6 Conclusion and Future Work Recent advances in diffusion-weighted imaging (DWI), such as High Angular Resolution Diffusion Imaging (HARDI), allows us to model the water diffusion at a voxel with an orientation distribution function (ODF) that can capture multiple orientation at a voxel Major research questions are: how would one analyze the HADRI data and make the correct inferences from the rich information provided? Whether new insights into the human brain, in particular white matter, would surface? Before HARDI can be useful in both diagnosis and clinical applications, an ODF-based computational framework, including registration, atlas generation and regression analysis, etc, is needed for HARDI-based analysis across populations Firstly, we proposed a novel large deformation diffeomorphic registration algorithm to align HARDI data characterized by ODFs The 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 reviewed the Riemannian manifold of ODFs We then defined the reorientation of an ODF when 111 CONCLUSION AND FUTURE WORK an affine transformation is applied and subsequently, defined the diffeomorphic group action to be applied to the ODFs based on this reorientation We incorporated the Riemannian metric of ODFs for quantifying the similarity of two HARDI images into a variational problem defined under the large deformation diffeomorphic metric mapping (LDDMM) framework We finally derived the gradient of the cost function in both Riemannian spaces of diffeomorphisms and the ODFs, and presented its numerical implementation Both synthetic and real brain HARDI data were used to illustrate the performance of our registration algorithm We also presented a Bayesian probabilistic model to estimate the HARDI atlas of the brain white matter First of all, we assumed that the HARDI atlas is generated from a known hyperatlas through a flow of diffeomorphisms A shape prior of the HARDI atlas can thus be constructed based on LDDMM LDDMM characterizes a nonlinear diffeomorphic shape space in a linear space of initial momentum that uniquely determines diffeomorphic geodesic flows from the hyperatlas Therefore, the shape prior of the HARDI atlas can be modeled using a centered Gaussian random field (GRF) model of the initial momentum In order to construct the likelihood of observed HARDI datasets, it is necessary to study the diffeomorphic transformation of individual observations relative to the atlas and the probabilistic distribution of ODFs To this end, we constructed the likelihood related to the transformation using the same construction as discussed for the shape prior of the atlas The probabilistic distribution of ODFs was then constructed based on the ODF Riemannian manifold We assumed that the observed ODFs are generated by an exponential map of random tangent vectors at the deformed atlas ODFs Hence, the likelihood of the ODFs can be modeled using a GRF of their tangent vectors in the Riemannian manifold of ODFs We solved for the maximum a posteriori using the Expectation-Maximization algorithm and derive the 112 corresponding update equations Finally, we illustrated the HARDI atlas constructed based on a Chinese aging cohort of 94 adults and compared it with that generated by averaging the coefficients of spherical harmonics of the ODFs across subjects We further proposed a geodesic regression algorithm on the Riemannian manifold of ODFs We derived the algorithm for the geodesic regression of ODFs and conducted the simulation experiment to evaluate its performance We then examined the effects of aging via geodesic regression of ODFs in a large group of healthy men and women, spanning the adult age range Results show that the proposed method is able to capture more regions with age effect on white matter changes as compared to the conventional regression based on DTI The evolution of ODF fields along the geodesic regression line depicts in great detail the changes of white matter including the breakdown of the myelin sheath with aging and the anterior-posterior gradient of corpus callosum In the investigation of the regional aging effects where corpus callosum and corticospinal tracts come across, the result suggests that the diffusivity in corpus callosum declines more than in corticospinal tracts in the selected region To sum up, experiments have shown that the HARDI-based computational framework offers valuable clues about the changes of white matter in population studies that were previously undetected with existing methods Future work will aim to find new biomarkers sensitive to white-matter pathologies related to neuropsychiatric disorders such as Alzheimer’s disease (AD) Together with the HARDI-based tractography e.g., [32], we can quantify in detail the strength of connections along a specific pathway, which can be useful in the diagnosis and prognosis of AD In addition, there were a few methods proposed for the reconstruction of ensemble average propagator (EAP) from original DWI images recently [e.g 114, 115, 116] In order to accurately reconstruct the diffusion signal and EAP, a thorough 113 CONCLUSION AND FUTURE WORK exploration of q-space is needed, which requires multiple b-value diffusion weighted imaging (mDWI) MDWI can characterize more complex neural fiber geometries when compared to single b-value techniques like diffusion tensor imaging (DTI) or high angular resolution diffusion imaging (HARDI) Hybrid diffusion imaging (HYDI) [117] is a mDWI technique that samples the diffusion signal along concentric spherical shells in q-space, with the number of encoding directions increased with each shell to increase the angular resolution with the level of diffusion weighting MDWI techniques like HYDI, however, have not been widely used by clinicians and neuroscientists partially due to their relatively long acquisition times In addition, there is a lack of fundamental image analysis tools, such as registration, that can fully utilize their information Our future work will also include the extension of our framework to HDYI dataset 114 References [1] A LVINA G OH , C HRISTOPHE L ENGLET, PAUL M T HOMPSON , AND R V IDAL A Nonparametric Riemannian Framework for Processing High Angular Resolution Diffusion Images and its Applications to 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