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

Orthogonal Mixed-Effects Modeling for High-Dimensional Longitudinal Data: An Unsupervised Learning Approach

The linear mixed-effects model is commonly utilized to interpret longitudinal data, characterizing both the global longitudinal trajectory across all observations and longitudinal trajectories within individuals. However, characterizing these trajectories in high-dimensional longitudinal data presents a challenge. To address this, our study proposes a novel approach, Unsupervised Orthogonal Mixed-Effects Trajectory Modeling (UOMETM), that leverages unsupervised learning to generate latent representations of both global and individual trajectories. We design an autoencoder with a latent space where an orthogonal constraint is imposed to separate the space of global trajectories from individual trajectories. We also devise a cross-reconstruction loss to ensure consistency of global trajectories and enhance the orthogonality between representation spaces. To evaluate UOMETM, we conducted simulation experiments on images to verify that every component functions as intended. Furthermore, we evaluated its performance and robustness using longitudinal brain cortical thickness from two Alzheimer's disease (AD) datasets. Comparative analyses with state-of-the-art methods revealed UOMETM's superiority in identifying global and individual longitudinal patterns, achieving a lower reconstruction error, superior orthogonality, and higher accuracy in AD classification and conversion forecasting. Remarkably, we found that the space of global trajectories did not significantly contribute to AD classification compared to the space of individual trajectories, emphasizing their clear separation. Moreover, our model exhibited satisfactory generalization and robustness across different datasets. The study shows the outstanding performance and potential clinical use of UOMETM in the context of longitudinal data analysis.

Hierarchical geodesic modeling on the diffusion orientation distribution function for longitudinal DW-MRI analysis

The analysis of anatomy that undergoes rapid changes, such as neuroimaging of the early developing brain, greatly benefits from spatio-temporal statistical analysis methods to represent population variations but also subject-wise characteristics over time. Methods for spatio-temporal modeling and for analysis of longitudinal shape and image data have been presented before, but, to our knowledge, not for diffusion weighted MR images (DW-MRI) fitted with higher-order diffusion models. To bridge the gap between rapidly evolving DW-MRI methods in longitudinal studies and the existing frameworks, which are often limited to the analysis of derived measures like fractional anisotropy (FA), we propose a new framework to estimate a population trajectory of longitudinal diffusion orientation distribution functions (dODFs) along with subject-specific changes by using hierarchical geodesic modeling. The dODF is an angular profile of the diffusion probability density function derived from high angular resolution diffusion imaging (HARDI) and we consider the dODF with the square-root representation to lie on the unit sphere in a Hilbert space, which is a well-known Riemannian manifold, to respect the nonlinear characteristics of dODFs. The proposed method is validated on synthetic longitudinal dODF data and tested on a longitudinal set of 60 HARDI images from 25 healthy infants to characterize dODF changes associated with early brain development.

A framework to construct a longitudinal DW-MRI infant atlas based on mixed effects modeling of dODF coefficients

Building of atlases plays a crucial role in the analysis of brain images. In scenarios where early growth, aging or disease trajectories are of key importance, longitudinal atlases become necessary as references, most often created from cross-sectional data. New opportunities will be offered by creating longitudinal brain atlases from longitudinal subject-specific image data, where explicit modeling of subject's variability in slope and intercept leads to a more robust estimation of average trajectories but also to estimates of confidence bounds. This work focuses on a framework to build a continuous 4D atlas from longitudinal high angular resolution diffusion images (HARDI) where, unlike atlases of derived scalar diffusion indices such as FA, statistics on dODFs is preserved. Multi-scalar images obtained from DW images are used for geometric alignment, and linear mixed-effects modeling from longitudinal diffusion orientation distribution functions (dODF) leads to estimation of continuous dODF changes. The proposed method is applied to a longitudinal dataset of HARDI images from healthy developing infants in the age range of 3 to 36 months. Verification of mixed-effects modeling is obtained by voxel-wise goodness of fit calculations. To demonstrate the potential of our method, we display changes of longitudinal atlas using dODF and derived generalized fractional anisotropy (GFA) of dODF. We also investigate white matter maturation patterns in genu, body, and splenium of the corpus callosum. The framework can be used to build an average dODF atlas from HARDI data and to derive subject-specific and population-based longitudinal change trajectories.

A Literature Review: Geometric Methods and Their Applications in Human-Related Analysis

Geometric features, such as the topological and manifold properties, are utilized to extract geometric properties. Geometric methods that exploit the applications of geometrics, e.g., geometric features, are widely used in computer graphics and computer vision problems. This review presents a literature review on geometric concepts, geometric methods, and their applications in human-related analysis, e.g., human shape analysis, human pose analysis, and human action analysis. This review proposes to categorize geometric methods based on the scope of the geometric properties that are extracted: object-oriented geometric methods, feature-oriented geometric methods, and routine-based geometric methods. Considering the broad applications of deep learning methods, this review also studies geometric deep learning, which has recently become a popular topic of research. Validation datasets are collected, and method performances are collected and compared. Finally, research trends and possible research topics are discussed.

Multivariate Regression with Gross Errors on Manifold-Valued Data

We consider the topic of multivariate regression on manifold-valued output, that is, for a multivariate observation, its output response lies on a manifold. Moreover, we propose a new regression model to deal with the presence of grossly corrupted manifold-valued responses, a bottleneck issue commonly encountered in practical scenarios. Our model first takes a correction step on the grossly corrupted responses via geodesic curves on the manifold, then performs multivariate linear regression on the corrected data. This results in a nonconvex and nonsmooth optimization problem on Riemannian manifolds. To this end, we propose a dedicated approach named PALMR, by utilizing and extending the proximal alternating linearized minimization techniques for optimization problems on euclidean spaces. Theoretically, we investigate its convergence property, where it is shown to converge to a critical point under mild conditions. Empirically, we test our model on both synthetic and real diffusion tensor imaging data, and show that our model outperforms other multivariate regression models when manifold-valued responses contain gross errors, and is effective in identifying gross errors.

Nonlinear regression on Riemannian manifolds and its applications to Neuro-image analysis

Regression in its most common form where independent and dependent variables are in ℝ ⁿ is a ubiquitous tool in Sciences and Engineering. Recent advances in Medical Imaging has lead to a wide spread availability of manifold-valued data leading to problems where the independent variables are manifold-valued and dependent are real-valued or vice-versa. The most common method of regression on a manifold is the geodesic regression, which is the counterpart of linear regression in Euclidean space. Often, the relation between the variables is highly complex, and existing most commonly used geodesic regression can prove to be inaccurate. Thus, it is necessary to resort to a non-linear model for regression. In this work we present a novel Kernel based non-linear regression method when the mapping to be estimated is either from M → ℝ ⁿ or ℝ ⁿ → M, where M is a Riemannian manifold. A key advantage of this approach is that there is no requirement for the manifold-valued data to necessarily inherit an ordering from the data in ℝ ⁿ . We present several synthetic and real data experiments along with comparisons to the state-of-the-art geodesic regression method in literature and thus validating the effectiveness of the proposed algorithm.

Multivariate General Linear Models (MGLM) on Riemannian Manifolds with Applications to Statistical Analysis of Diffusion Weighted Images

Linear regression is a parametric model which is ubiquitous in scientific analysis. The classical setup where the observations and responses, i.e., (x_i , y_i ) pairs, are Euclidean is well studied. The setting where y_i is manifold valued is a topic of much interest, motivated by applications in shape analysis, topic modeling, and medical imaging. Recent work gives strategies for max-margin classifiers, principal components analysis, and dictionary learning on certain types of manifolds. For parametric regression specifically, results within the last year provide mechanisms to regress one real-valued parameter, x_i ∈ R, against a manifold-valued variable, y_i ∈ . We seek to substantially extend the operating range of such methods by deriving schemes for multivariate multiple linear regression -a manifold-valued dependent variable against multiple independent variables, i.e., f : Rⁿ → . Our variational algorithm efficiently solves for multiple geodesic bases on the manifold concurrently via gradient updates. This allows us to answer questions such as: what is the relationship of the measurement at voxel y to disease when conditioned on age and gender. We show applications to statistical analysis of diffusion weighted images, which give rise to regression tasks on the manifold GL(n)/O(n) for diffusion tensor images (DTI) and the Hilbert unit sphere for orientation distribution functions (ODF) from high angular resolution acquisition. The companion open-source code is available on nitrc.org/projects/riem_mglm.