Manifold Learning in Medical Imaging

Manifold learning theory has seen a surge of interest in the modeling of large and extensive datasets in medical imaging since they capture the essence of data in a way that fundamentally outperforms linear methodologies, the purpose of which is to essentially describe things that are flat. This problematic is particularly relevant with medical imaging data, where linear techniques are frequently unsuitable for capturing variations in anatomical structures. In many cases, there is enough structure in the data (CT, MRI, ultrasound) so a lower dimensional object can describe the degrees of freedom, such as in a manifold structure. Still, complex, multivariate distributions tend to demonstrate highly variable structural topologies that are impossible to capture with a single manifold learning algorithm. This chapter will present recent techniques developed in manifold theory for medical imaging analysis, to allow for statistical organ shape modeling, image segmentation and registration from the concept of navigation of manifolds, classification, as well as disease prediction models based on discriminant manifolds. We will present the theoretical basis of these works, with illustrative results on their applications from various organs and pathologies, including neurodegenerative diseases and spinal deformities

Learning the clustering of longitudinal shape data sets into a mixture of independent or branching trajectories

Given repeated observations of several subjects over time, i.e. a longitudinal data set, this paper introduces a new model to learn a classification of the shapes progression in an unsupervised setting: we automatically cluster a longitudinal data set in different classes without labels. Our method learns for each cluster an average shape trajectory (or representative curve) and its variance in space and time. Representative trajectories are built as the combination of pieces of curves. This mixture model is flexible enough to handle independent trajectories for each cluster as well as fork and merge scenarios. The estimation of such non linear mixture models in high dimension is known to be difficult because of the trapping states effect that hampers the optimisation of cluster assignments during training. We address this issue by using a tempered version of the stochastic EM algorithm. Finally, we apply our algorithm on different data sets. First, synthetic data are used to show that a tempered scheme achieves better convergence. We then apply our method to different real data sets: 1D RECIST score used to monitor tumors growth, 3D facial expressions and meshes of the hippocampus. In particular, we show how the method can be used to test different scenarios of hip-pocampus atrophy in ageing by using an heteregenous population of normal ageing individuals and mild cog-nitive impaired subjects

Image and Shape Analysis for Spatiotemporal Data

In analyzing brain development or identifying disease it is important to understand anatomical age-related changes and shape differences. Data for these studies is frequently spatiotemporal and collected from normal and/or abnormal subjects. However, images and shapes over time often have complex structures and are best treated as elements of non-Euclidean spaces. This dissertation tackles problems of uncovering time-varying changes and statistical group differences in image or shape time-series. There are three major contributions: 1) a framework of parametric regression models on manifolds to capture time-varying changes. These include a metamorphic geodesic regression approach for image time-series and standard geodesic regression, time-warped geodesic regression, and cubic spline regression on the Grassmann manifold; 2) a spatiotemporal statistical atlas approach, which augments a commonly used atlas such as the median with measures of data variance via a weighted functional boxplot; 3) hypothesis testing for shape analysis to detect group differences between populations. The proposed method for cross-sectional data uses shape ordering and hence does not require dense shape correspondences or strong distributional assumptions on the data. For longitudinal data, hypothesis testing is performed on shape trajectories which are estimated from individual subjects. Applications of these methods include 1) capturing brain development and degeneration; 2) revealing growth patterns in pediatric upper airways and the scoring of airway abnormalities; 3) detecting group differences in longitudinal corpus callosum shapes of subjects with dementia versus normal controls.Doctor of Philosoph

Invariant higher-order variational problems II

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesics on the group of transformations project to cubics. Finally, we apply second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.Comment: 40 pages, 1 figure. First version -- comments welcome

A model of brain morphological changes related to aging and Alzheimer's disease from cross-sectional assessments

In this study we propose a deformation-based framework to jointly model the influence of aging and Alzheimer's disease (AD) on the brain morphological evolution. Our approach combines a spatio-temporal description of both processes into a generative model. A reference morphology is deformed along specific trajectories to match subject specific morphologies. It is used to define two imaging progression markers: 1) a morphological age and 2) a disease score. These markers can be computed locally in any brain region. The approach is evaluated on brain structural magnetic resonance images (MRI) from the ADNI database. The generative model is first estimated on a control population, then, for each subject, the markers are computed for each acquisition. The longitudinal evolution of these markers is then studied in relation with the clinical diagnosis of the subjects and used to generate possible morphological evolution. In the model, the morphological changes associated with normal aging are mainly found around the ventricles, while the Alzheimer's disease specific changes are more located in the temporal lobe and the hippocampal area. The statistical analysis of these markers highlights differences between clinical conditions even though the inter-subject variability is quiet high. In this context, the model can be used to generate plausible morphological trajectories associated with the disease. Our method gives two interpretable scalar imaging biomarkers assessing the effects of aging and disease on brain morphology at the individual and population level. These markers confirm an acceleration of apparent aging for Alzheimer's subjects and can help discriminate clinical conditions even in prodromal stages. More generally, the joint modeling of normal and pathological evolutions shows promising results to describe age-related brain diseases over long time scales.Comment: NeuroImage, Elsevier, In pres

Doctor of Philosophy

dissertationAn important aspect of medical research is the understanding of anatomy and its relation to function in the human body. For instance, identifying changes in the brain associated with cognitive decline helps in understanding the process of aging and age-related neurological disorders. The field of computational anatomy provides a rich mathematical setting for statistical analysis of complex geometrical structures seen in 3D medical images. At its core, computational anatomy is based on the representation of anatomical shape and its variability as elements of nonflat manifold of diffeomorphisms with an associated Riemannian structure. Although such manifolds effectively represent natural biological variability, intrinsic methods of statistical analysis within these spaces remain deficient at large. This dissertation contributes two critical missing pieces for statistics in diffeomorphisms: (1) multivariate regression models for cross-sectional study of shapes, and (2) generalization of classical Euclidean, mixed-effects models to manifolds for longitudinal studies. These models are based on the principle that statistics on manifold-valued information must respect the intrinsic geometry of that space. The multivariate regression methods provide statistical descriptors of the relationships of anatomy with clinical indicators. The novel theory of hierarchical geodesic models (HGMs) is developed as a natural generalization of hierarchical linear models (HLMs) to describe longitudinal data on curved manifolds. Using a hierarchy of geodesics, the HGMs address the challenge of modeling the shape-data with unbalanced designs typically arising as a result of follow-up medical studies. More generally, this research establishes a mathematical foundation to study dynamics of changes in anatomy and the associated clinical progression with time. This dissertation also provides efficient algorithms that utilize state-of-the-art high performance computing architectures to solve models on large-scale, longitudinal imaging data. These manifold-based methods are applied to predictive modeling of neurological disorders such as Alzheimer's disease. Overall, this dissertation enables clinicians and researchers to better utilize the structural information available in medical images

Proceedings of the fifth international workshop on Mathematical Foundations of Computational Anatomy (MFCA 2015)

International audienceComputational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information.The Mathematical Foundations of Computational Anatomy (MFCA) workshop aims at fostering the interactions between the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum for the exchange of the theoretical ideas and aims at being a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations.Following the first edition of this workshop in 20061, the second edition in New-York in 20082, the third edition in Toronto in 20113, the forth edition in Nagoya Japan on September 22 20134, the fifth edition was held in Munich on October 9 20155.Contributions were solicited in Riemannian, sub-Riemannian and group theoretical methods, advanced statistics on deformations and shapes, metrics for computational anatomy, statistics of surfaces, time-evolving geometric processes, stratified spaces, optimal transport, approximation methods in statistical learning and related subjects. Among the submitted papers, 14 were selected andorganized in 4 oral sessions

Rate-invariant analysis of covariance trajectories

Statistical analysis of dynamic systems, such as videos and dynamic functional connectivity, is often translated into a problem of analyzing trajectories of relevant features, particularly covariance matrices. As an example, in video-based action recognition, a natural mathematical representation of activity videos is as parameterized trajectories on the set of symmetric, positive-definite matrices (SPDMs). The variable execution-rates of actions, implying arbitrary parameterizations of trajectories, complicates their analysis and classification. To handle this challenge, we represent covariance trajectories using transported square-root vector fields (TSRVFs), constructed by parallel translating scaled-velocity vectors of trajectories to their starting points. The space of such representations forms a vector bundle on the SPDM manifold. Using a natural Riemannian metric on this vector bundle, we approximate geodesic paths and geodesic distances between trajectories in the quotient space of this vector bundle. This metric is invariant to the action of the reparameterization group, and leads to a rate-invariant analysis of trajectories. In the process, we remove the parameterization variability and temporally register trajectories during analysis. We demonstrate this framework in multiple contexts, using both generative statistical models and discriminative data analysis. The latter is illustrated using several applications involving video-based action recognition and dynamic functional connectivity analysis