Registration, Atlas Generation, and Statistical Analysis of High Angular Resolution Diffusion Imaging based on Riemannian Structure of Orientation Distribution Functions Jia Du Department of Bioengineering National University of Singapore A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF PhilosophiæDoctor (PhD) June 18, 2013 Reviewer: Reviewer: Reviewer: ii Declaration I hereby declare that the thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously Jia Du, June 18, 2013 Acknowledgements I would like to express my gratitude to my advisor, Dr Anqi Qiu, for giving me professional guidance, unending encouragement and full research support I appreciate all her contributions of time and ideas over the course of my Ph.D studies I am also deeply grateful to our collaborator, Dr Alvina Goh, in the Department of Mathematics, National University of Singapore I thank her for the technical support and constructive advice throughout this work I would also like to thank my lab mates and friends who have been giving me advice and support during those four years: Dr Jidan Zhong, Jordan Bingren Bai, Ta Anh Tuan, Hock Wei Soon, Dr Sergey Kushnarev, Yanbo Wang, Kelei Chen, Jiajing Li, Mengqiao Dai, and Liuya Min Publication List Journals • Jia Du, Laurent Younes, Anqi Qiu, Whole Brain Diffeomorphic Metric Mapping via Integration of Sulcal and Gyral Curves, Cortical Surfaces, and Images NeuroImage, 56(1):162-173, 2011 • Jia Du, Alvina Goh, Anqi Qiu, Diffeomorphic Metric Mapping of High Angular Resolution Diffusion Imaging based on Riemannian Structure of Orientation Distribution Functions IEEE Transactions on Medical Imaging (TMI), 31(5):1021 - 1033 ,2012 • Jia Du, Alvina Goh, Anqi Qiu, Bayesian Estimation of White Matter Atlas from High Angular Resolution Diffusion Imaging IEEE Transactions on Image Processing, submitted • Jia Du, Alvina Goh, Sergey Kushnarev, Anqi Qiu, Geodesic Linear Regression on Orientation Distribution Function with its Application to Aging Study NeuroImage, submitted Conference Proceedings • Jia Du, Alvina Goh, Anqi Qiu, Large Deformation Diffeomorphic Metric Mapping of Orientation Distribution Functions, Information Processing in Medical Imaging (IPMI), 2011, (Top conference in medical imaging analysis, acceptance rate is less than 25%) (oral presentation, out of total 24 oral presentations) • Jia Du, Anqi Qiu, Integrative Diffeomorphic Metric Mapping Based on Image and Unlabeled Points, IEEE International Conference on Complex Medical Engineering, 2011 (oral presentation) • Jia Du, A Pasha Hosseinbor, Moo K Chung, Andrew L Alexander, Anqi Qiu, Diffeomorphic Metric Mapping of Hybrid Diffusion Imaging based on BFOR iii Signal Basis, Information Processing in Medical Imaging (IPMI), 2013 • Jia Du, Alvina Goh, Anqi Qiu, Bayesian Atlas Estimation from High Angular Resolution Diffusion Imaging (HARDI), Geometric Science of Information (GSI), 2013 (oral presentation) Conference Abstracts • Jia Du, Anqi Qiu, Whole Brain Diffeomorphic Mapping via the Integration of Sulcal Curves, Cortical Surfaces, and Images, Organization for Human Brain Mapping (OHBM), 2011, Quebec City (Trainee Abstract Travel Award, Interactive Poster, highlighting as top ranked abstracts) • Jia Du, Alvina Goh, Anqi Qiu, Bayesian Atlas Estimation for High Angular Resolution Diffusion Imaging, Organization for Human Brain Mapping (OHBM), 2012, Beijing • Jia Du, Alvina Goh, Anqi Qiu, Large Deformation Diffeomorphic Metric Mapping of High Angular Resolution Diffusion Imaging, Organization for Human Brain Mapping (OHBM), 2012, Beijing iv Contents List of Tables xi List of Figures xiii List of Symbols xix 1 1.1 Motivation 1.2 Research Challenges and Thesis Contributions 1.2.1 Registation 1.2.2 Atlas Generation 10 1.2.3 Introduction Statistical Analysis 12 15 2.1 Riemannian Manifold of ODFs 15 2.2 Large Deformation Diffeomorphic Metric Mapping 18 2.2.1 Diffeomorphic Metric 18 2.2.2 Background Conservation Law of Momentum 20 Diffeomorphic Metric Mapping of High Angular Resolution Diffusion Imaging based on Riemannian Structure of Orientation Distribution Functions 23 3.1 Affine Transformation on Square-Root ODFs 25 3.2 Diffeomorphic Group Action on Square-Root ODFs 29 3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs 32 Gradient of J with respect to mt 34 Derivation of the gradient of Ex with respect to φ1 38 3.3.1 3.3.1.1 v CONTENTS 3.3.2 Euler-Lagrange Equation for LDDMM-ODF 41 3.3.3 Numerical Implementation 41 3.4 Synthetic Data 44 3.5 HARDI Data of Children Brains 47 3.5.1 Comparison of LDDMM-FA, LDDMM-DTI and LDDMM-ODF 47 3.5.2 Comparison with existing ODF registration algorithm 3.5.3 Computational complexity of LDDMM-FA, LDDMM-ODF 52 and LDDMM-Raffelt 3.6 54 Summary 55 Bayesian Estimation of White Matter Atlas from High Angular Resolution Diffusion Imaging 59 4.1 General Framework of Bayesian HARDI Atlas Estimation 4.2 The Shape Prior of the Atlas and the Distribution of Random Diffeomorphisms 61 63 4.3 The Conditional Likelihood of the ODF Data 65 4.4 Expectation-Maximization Algorithm 67 Derivation of Update Equations of σ and m0 in EM 70 Results 71 4.5.1 HARDI Atlas Generation 73 4.5.2 Convergence and Effects of Hyperatlas Choice of the HARDI 4.4.1 4.5 Atlas Estimation 4.5.3 Aging HARDI atlases 81 4.5.4 4.6 74 Comparison with existing method 82 Summary 83 Geodesic Regression of Orientation Distribution Functions with its Application to Aging Study 85 5.1 Geodesic Regression on the ODF manifold 86 5.1.1 Least-Squares Estimation and Algorithm 89 5.1.1.1 Derivation of the Least-Squares Estimation 90 Statistical Testing 94 Experiments 95 5.2.1 95 5.1.2 5.2 Experiments on Synthetic ODF Data vi CONTENTS 5.2.2 5.3 Experiments on Real Human Brain Data: Aging Study 101 5.2.2.1 Image Acquisition and Preprocessing 101 5.2.2.2 Geodesic Regression of ODFs and Aging Effect 103 Summary 106 Conclusion and Future Work 111 References 115 vii Summary Progress in the diffusion weighted magnetic resonance imaging (DW-MRI) techniques over the last two decades has enabled neuroscientists to image the human brain white matter in-vivo and from large populations Diffusion tensor imaging (DTI), which models the axonal orientations of neurons using a three-dimensional ellipsoid tensor, has become one of the most popular methods to study the white matter micro-structure for identifying neuropathology of mental illnesses and understanding fundamental neuroscience questions on brain connections However, a major shortcoming of DTI is that it can only reveal one dominant axonal orientation at each location while between one to two thirds of the human brain white matter are thought to contain multiple axonal bundles crossing each other Recent advances in DW-MRI, such as high angular resolution diffusion imaging (HARDI), address this well-known limitation of DTI by modeling the water diffusion with an orientation distribution function (ODF) that can capture multiple axonal orientations at a voxel For both scientific and clinical applications, it is necessary to develop methods to represent, compare and make correct inferences from the rich information provided by HARDI data However, the main challenge arises from the complexity of HARDI data as the existing analysis frameworks based on scalar images or DTI are unable to handle such data The main contribution of this thesis is providing an HARDI-based analysis framework for the studies of white matter similarities and differences across large populations Under a unified Riemannian manifold of ODF, the framework includes three components: registration, atlas generation and statistical analysis Firstly, we propose a novel ODF-based registration algorithm, which seeks an optimal diffeomorphism between ODFs of two LIST OF FIGURES φ Diffeomorphism φt Time-dependent Diffeomorphism vt Time-dependent Velocity mt Time-dependent Momentum Itemp Object as Template or Atlas Itarg Object as Target or Subject Ψ Riemannian Manifold of ODFs ψ square-root ODFs xx Introduction 1.1 Motivation The white matter region of the human brain is composed of neuronal axons that provide insights on brain connections Such information is very useful for identifying neuropathology of mental illnesses and understanding fundamental neuroscience questions on how the brain regions interact each other In the last decade, diffusion weighted magnetic resonance imaging (DW-MRI) technique has exploited the property that water molecules move faster along neural axons than against them By measuring water diffusion in the brain, the location and trajectories of axons can be visualized and the axonal pathways can be reconstructed using DW-MRI Several techniques may be used to reconstruct the local orientation of brain tissue from DW-MRI data A classical method is known as Diffusion Tensor Imaging (DTI) [3], which characterizes the diffusivity profile of water molecules in brain tissue by a single oriented 3D Gaussian probability distribution function (PDF) In DTI, the diffusivity profile is often represented mathematically by a symmetric positive definite (SPD) tensor field D : R3 → SPD(3) ⊂ R3×3 that 1 INTRODUCTION measures the extent of diffusion in any direction v ∈ R3 as v Dv The geometry of SPD(3) is well-studied and several metrics for comparing tensors have been proposed [4, 5, 6, 7] Based on these metrics, statistical tests such as voxel-based analysis of diffusion tensors have been developed [8, 9, 10, 11] Before such population studies can been carried out, there is a essential need to perform DTI registration, that is, to align tensor data across subjects to a standard coordinate space Compared to the classical image registration problem, the registration of DTI fields is more complicated since DTI data contains structural information affected by the transformation Two key transformations need to be defined: a transformation to spatially align anatomical structures between two brains in a 3D volume domain, and a transformation to align the local diffusivity profiles defined at each voxel of two brains More precisely, a transformation φ of the image domain induces a reorientation of the DTI as the direction of diffusion depends on the coordinate system Thus, for two diffusion tensors D1 (x) and D2 (x) at voxel x, it is no longer true that D1 (x) ≈ D2 (φ(x)) and each tensor must be reoriented in such a way that it remains consistent with the surrounding anatomical structure There exist several approaches for reorientation that are used in DTI [12] For instance, the Finite Strain (FS) scheme decomposes an affine transformation matrix A into A = RS, where R is the rigid rotation and S is the deformation, and reorients the tensor D as RDR An alternative strategy is the Preservation of Principal Direction (PPD), in which the reoriented tensor keeps its eigenvalues, yet its principal eigenvector v1 is transformed as Av1 / Av1 The reader is referred to [13, 14, 15, 16, 17, 18] and references therein for the existing DTI registration methods The natural question after registration is how does one generate the DTI atlas of the population that well characterizes the in-vivo white matter anatomy of the human 1.1 Motivation brain To this end, there are several works on DTI-based atlas generation Park et al constructed an atlas with full tensor information in [19] Since then there have been several different approaches to DTI atlas construction, either using scalar registration [20, 21], multi-channel methods [19] or by directly optimizing tensor similarity [22] Since then, there have been works that leveraged the DTI template information and some of these include several exploratory works that provided anatomical validation [23] and anatomical labeling of fiber tracts [24, 25], and evaluation The comprehensive work by Mori et al [24] provides a three-dimensional and two-dimensional in-vivo atlas of various white matter tracts in the human brain based on DTI and has become an essential resource for neuroimaging researchers Hua et al [25] create a white matter parcellation atlas based on probabilistic maps of the major white matter tracts and show that there is an excellent correlation of fractional anisotropy and mean diffusivity between the automated and the individual tractography-based results Lawes et al [23] show that it is possible to establish a close correspondence of the fiber tracts generated from the DTI atlas with the tracts isolated with classical dissection of post-mortem brain tissue A DTI atlas containing the complete diffusion tensor information is constructed by Verhoeven et al in [21] Using robust fiber tracking methods on this DTI atlas, Verhoeven et al reconstruct a large number of white matter tracts and show that their framework yields highly reproducible and reliable fiber tracts Schotten et al [26] produce a white matter atlas that describes the in-vivo variability of the major association, commissural, and projection connections and study the inter-subject variability between left and right hemispheres in relation to gender based on this atlas O’Donnell et al [27] and Yushkevich et al [28] use DTI atlases directly to study white matter fiber tracts With all the subjects aligned into the common coordinate, the last step is the DTIbased statistical analysis As mentioned previously, the diffusivity profile in DTI INTRODUCTION is represented mathematically by a symmetric positive definite (SPD) tensor field D : R3 → SPD(3) ⊂ R3×3 The geometry of SPD(3) is well-studied and several metrics such as fractional anisotropy (FA) and mean diffusivity (MD) for comparing tensors have been proposed [4, 5, 6, 7, 29] Based on these metrics, statistical tests such as voxel-based analysis of diffusion tensors have been developed [8, 9, 10, 11] With the wide use of DTI combined with the development of the associated brain analysis technologies [e.g 30, 31, 32], the last decade has witnessed an explosion of studies of changes in white matter micro-structure Changes in white matter micro-structure have been associated with healthy aging [for review see 33] and developmental maturation [for review see 34] For clinical populations, such changes have also been demonstrated in patients with schizophrenia [for review see 35, 36], depression [for review see 37] and neurodegenerative diseases such as Alzheimer’s disease [for review see 38] and Parkinson’s disease [for review see 39] While it has been demonstrated that DTI is valuable for studying brain white matter development in children and detecting abnormalities in patients with neuropsychiatric disorders and neurodegenerative diseases, a major shortcoming of DTI is that it can only reveal one dominant fiber orientation at each location, when between one and two thirds of the voxels in the human brain white matter are thought to contain multiple fiber bundles crossing each other [40] High angular resolution diffusion imaging (HARDI) [41] addresses this well-known limitation of DTI HARDI measures diffusion along n uniformly distributed directions on the sphere and can characterize more complex fiber geometries Several reconstruction techniques can be used to characterize diffusion based on the HARDI signals One class is based on higher-order tensors [42, 43] and leverage prior work on DTI Another method is Q-ball Imaging, which uses the Funk-Radon transform to reconstruct an orientation distribution function (ODF) The 1.1 Motivation model-free ODF is the angular profile of the diffusion PDF of water molecules and has been approximated using different sets of basis functions such as spherical harmonics (SH) [44, 45, 46, 47, 48, 49] Such methods are relatively fast to implement because the ODF is computed analytically Unlike the tensor model used in DTI, the ODF has no restriction on the number of axons present in a specific anatomical location and thus, can well characterize the true underlying white matter architecture, as shown in Figure 1.1 By quantitatively comparing fiber orientations retrieved from ODFs against histological measurements, Leergaard et al [50] shows that accurate fiber estimates can be obtained from HARDI data, further validating its usage in brain studies (a) T1 (b) DTI (Tensor) (c) HARDI (ODF) Figure 1.1: Illustration of DTI versus HARDI The colors in panel (b) indicts the fractional anisotropy (FA) of each tensor, where blue stands for low FA value and red for high value Similar to the shape of ODF, the colors in panel (c) also indices the relative values of ODF in each direction, where blue stands for low ODF value and red for high value Before HARDI can be useful in diagnosis and clinical applications, there is a need to address several major research questions These include how one would analyze the HARDI data and make the correct inferences from the rich information provided and whether new insights into the human brain, in particular white matter, would surface To answer these questions, an analysis framework similar to the one of DTI, including HARDI-based registration, HARDI-based atlas generation and HARDI-based statistical INTRODUCTION analysis, is needed for the studies of white matter similarities and differences across large populations 1.2 Research Challenges and Thesis Contributions For HARDI-based brain analysis, the main challenge arises from the complexity of HARDI data as the existing analysis frameworks based on scalar images or DTI are unable to handle such data For both scientific and clinical applications, it is necessary to develop methods to represent, compare and make correct inferences from the rich information provided by HARDI data While many current efforts focus on computing the representation such as orientation distribution functions (ODF) from HARDI signals [e.g 44, 45, 47, 48, 49], much less work has been done on developing methods for comparison and analysis of these high-dimensional datasets, especially among a large group of subjects There remain several challenges in using HARDI-based brain analysis across populations The main contribution of this thesis is providing an ODF-based analysis framework for the HARDI-based studies of white matter similarities and differences across large populations, as summarized in Figure 1.2 Under a unified Riemannian manifold of ODFs proposed in [1], the framework addresses the three challenges as mentioned above: ODF-based registration, ODF-based atlas generation and ODF-based statistical analysis 1.2.1 Registation The first open challenge in the analysis of mathematically complex HARDI data is registration, which aims to align ODF fields reconstructed from HARDI data across 1.2 Research Challenges and Thesis Contributions Scope of this thesis HARDI data ODF Reconstruction Data Acquisition ODF images Atlas Generation ODF atlas Subjects Registration serve as common space in registration ODF images in common space Statistical Analysis Biomarkers/ Inference Figure 1.2: The ODF-based analysis framework for the HARDI-based studies of white matter similarities and differences across large populations subjects to a standard coordinate space Compared to the classical image registration problem, the registration of ODF fields is more challenging since ODF fields contain structural information affected by the transformation Three key components are needed for the ODF-based registration: (i) a metric to measure the distance between ODFs, (ii) a reorientation scheme to reorient ODFs at each voxel based on a local spatial transformation, and (iii) a transformation to spatially align ODF fields between two brains in a 3D volume domain Several HARDI registration algorithms have been recently proposed under a specific model of local diffusivity Chiang et al [51] propose an information-theoretic approach for fluid registration of ODFs An inverseconsistent fluid registration algorithm that minimizes the symmetrized Kullback-Leibler INTRODUCTION divergence (sKL) or J-divergence of the two DT images [18] is first performed and the ODF fields are registered by applying the corresponding DTI mapping The ODFs are reoriented using the PPD method where the principal direction of the ODF is determined by principal component analysis Cheng et al [52] take the approach of representing HARDI by Gaussian mixture fields (GMF) and assumes a thin-plate spline deformation The L2 metric of GMFs is minimized, and reorientation is performed on the individual Gaussian components, each representing a major fiber direction Barmpoutis et al [53] use a 4th order tensor model and assumes a region-based nonrigid deformation The rotationally invariant Hellinger distance is considered and an affine tensor reorientation, which accounts for rotation, scaling and shearing effects, is applied Geng et al [54] perform a diffeomorphic registration with the L2 metric on ODFs represented by spherical harmonics Reorientation is done by altering the SH coefficients in a manner similar to the FS method in DTI where only the rotation is extracted and applied Bloy et al [55] perform alignment of ODF fields by using a multi-channel diffeomorphic demons registration algorithm on rotationally invariant feature maps and uses the FS scheme in reorientation Hong et al [56] are the first to introduce affine transformation of ODFs [57] presents an alternative definition by making use of a more “continuous” representation of the ODF in terms of SH delta functions and show that their proposed reorientation scheme is computationally advantageous and having more robust results Yap et al [58] use the SH-based ODF representation and propose a hierarchical registration scheme, where the alignment is updated by using anatomical features extracted from the increasing order of the SH representation Reorientation is done by making use of the scheme proposed in [57] Now, the L2 metric on ODFs represented by SH, which is adopted in all these approaches [54, 55, 56, 57], is an ambient distance, not a geodesic distance as the 1.2 Research Challenges and Thesis Contributions ODFs lie on a Riemannian manifold [1, 59] In addition, while a majority of recent ODF-based registration approaches seeks small deformation between two brains [e.g 51, 52, 53, 56], other studies [60] have suggested that the transformation from one brain to another can be really large and therefore small deformation models may not be enough Thus, there is a need to develop an algorithm for the ODF-based registration, which incorporates a Riemannian metric of ODFs, a reorientation scheme with local affine transformation of ODFs, and a spatial transformation which allows for large deformation Contribution In this thesis, we present a novel registration algorithm for HARDI data represented by ODFs under the framework of large deformation diffeomorphic metric mapping (LDDMM) such that the deformation of two brains is diffeomorphic (one-toone, smooth, and invertible) and can be in a large scale Our proposed algorithm seeks an optimal diffeomorphism of large deformation between two ODF fields across a spatial volume domain and at the same time, locally reorients an ODF in a manner that remains consistent with the surrounding anatomical structure We define the reorientation of an ODF when an affine transformation is applied and subsequently, define the diffeomorphic group action to be applied to the ODF based on this reorientation The ODF reorientation used in this paper ensures that the transformed ODF remains consistent with the surrounding anatomical structure and at the same time, not solely dependent on the rotation Rather, the reorientation takes into account the effects of the affine transformation and ensures the volume fraction of fibers oriented toward a small patch must remain the same after the patch is transformed As we will show later in the paper, our proposed algorithm also takes into consideration the influence of the fiber orientation difference on the deformation The Riemannian metric for the similarity of INTRODUCTION ODFs is then incorporated into a variational problem in LDDMM Finally, we derive the gradient of the cost function in both Riemannian spaces of diffeomorphisms and the ODFs and present its numerical implementation Our experiments are shown on synthetic and real HARDI brain data, and its advantages over the DTI-based registration are well-validated 1.2.2 Atlas Generation While ODF-based registration methods, it is now possible to warp anatomical structures of white matter across subjects into a common coordinate space, referred as the atlas One possible choice of the atlas is to use a subject from the population being studied The difficulties with this approach are that the atlas may not be truly representative of the population, particularly when severe neurodegenerative disorders or brain development are studied [61] Wide variations of the anatomy across subjects relative to the atlas may cause the failure of the mapping Thus, one of the fundamental limitations of choosing the anatomy of a single subject as an atlas is the introduction of a statistical bias based on the arbitrary choice of the atlas anatomy Therefore, the next challenge lies in how to find the atlas that characterizes the in-vivo white matter anatomy for a specified clinical population, which referred as HARDI-based atlas generation While the atlas generation techniques based on intensity images, including those based only on affine or non-linear registration methods [62, 63] and probabilistic models coupled to the Expectation-Maximization (EM) algorithm to estimate both a shape prior of the atlas and an intensity image likelihood function [61, 64, 65], have matured significantly over the last decade, the white matter atlas generation based on HARDI is still very much in its infancy Bouix et al [66] employ an image registration approach 10 1.2 Research Challenges and Thesis Contributions that first seeks the transformation between fractional anisotropic (FA) images and then resample the HARDI signals of each subject into a common coordinate according to this transformation The HARDI atlas is then generated by averaging the coefficients of spherical harmonics of the ODF across subjects Yeh and Tseng [67] construct the spatial normalization of the diffusion information using a q-space diffeomorphic reconstruction method and reconstruct the spin distribution function (SDF) in the ICBM152 space from the diffusion MR signals Finally, the white matter atlas is computed by averaging the SDF over individual subjects Bloy et al [68] perform alignment of ODF fields by using a multi-channel diffeomorphic demons registration algorithm on rotationally invariant feature maps and white matter parcellation is done via a spatially coherent normalized cuts algorithms Those existing HARDI atlas generation methods suffer from blurring due to averaging and dependence on the selection of initial atlas To the best of our knowledge, an atlas generation framework, that incorporates both a shape prior of the white matter anatomy and a probabilistic model of the ODFs, remains lacking Contribution In this thesis, we extend the previous Bayesian model for the intensity image atlas generation proposed in [61, 65] to that for HARDI Briefly, we derive a Bayesian model with a shape priori of the HARDI atlas in terms of diffeomorphic transformations and a likelihood function of the ODFs in terms of their tangent vectors on an ODF Riemannian manifold As we will see later, the extension of the Bayesian model from intensity images to HARDI is non-trivial Our main contribution of this work is to construct the likelihood function of the ODFs based on their Riemannian structure and derive the Expectation-Maximization algorithm and the update equations to solve the Bayesian HARDI atlas estimation Our findings show that the atlas estimated 11 INTRODUCTION using our algorithm preserves anatomical details of the white matter The results on different age groups are consistent with existing literature, e.g., [69, 70, 71] the corpus callosum thinning as age increases 1.2.3 Statistical Analysis After the HARDI datasets across a population have been warped into the common HARDI atlas, the non-linear nature of HARDI data presents a challenge to HARDIbased statistical analysis To this end, there has been great emphasis on deriving scalarbased metrics from ODFs so that fundamental statistical tools, such as linear regression, can be easily employed One of the earliest scalar measures is the generalized fractional anisotropy (GFA) proposed by [72] GFA is defined as the ratio of standard deviation of the ODF to its root mean square Similar to fractional anisotropy (FA) derived from DTI, GFA takes a value between zero and one and describes the degree of anisotropy of a diffusion process GFA has thus far been used in studies on subcortical ischemic stroke [73], impulsivity [74], and genetic influence on the brain white matter [51, 75], yielding promising results In addition to GFA, other scalar measures have also been proposed [76] propose a scalar measure known as peak geodesic concentration (GC), which is defined as the concentration relative to the peak fiber orientation identified from ODF and thus reflects the degree of directionally coherent diffusion [76] claim that GC is sensitive to the presence of single or multiple fiber populations within a voxel and therefore, is a unique scalar measure that can be used for evaluation of pathology [77] use a polynomial approach to extract geometric characteristics from ODFs and define peak fractional anisotropy (PFA) and total-PFA at the fiber orientation with the extrema and principal curvatures Finally, [78] propose and compute the apparent intravoxel fiber 12 1.2 Research Challenges and Thesis Contributions population dispersion (FPD) It conveys the manner in which distinct fiber populations are partitioned within the same voxel and is more effective in revealing regions with crossing tracts than FA Despite ease of statistical analysis on the aforementioned scalar measures, they discard the information that is inherent in ODF and may be of interest in detecting underlying axonal organization Recently, multivariate statistical analysis has been directly applied to ODFs [e.g 1, 79, 80] For example, [79] perform a multivariate group-wise genetic analysis of white matter integrity by adapting the multivariate intraclass correlation value (ICC) to ODFs The ICC is obtained from the coefficients of the spherical harmonics of ODFs at each voxel [79] show that the ICC increases the detection power in finding genetic influence on the white matter architecture when compared to statistics derived from GFA Instead of working with ODFs in the Euclidean space, several recent works have proposed a Riemannian framework for analyzing ODFs [1] use the square-root representation for the ODF Riemannian manifold Under this representation, Riemannian operations, such as the geodesics, exponential and logarithm maps, are available in closed form [1] develop principal geodesic analysis on tangent vectors of ODFs on the manifold and generalize the Hotelling’s T-squared statistic for the comparison of ODFs between two populations [80] approximate the square root of the ODF as a linear combination of orthonormal basis functions Since the coefficients of this expansion live in a finitedimensional sphere, processing operations can be performed in the space of coefficients with reduced computational complexity Since the Riemannian framework facilitates the utilization of the full information of ODFs, the natural question is whether fundamental statistical tools, such as regression, can be adapted from the Euclidean space to the manifold setting Regression analysis is a fundamental statistical tool to determine how a measured variable is related to 13 INTRODUCTION one or more independent variables The most widely used regression model is linear regression because of its simplicity, ease of interpretation, and ability to model many phenomena However, if the response variable takes values in a nonlinear manifold, a linear model is not applicable Such manifold-valued measurements arise in many applications, including those that involve directional data, transformations, tensors, and shapes and in our case, ODFs Contribution In this thesis, we adapt the framework of geodesic regression, proposed in [81], to the HARDI data To this end, we derive the algorithm for the geodesic regression on the Riemannian manifold of ODFs Similar to [81], we define a least-squares problem that minimizes the sum-of-squared geodesic distances between observed ODFs and their model fitted data in order to find the optimal regression model We derive the appropriate gradient terms later in this paper and use the gradient descent to seek the minimizer of this least-squares 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 We apply the ODF regression algorithm and statistical testing to synthetic and real human data Especially, we examine the effects of aging via geodesic regression of ODFs in normal adults aged 21 years old and above 14 ... Background Conservation Law of Momentum 20 Diffeomorphic Metric Mapping of High Angular Resolution Diffusion Imaging based on Riemannian Structure of Orientation Distribution Functions. .. 56 (1) :16 2 -17 3, 2 011 • Jia Du, Alvina Goh, Anqi Qiu, Diffeomorphic Metric Mapping of High Angular Resolution Diffusion Imaging based on Riemannian Structure of Orientation Distribution Functions IEEE... Transactions on Medical Imaging (TMI), 31( 5) :10 21 - 10 33 ,2 012 • Jia Du, Alvina Goh, Anqi Qiu, Bayesian Estimation of White Matter Atlas from High Angular Resolution Diffusion Imaging IEEE Transactions