Bayesian atlas estimation for the variability analysis of shape complexes

An Iterative Centroid Approach for Diffeomorphic Online Atlasing

Online atlasing, i.e., incrementing an atlas with new images as they are acquired, is key when performing studies on very large, or still being gathered, databases. Regular approaches to atlasing however do not focus on this aspect and impose a complete reconstruction of the atlas when adding images. We propose instead a diffeomorphic online atlasing method that allows gradual updates to an atlas. In this iterative centroid approach, we integrate new subjects in the atlas in an iterative manner, gradually moving the centroid of the images towards its final position. This leads to a computationally cheap approach since it only necessitates one additional registration per new subject added. We validate our approach on several experiments with three main goals: 1- to evaluate atlas image quality of the obtained atlases with sharpness and overlap measures, 2- to assess the deviation in terms of transformations with respect to a conventional atlasing method and 3- to compare its computational time with regular approaches of the literature. We demonstrate that the transformations divergence with respect to a state-of-the-art atlas construction method is small and reaches a plateau, that the two construction methods have the same ability to map subject homologous regions onto a common space and produce images of equivalent quality. The computational time of our approach is also drastically reduced for regular updates. Finally, we also present a direct extension of our method to update spatio-temporal atlases, especially useful for developmental studies.

Double Diffeomorphism: Combining Morphometry and Structural Connectivity Analysis

The brain is composed of several neural circuits which may be seen as anatomical complexes composed of grey matter structures interconnected by white matter tracts. Grey and white matter components may be modeled as 3-D surfaces and curves, respectively. Neurodevelopmental disorders involve morphological and organizational alterations which cannot be jointly captured by usual shape analysis techniques based on single diffeomorphisms. We propose a new deformation scheme, called double diffeomorphism, which is a combination of two diffeomorphisms. The first one captures changes in structural connectivity, whereas the second one recovers the global morphological variations of both grey and white matter structures. This deformation model is integrated into a Bayesian framework for atlas construction. We evaluate it on a data-set of 3-D structures representing the neural circuits of patients with Gilles de la Tourette syndrome (GTS). We show that this approach makes it possible to localise, quantify, and easily visualise the pathological anomalies altering the morphology and organization of the neural circuits. Furthermore, results also indicate that the proposed deformation model better discriminates between controls and GTS patients than a single diffeomorphism.

Probabilistic modeling of anatomical variability using a low dimensional parameterization of diffeomorphisms

We present an efficient probabilistic model of anatomical variability in a linear space of initial velocities of diffeomorphic transformations and demonstrate its benefits in clinical studies of brain anatomy. To overcome the computational challenges of the high dimensional deformation-based descriptors, we develop a latent variable model for principal geodesic analysis (PGA) based on a low dimensional shape descriptor that effectively captures the intrinsic variability in a population. We define a novel shape prior that explicitly represents principal modes as a multivariate complex Gaussian distribution on the initial velocities in a bandlimited space. We demonstrate the performance of our model on a set of 3D brain MRI scans from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. Our model yields a more compact representation of group variation at substantially lower computational cost than the state-of-the-art method such as tangent space PCA (TPCA) and probabilistic principal geodesic analysis (PPGA) that operate in the high dimensional image space.

Parsimonious Approximation of Streamline Trajectories in White Matter Fiber Bundles

Fiber bundles stemming from tractography algorithms contain many streamlines. They require therefore a great amount of computer memory and computational resources to be stored, visualised and processed. We propose an approximation scheme for fiber bundles which results in a parsimonious representation of weighted prototypes. Prototypes are chosen among the streamlines and they represent groups of similar streamlines. Their weight is related to the number of approximated streamlines. Both streamlines and prototypes are modelled as weighted currents. This computational model does not need point-to-point correspondences and two streamlines are considered similar if their endpoints are close to each other and if their pathways follow similar trajectories. Moreover, the space of weighted currents is a vector space with a closed-form metric. This permits easy computation of the approximation error and the selection of the prototypes is based on the minimisation of this error. We propose an iterative algorithm which approximates independently and simultaneously all the fascicles of the bundle in a fast and accurate way. We show that the resulting representation preserves the shape of the bundle and it can be used to accurately reconstruct the original structural connectivity. We evaluate our algorithm on bundles obtained from both deterministic and probabilistic tractography algorithms. The resulting approximations use on average only 2% of the original streamlines as prototypes. This drastically reduces the computational burden of the processes where the geometry of the streamlines is considered. We demonstrate its effectiveness using as example the registration between two fiber bundles.

Low-Dimensional Statistics of Anatomical Variability via Compact Representation of Image Deformations

Using image-based descriptors to investigate clinical hypotheses and therapeutic implications is challenging due to the notorious "curse of dimensionality" coupled with a small sample size. In this paper, we present a low-dimensional analysis of anatomical shape variability in the space of diffeomorphisms and demonstrate its benefits for clinical studies. To combat the high dimensionality of the deformation descriptors, we develop a probabilistic model of principal geodesic analysis in a bandlimited low-dimensional space that still captures the underlying variability of image data. We demonstrate the performance of our model on a set of 3D brain MRI scans from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. Our model yields a more compact representation of group variation at substantially lower computational cost than models based on the high-dimensional state-of-the-art approaches such as tangent space PCA (TPCA) and probabilistic principal geodesic analysis (PPGA).

A Hybrid Multishape Learning Framework for Longitudinal Prediction of Cortical Surfaces and Fiber Tracts Using Neonatal Data

Dramatic changes of the human brain during the first year of postnatal development are poorly understood due to their multifold complexity. In this paper, we present the first attempt to jointly predict, using neonatal data, the dynamic growth pattern of brain cortical surfaces (collection of 3D triangular faces) and fiber tracts (collection of 3D lines). These two entities are modeled jointly as a multishape (a set of interlinked shapes). We propose a hybrid learning-based multishape prediction framework that captures both the diffeomorphic evolution of the cortical surfaces and the non-diffeomorphic growth of fiber tracts. In particular, we learn a set of geometric and dynamic cortical features and fiber connectivity features that characterize the relationships between cortical surfaces and fibers at different timepoints (0, 3, 6, and 9 months of age). Given a new neonatal multishape at 0 month of age, we hierarchically predict, at 3, 6 and 9 months, the postnatal cortical surfaces vertex-by-vertex along with fibers connected to adjacent faces to these vertices. This is achieved using a new fiber-to-face metric that quantifies the similarity between multishapes. For validation, we propose several evaluation metrics to thoroughly assess the performance of our framework. The results confirm that our framework yields good prediction accuracy of complex neonatal multishape development within a few seconds.

Multidirectional and Topography-based Dynamic-scale Varifold Representations with Application to Matching Developing Cortical Surfaces

The human cerebral cortex is marked by great complexity as well as substantial dynamic changes during early postnatal development. To obtain a fairly comprehensive picture of its age-induced and/or disorder-related cortical changes, one needs to match cortical surfaces to one another, while maximizing their anatomical alignment. Methods that geodesically shoot surfaces into one another as currents (a distribution of oriented normals) and varifolds (a distribution of non-oriented normals) provide an elegant Riemannian framework for generic surface matching and reliable statistical analysis. However, both conventional current and varifold matching methods have two key limitations. First, they only use the normals of the surface to measure its geometry and guide the warping process, which overlooks the importance of the orientations of the inherently convoluted cortical sulcal and gyral folds. Second, the 'conversion' of a surface into a current or a varifold operates at a fixed scale under which geometric surface details will be neglected, which ignores the dynamic scales of cortical foldings. To overcome these limitations and improve varifold-based cortical surface registration, we propose two different strategies. The first strategy decomposes each cortical surface into its normal and tangent varifold representations, by integrating principal curvature direction field into the varifold matching framework, thus providing rich information of the orientation of cortical folding and better characterization of the complex cortical geometry. The second strategy explores the informative cortical geometric features to perform a dynamic-scale measurement of the cortical surface that depends on the local surface topography (e.g., principal curvature), thereby we introduce the concept of a topography-based dynamic-scale varifold. We tested the proposed varifold variants for registering 12 pairs of dynamically developing cortical surfaces from 0 to 6 months of age. Both variants improved the matching accuracy in terms of closeness to the target surface and the goodness of alignment with regional anatomical boundaries, when compared with three state-of-the-art methods: (1) diffeomorphic spectral matching, (2) conventional current-based surface matching, and (3) conventional varifold-based surface matching.

Prediction of Longitudinal Development of Infant Cortical Surface Shape Using a 4D Current-Based Learning Framework

Understanding the early dynamics of the highly folded human cerebral cortex is still an actively evolving research field teeming with unanswered questions. Longitudinal neuroimaging analysis and modeling have become the new trend to advance research in this field. However, this is challenged by a limited number of acquisition timepoints and the absence of inter-subject matching between timepoints. In this paper, we propose a novel framework that unprecedentedly solves the problem of predicting the dynamic evolution of infant cortical surface shape solely from a single baseline shape based on a spatiotemporal (4D) current-based learning approach. Specifically, our method learns from longitudinal data both the geometric (vertices positions) and dynamic (temporal evolution trajectories) features of the infant cortical surface, comprising a training stage and a prediction stage. In the training stage, we first use the current-based shape regression model to set up the inter-subject cortical surface correspondences at baseline of all training subjects. We then estimate for each training subject the diffeomorphic temporal evolution trajectories of the cortical surface shape and build an empirical mean spatiotemporal surface atlas. In the prediction stage, given an infant, we first warp all training subjects onto its baseline cortical surface. Second, we select the most appropriate learnt features from training subjects to simultaneously predict the cortical surface shapes at all later timepoints from its baseline cortical surface, based on closeness metrics between this baseline surface and the learnt baseline population average surface atlas. We used the proposed framework to predict the inner cortical surface shape at 3, 6 and 9 months from the cortical shape at birth in 9 healthy infants. Our method predicted with good accuracy the spatiotemporal dynamic change of the highly folded cortex.

Joint Morphometry of Fiber Tracts and Gray Matter Structures Using Double Diffeomorphisms

This work proposes an atlas construction method to jointly analyse the relative position and shape of fiber tracts and gray matter structures. It is based on a double diffeomorphism which is a composition of two diffeomorphisms. The first diffeomorphism acts only on the white matter keeping fixed the gray matter of the atlas. The resulting white matter, together with the gray matter, are then deformed by the second diffeomorphism. The two diffeomorphisms are related and jointly optimised. In this way, the, first diffeomorphisms explain the variability in structural connectivity within the population, namely both changes in the connected areas of the gray matter and in the geometry of the pathway of the tracts. The second diffeomorphisms put into correspondence the homologous anatomical structures across subjects. Fiber bundles are approximated with weighted prototypes using the metric of weighted currents. The atlas, the covariance matrix of deformation parameters and the noise variance of each structure are automatically estimated using a Bayesian approach. This method is applied to patients with Tourette syndrome and controls showing a variability in the structural connectivity of the left cortico-putamen circuit.

Bayesian principal geodesic analysis in diffeomorphic image registration

Computing a concise representation of the anatomical variability found in large sets of images is an important first step in many statistical shape analyses. In this paper, we present a generative Bayesian approach for automatic dimensionality reduction of shape variability represented through diffeomorphic mappings. To achieve this, we develop a latent variable model for principal geodesic analysis (PGA) that provides a probabilistic framework for factor analysis on diffeomorphisms. Our key contribution is a Bayesian inference procedure for model parameter estimation and simultaneous detection of the effective dimensionality of the latent space. We evaluate our proposed model for atlas and principal geodesic estimation on the OASIS brain database of magnetic resonance images. We show that the automatically selected latent dimensions from our model are able to reconstruct unseen brain images with lower error than equivalent linear principal components analysis (LPCA) models in the image space, and it also outperforms tangent space PCA (TPCA) models in the diffeomorphism setting.

Fusion of white and gray matter geometry: a framework for investigating brain development

Current neuroimaging investigation of the white matter typically focuses on measurements derived from diffusion tensor imaging, such as fractional anisotropy (FA). In contrast, imaging studies of the gray matter oftentimes focus on morphological features such as cortical thickness, folding and surface curvature. As a result, it is not clear how to combine findings from these two types of approaches in order to obtain a consistent picture of morphological changes in both gray and white matter. In this paper, we propose a joint investigation of gray and white matter morphology by combining geometrical information from white and the gray matter. To achieve this, we first introduce a novel method for computing multi-scale white matter tract geometry. Its formulation is based on the differential geometry of curve sets and is easily incorporated into a continuous scale-space framework. We then incorporate this method into a novel framework for "fusing" white and gray matter geometrical information. Given a set of fiber tracts originating in a particular cortical region, the key idea is to compute two scalar fields that represent geometrical characteristics of the white matter and of the surface of the cortical region. A quantitative marker is created by combining the distributions of these scalar values using Mutual Information. This marker can be then used in the study of normal and pathological brain structure and development. We apply this framework to a study on autism spectrum disorder in children. Our preliminary results support the view that autism may be characterized by early brain overgrowth, followed by reduced or arrested growth (Courchesne, 2004).