A Statistical Model for Simultaneous Template Estimation, Bias Correction, and Registration of 3D Brain Images

Template estimation plays a crucial role in computational anatomy since it provides reference frames for performing statistical analysis of the underlying anatomical population variability. While building models for template estimation, variability in sites and image acquisition protocols need to be accounted for. To account for such variability, we propose a generative template estimation model that makes simultaneous inference of both bias fields in individual images, deformations for image registration, and variance hyperparameters. In contrast, existing maximum a posterori based methods need to rely on either bias-invariant similarity measures or robust image normalization. Results on synthetic and real brain MRI images demonstrate the capability of the model to capture heterogeneity in intensities and provide a reliable template estimation from registration

Symmetric diffeomorphic modeling of longtudinal structural MRI

This technology report describes the longitudinal registration approach that we intend to incorporate into SPM12. It essentially describes a group-wise intra-subject modeling framework, which combines diffeomorphic and rigid-body registration, incorporating a correction for the intensity inhomogeneity artifact usually seen in MRI data. Emphasis is placed on achieving internal consistency and accounting for many of the mathematical subtleties that most implementations overlook. The implementation was evaluated using examples from the OASIS Longitudinal MRI Data in Non-demented and Demented Older Adults

Hierarchical unbiased graph shrinkage (HUGS): A novel groupwise registration for large data set

Normalizing all images in a large data set into a common space is a key step in many clinical and research studies, e.g., for brain development, maturation, and aging. Recently, groupwise registration has been developed for simultaneous alignment of all images without selecting a particular image as template, thus potentially avoiding bias in the registration. However, most conventional groupwise registration methods do not explore the data distribution during the image registration. Thus, their performance could be affected by large inter-subject variations in the data set under registration. To solve this potential issue, we propose to use a graph to model the distribution of all image data sitting on the image manifold, with each node representing an image and each edge representing the geodesic pathway between two nodes (or images). Then, the procedure of warping all images to their population center turns to the dynamic shrinking of the graph nodes along their graph edges until all graph nodes become close to each other. Thus, the topology of image distribution on the image manifold is always preserved during the groupwise registration. More importantly, by modeling the distribution of all images via a graph, we can potentially reduce registration error since every time each image is warped only according to its nearby images with similar structures in the graph. We have evaluated our proposed groupwise registration method on both infant and adult data sets, by also comparing with the conventional group-mean based registration and the ABSORB methods. All experimental results show that our proposed method can achieve better performance in terms of registration accuracy and robustness

Doctor of Philosophy in Computing

dissertationAn important area of medical imaging research is studying anatomical diffeomorphic shape changes and detecting their relationship to disease processes. For example, neurodegenerative disorders change the shape of the brain, thus identifying differences between the healthy control subjects and patients affected by these diseases can help with understanding the disease processes. Previous research proposed a variety of mathematical approaches for statistical analysis of geometrical brain structure in three-dimensional (3D) medical imaging, including atlas building, brain variability quantification, regression, etc. The critical component in these statistical models is that the geometrical structure is represented by transformations rather than the actual image data. Despite the fact that such statistical models effectively provide a way for analyzing shape variation, none of them have a truly probabilistic interpretation. This dissertation contributes a novel Bayesian framework of statistical shape analysis for generic manifold data and its application to shape variability and brain magnetic resonance imaging (MRI). After we carefully define the distributions on manifolds, we then build Bayesian models for analyzing the intrinsic variability of manifold data, involving the mean point, principal modes, and parameter estimation. Because there is no closed-form solution for Bayesian inference of these models on manifolds, we develop a Markov Chain Monte Carlo method to sample the hidden variables from the distribution. The main advantages of these Bayesian approaches are that they provide parameter estimation and automatic dimensionality reduction for analyzing generic manifold-valued data, such as diffeomorphisms. Modeling the mean point of a group of images in a Bayesian manner allows for learning the regularity parameter from data directly rather than having to set it manually, which eliminates the effort of cross validation for parameter selection. In population studies, our Bayesian model of principal modes analysis (1) automatically extracts a low-dimensional, second-order statistics of manifold data variability and (2) gives a better geometric data fit than nonprobabilistic models. To make this Bayesian framework computationally more efficient for high-dimensional diffeomorphisms, this dissertation presents an algorithm, FLASH (finite-dimensional Lie algebras for shooting), that hugely speeds up the diffeomorphic image registration. Instead of formulating diffeomorphisms in a continuous variational problem, Flash defines a completely new discrete reparameterization of diffeomorphisms in a low-dimensional bandlimited velocity space, which results in the Bayesian inference via sampling on the space of diffeomorphisms being more feasible in time. Our entire Bayesian framework in this dissertation is used for statistical analysis of shape data and brain MRIs. It has the potential to improve hypothesis testing, classification, and mixture models

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

Medical image analysis via Fréchet means of diffeomorphisms

The construction of average models of anatomy, as well as regression analysis of anatomical structures, are key issues in medical research, e.g., in the study of brain development and disease progression. When the underlying anatomical process can be modeled by parameters in a Euclidean space, classical statistical techniques are applicable. However, recent work suggests that attempts to describe anatomical differences using flat Euclidean spaces undermine our ability to represent natural biological variability. In response, this dissertation contributes to the development of a particular nonlinear shape analysis methodology. This dissertation uses a nonlinear deformable model to measure anatomical change and define geometry-based averaging and regression for anatomical structures represented within medical images. Geometric differences are modeled by coordinate transformations, i.e., deformations, of underlying image coordinates. In order to represent local geometric changes and accommodate large deformations, these transformations are taken to be the group of diffeomorphisms with an associated metric. A mean anatomical image is defined using this deformation-based metric via the Fréchet mean—the minimizer of the sum of squared distances. Similarly, a new method called manifold kernel regression is presented for estimating systematic changes—as a function of a predictor variable, such as age—from data in nonlinear spaces. It is defined by recasting kernel regression in terms of a kernel-weighted Fréchet mean. This method is applied to determine systematic geometric changes in the brain from a random design dataset of medical images. Finally, diffeomorphic image mapping is extended to accommodate extraneous structures—objects that are present in one image and absent in another and thus change image topology—by deflating them prior to the estimation of geometric change. The method is applied to quantify the motion of the prostate in the presence of transient bowel gas

Proceedings of the Third International Workshop on Mathematical Foundations of Computational Anatomy - Geometrical and Statistical Methods for Modelling Biological Shape Variability

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 successful rst edition of this workshop in 20061 and second edition in New-York in 20082, the third edition was held in Toronto on September 22 20113. Contributions were solicited in Riemannian and group theoretical methods, geometric measurements of the anatomy, advanced statistics on deformations and shapes, metrics for computational anatomy, statistics of surfaces, modeling of growth and longitudinal shape changes. 22 submissions were reviewed by three members of the program committee. To guaranty a high level program, 11 papers only were selected for oral presentation in 4 sessions. Two of these sessions regroups classical themes of the workshop: statistics on manifolds and diff eomorphisms for surface or longitudinal registration. One session gathers papers exploring new mathematical structures beyond Riemannian geometry while the last oral session deals with the emerging theme of statistics on graphs and trees. Finally, a poster session of 5 papers addresses more application oriented works on computational anatomy

Diffeomorphic image registration with applications to deformation modelling between multiple data sets

Over last years, the diffeomorphic image registration algorithms have been successfully introduced into the field of the medical image analysis. At the same time, the particular usability of these techniques, in majority derived from the solid mathematical background, has been only quantitatively explored for the limited applications such as longitudinal studies on treatment quality, or diseases progression. The thesis considers the deformable image registration algorithms, seeking out those that maintain the medical correctness of the estimated dense deformation fields in terms of the preservation of the object and its neighbourhood topology, offer the reasonable computational complexity to satisfy time restrictions coming from the potential applications, and are able to cope with low quality data typically encountered in Adaptive Radiotherapy (ART). The research has led to the main emphasis being laid on the diffeomorphic image registration to achieve one-to-one mapping between images. This involves introduction of the log-domain parameterisation of the deformation field by its approximation via a stationary velocity field. A quantitative and qualitative examination of existing and newly proposed algorithms for pairwise deformable image registration presented in this thesis, shows that the log-Euclidean parameterisation can be successfully utilised in the biomedical applications. Although algorithms utilising the log-domain parameterisation have theoretical justification for maintaining diffeomorphism, in general, the deformation fields produced by them have similar properties as these estimated by classical methods. Having this in mind, the best compromise in terms of the quality of the deformation fields has been found for the consistent image registration framework. The experimental results suggest also that the image registration with the symmetrical warping of the input images outperforms the classical approaches, and simultaneously can be easily introduced to most known algorithms. Furthermore, the log-domain implicit group-wise image registration is proposed. By linking the various sets of images related to the different subjects, the proposed image registration approach establishes a common subject space and between-subject correspondences therein. Although the correspondences between groups of images can be found by performing the classic image registration, the reference image selection (not required in the proposed implementation), may lead to a biased mean image being estimated and the corresponding common subject space not adequate to represent the general properties of the data sets. The approaches to diffeomorphic image registration have been also utilised as the principal elements for estimating the movements of the organs in the pelvic area based on the dense deformation field prediction system driven by the partial information coming from the specific type of the measurements parameterised using the implicit surface representation, and recognising facial expressions where the stationary velocity fields are used as the facial expression descriptors. Both applications have been extensively evaluated based on the real representative data sets of three-dimensional volumes and two-dimensional images, and the obtained results indicate the practical usability of the proposed techniques