is to relate the statistical expressions to the amount of information that populations accumulate by selection. However, the connection between selection and information theory has never been compelling. Here, I show the correct relations between statistical expressions for selection and information theory expressions for selection. Those relations link selection to the fundamental concepts of entropy and information in the theories of physics, statistics and communication. We can now read the equations of selection in terms of their natural meaning. Selection causes populations to accumulate information about the environment.

Estimation of non-negative ODFs using the eigenvalue distribution of spherical functions

Current methods in high angular resolution diffusion imaging (HARDI) estimate the probability density function of water diffusion as a continuous-valued orientation distribution function (ODF) on the sphere. However, such methods could produce an ODF with negative values, because they enforce non-negativity only at finitely many directions. In this paper, we propose to enforce non-negativity on the continuous domain by enforcing the positive semi-definiteness of Toeplitz-like matrices constructed from the spherical harmonic representation of the ODF. We study the distribution of the eigenvalues of these matrices and use it to derive an iterative semi-definite program that enforces non-negativity on the continuous domain. We illustrate the performance of our method and compare it to the state-of-the-art with experiments on synthetic and real data.

GROUP ACTION INDUCED AVERAGING FOR HARDI PROCESSING

We consider the problem of processing high angular resolution diffusion images described by orientation distribution functions (ODFs). Prior work showed that several processing operations, e.g., averaging, interpolation and filtering, can be reduced to averaging in the space of ODFs. However, this approach leads to anatomically erroneous results when the ODFs to be processed have very different orientations. To address this issue, we propose a group action induced distance for averaging ODFs, which leads to a novel processing framework on the spaces of orientation (the space of 3D rotations) and shape (the space of ODFs with the same orientation). Experiments demonstrate that our framework produces anatomically meaningful results.

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.

Non-local robust detection of DTI white matter differences with small databases

Diffusion imaging, through the study of water diffusion, allows for the characterization of brain white matter, both at the population and individual level. In recent years, it has been employed to detect brain abnormalities in patients suffering from a disease, e.g., from multiple sclerosis (MS). State-of-the-art methods usually utilize a database of matched (age, sex, ...) controls, registered onto a template, to test for differences in the patient white matter. Such approaches however suffer from two main drawbacks. First, registration algorithms are prone to local errors, thereby degrading the comparison results. Second, the database needs to be large enough to obtain reliable results. However, in medical imaging, such large databases are hardly available. In this paper, we propose a new method that addresses these two issues. It relies on the search for samples in a local neighborhood of each pixel to increase the size of the database. Then, we propose a new test based on these samples to perform a voxelwise comparison of a patient image with respect to a population of controls. We demonstrate on simulated and real MS patient data how such a framework allows for an improve detection power and a better robustness and reproducibility, even with a small database.

Diffeomorphism invariant Riemannian framework for ensemble average propagator computing

BACKGROUND

In Diffusion Tensor Imaging (DTI), Riemannian framework based on Information Geometry theory has been proposed for processing tensors on estimation, interpolation, smoothing, regularization, segmentation, statistical test and so on. Recently Riemannian framework has been generalized to Orientation Distribution Function (ODF) and it is applicable to any Probability Density Function (PDF) under orthonormal basis representation. Spherical Polar Fourier Imaging (SPFI) was proposed for ODF and Ensemble Average Propagator (EAP) estimation from arbitrary sampled signals without any assumption.

PURPOSE

Tensors only can represent Gaussian EAP and ODF is the radial integration of EAP, while EAP has full information for diffusion process. To our knowledge, so far there is no work on how to process EAP data. In this paper, we present a Riemannian framework as a mathematical tool for such task.

METHOD

We propose a state-of-the-art Riemannian framework for EAPs by representing the square root of EAP, called wavefunction based on quantum mechanics, with the Fourier dual Spherical Polar Fourier (dSPF) basis. In this framework, the exponential map, logarithmic map and geodesic have closed forms, and weighted Riemannian mean and median uniquely exist. We analyze theoretically the similarities and differences between Riemannian frameworks for EAPs and for ODFs and tensors. The Riemannian metric for EAPs is diffeomorphism invariant, which is the natural extension of the affine-invariant metric for tensors. We propose Log-Euclidean framework to fast process EAPs, and Geodesic Anisotropy (GA) to measure the anisotropy of EAPs. With this framework, many important data processing operations, such as interpolation, smoothing, atlas estimation, Principal Geodesic Analysis (PGA), can be performed on EAP data.

RESULTS AND CONCLUSIONS

The proposed Riemannian framework was validated in synthetic data for interpolation, smoothing, PGA and in real data for GA and atlas estimation. Riemannian median is much robust for atlas estimation.

The Center for Computational Biology: resources, achievements, and challenges

The Center for Computational Biology (CCB) is a multidisciplinary program where biomedical scientists, engineers, and clinicians work jointly to combine modern mathematical and computational techniques, to perform phenotypic and genotypic studies of biological structure, function, and physiology in health and disease. CCB has developed a computational framework built around the Manifold Atlas, an integrated biomedical computing environment that enables statistical inference on biological manifolds. These manifolds model biological structures, features, shapes, and flows, and support sophisticated morphometric and statistical analyses. The Manifold Atlas includes tools, workflows, and services for multimodal population-based modeling and analysis of biological manifolds. The broad spectrum of biomedical topics explored by CCB investigators include the study of normal and pathological brain development, maturation and aging, discovery of associations between neuroimaging and genetic biomarkers, and the modeling, analysis, and visualization of biological shape, form, and size. CCB supports a wide range of short-term and long-term collaborations with outside investigators, which drive the center's computational developments and focus the validation and dissemination of CCB resources to new areas and scientific domains.

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.

Spatial HARDI: improved visualization of complex white matter architecture with Bayesian spatial regularization

Imaging of water diffusion using magnetic resonance imaging has become an important noninvasive method for probing the white matter connectivity of the human brain for scientific and clinical studies. Current methods, such as diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI) such as q-ball imaging, and diffusion spectrum imaging (DSI), are limited by low spatial resolution, long scan times, and low signal-to-noise ratio (SNR). These methods fundamentally perform reconstruction on a voxel-by-voxel level, effectively discarding the natural coherence of the data at different points in space. This paper attempts to overcome these tradeoffs by using spatial information to constrain the reconstruction from raw diffusion MRI data, and thereby improve angular resolution and noise tolerance. Spatial constraints are specified in terms of a prior probability distribution, which is then incorporated in a Bayesian reconstruction formulation. By taking the log of the resulting posterior distribution, optimal Bayesian reconstruction is reduced to a cost minimization problem. The minimization is solved using a new iterative algorithm based on successive least squares quadratic descent. Simulation studies and in vivo results are presented which indicate significant gains in terms of higher angular resolution of diffusion orientation distribution functions, better separation of crossing fibers, and improved reconstruction SNR over the same HARDI method, spherical harmonic q-ball imaging, without spatial regularization. Preliminary data also indicate that the proposed method might be better at maintaining accurate ODFs for smaller numbers of diffusion-weighted acquisition directions (hence faster scans) compared to conventional methods. Possible impacts of this work include improved evaluation of white matter microstructural integrity in regions of crossing fibers and higher spatial and angular resolution for more accurate tractography.

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 Collapse

Key Words

• high-angular resolution diffusion imaging
• anisotropic scalar
• generalized fractional anisotropy
• tensor
• signal-to-noise ratio
• kullback-leibler divergence

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MESH Headings

• Algorithms
• Anisotropy
• Computer Simulation
• Corpus Callosum/physiology
• Diffusion
• Diffusion Magnetic Resonance Imaging/methods
• Female
• Humans
• Imaging, Three-Dimensional/methods
• Male
• Models, Neurological
• Young Adult

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Grants

• EB007813 NIBIB NIH HHS
• R01 EB007813 NIBIB NIH HHS
• AG020098 NIA NIH HHS
• R56 AG020098 NIA NIH HHS
• HD050735 NICHD NIH HHS
• EB008432 NIBIB NIH HHS
• R01 EB008281-14 NIBIB NIH HHS
• R01 AG020098-09 NIA NIH HHS
• R01 EB008281 NIBIB NIH HHS
• R01 EB007813-04 NIBIB NIH HHS
• R01 HD050735 NICHD NIH HHS
• R01 EB008432 NIBIB NIH HHS
• EB008281 NIBIB NIH HHS
• R01 AG020098 NIA NIH HHS
• R01 EB008432-03 NIBIB NIH HHS

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Affiliation(s)

Liang Zhan Laboratory of Neuro Imaging, Department of Neurology, UCLA School of Medicine, 635 Charles E. Young Drive South, Suite 225E, Los Angeles, CA 90095-7332, USA
Alex D Leow
Neda Jahanshad
Ming-Chang Chiang
Marina Barysheva
Agatha D Lee
Arthur W Toga
Katie L McMahon
Greig I de Zubicaray
Margaret J Wright
Paul M Thompson

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