Splines for diffeomorphisms

LESA: Longitudinal Elastic Shape Analysis of Brain Subcortical Structures

Over the past 30 years, magnetic resonance imaging has become a ubiquitous tool for accurately visualizing the change and development of the brain's subcortical structures (e.g., hippocampus). Although subcortical structures act as information hubs of the nervous system, their quantification is still in its infancy due to many challenges in shape extraction, representation, and modeling. Here, we develop a simple and efficient framework of longitudinal elastic shape analysis (LESA) for subcortical structures. Integrating ideas from elastic shape analysis of static surfaces and statistical modeling of sparse longitudinal data, LESA provides a set of tools for systematically quantifying changes of longitudinal subcortical surface shapes from raw structure MRI data. The key novelties of LESA include: (i) it can efficiently represent complex subcortical structures using a small number of basis functions and (ii) it can accurately delineate the spatiotemporal shape changes of the human subcortical structures. We applied LESA to analyze three longitudinal neuroimaging data sets and showcase its wide applications in estimating continuous shape trajectories, building life-span growth patterns, and comparing shape differences among different groups. In particular, with the Alzheimer's Disease Neuroimaging Initiative (ADNI) data, we found that the Alzheimer's Disease (AD) can significantly speed the shape change of ventricle and hippocampus from 60 to 75 years old compared with normal aging.

Fast predictive simple geodesic regression

Deformable image registration and regression are important tasks in medical image analysis. However, they are computationally expensive, especially when analyzing large-scale datasets that contain thousands of images. Hence, cluster computing is typically used, making the approaches dependent on such computational infrastructure. Even larger computational resources are required as study sizes increase. This limits the use of deformable image registration and regression for clinical applications and as component algorithms for other image analysis approaches. We therefore propose using a fast predictive approach to perform image registrations. In particular, we employ these fast registration predictions to approximate a simplified geodesic regression model to capture longitudinal brain changes. The resulting method is orders of magnitude faster than the standard optimization-based regression model and hence facilitates large-scale analysis on a single graphics processing unit (GPU). We evaluate our results on 3D brain magnetic resonance images (MRI) from the ADNI datasets.

ACCELERATION CONTROLLED DIFFEOMORPHISMS FOR NONPARAMETRIC IMAGE REGRESSION

The analysis of medical image time-series is becoming increasingly important as longitudinal imaging studies are maturing and large scale open imaging databases are becoming available. Image regression is widely used for several purposes: as a statistical representation for hypothesis testing, to bring clinical scores and images not acquired at the same time into temporal correspondence, or as a consistency filter to enforce temporal correlation. Geodesic image regression is the most prominent method, but the geodesic constraint limits the flexibility and therefore the application of the model, particularly when the observation time window is large or the anatomical changes are non-monotonic. In this paper, we propose to parameterize diffeomorphic flow by acceleration rather than velocity, as in the geodesic model. This results in a nonparametric image regression model which is completely flexible to capture complex change trajectories, while still constrained to be diffeomorphic and with a guarantee of temporal smoothness. We demonstrate the application of our model on synthetic 2D images as well as real 3D images of the cardiac cycle.

Fast Geodesic Regression for Population-Based Image Analysis

Geodesic regression on images enables studies of brain development and degeneration, disease progression, and tumor growth. The high-dimensional nature of image data presents significant computational challenges for the current regression approaches and prohibits large scale studies. In this paper, we present a fast geodesic regression method that dramatically decreases the computational cost of the inference procedure while maintaining prediction accuracy. We employ an efficient low dimensional representation of diffeomorphic transformations derived from the image data and characterize the regressed trajectory in the space of diffeomorphisms by its initial conditions, i.e., an initial image template and an initial velocity field computed as a weighted average of pairwise diffeomorphic image registration results. This construction is achieved by using a first-order approximation of pairwise distances between images. We demonstrate the efficiency of our model on a set of 3D brain MRI scans from the OASIS dataset and show that it is dramatically faster than the state-of-the-art regression methods while producing equally good regression results on the large subject cohort.

Frequency Diffeomorphisms for Efficient Image Registration

This paper presents an efficient algorithm for large deformation diffeomorphic metric mapping (LDDMM) with geodesic shooting for image registration. We introduce a novel finite dimensional Fourier representation of diffeomorphic deformations based on the key fact that the high frequency components of a diffeomorphism remain stationary throughout the integration process when computing the deformation associated with smooth velocity fields. We show that manipulating high dimensional diffeomorphisms can be carried out entirely in the bandlimited space by integrating the nonstationary low frequency components of the displacement field. This insight substantially reduces the computational cost of the registration problem. Experimental results show that our method is significantly faster than the state-of-the-art diffeomorphic image registration methods while producing equally accurate alignment. We demonstrate our algorithm in two different applications of image registration: neuroimaging and in-utero imaging.

Geodesic shape regression with multiple geometries and sparse parameters

Many problems in medicine are inherently dynamic processes which include the aspect of change over time, such as childhood development, aging, and disease progression. From medical images, numerous geometric structures can be extracted with various representations, such as landmarks, point clouds, curves, and surfaces. Different sources of geometry may characterize different aspects of the anatomy, such as fiber tracts from DTI and subcortical shapes from structural MRI, and therefore require a modeling scheme which can include various shape representations in any combination. In this paper, we present a geodesic regression model in the large deformation (LDDMM) framework applicable to multi-object complexes in a variety of shape representations. Our model decouples the deformation parameters from the specific shape representations, allowing the complexity of the model to reflect the nature of the shape changes, rather than the sampling of the data. As a consequence, the sparse representation of diffeomorphic flow allows for the straightforward embedding of a variety of geometry in different combinations, which all contribute towards the estimation of a single deformation of the ambient space. Additionally, the sparse representation along with the geodesic constraint results in a compact statistical model of shape change by a small number of parameters defined by the user. Experimental validation on multi-object complexes demonstrate robust model estimation across a variety of parameter settings. We further demonstrate the utility of our method to support the analysis of derived shape features, such as volume, and explore shape model extrapolation. Our method is freely available in the software package deformetrica which can be downloaded at www.deformetrica.org.

Detection of Conversion from Mild Cognitive Impairment to Alzheimer's Disease Using Longitudinal Brain MRI

Mild Cognitive Impairment (MCI) is an intermediate stage between healthy and Alzheimer's disease (AD). To enable early intervention it is important to identify the MCI subjects that will convert to AD in an early stage. In this paper, we provide a new method to distinguish between MCI patients that either convert to Alzheimer's Disease (MCIc) or remain stable (MCIs), using only longitudinal T1-weighted MRI. Currently, most longitudinal studies focus on volumetric comparison of a few anatomical structures, thereby ignoring more detailed development inside and outside those structures. In this study we propose to exploit the anatomical development within the entire brain, as found by a non-rigid registration approach. Specifically, this anatomical development is represented by the Stationary Velocity Field (SVF) from registration between the baseline and follow-up images. To make the SVFs comparable among subjects, we use the parallel transport method to align them in a common space. The normalized SVF together with derived features are then used to distinguish between MCIc and MCIs subjects. This novel feature space is reduced using a Kernel Principal Component Analysis method, and a linear support vector machine is used as a classifier. Extensive comparative experiments are performed to inspect the influence of several aspects of our method on classification performance, specifically the feature choice, the smoothing parameter in the registration and the use of dimensionality reduction. The optimal result from a 10-fold cross-validation using 36 month follow-up data shows competitive results: accuracy 92%, sensitivity 95%, specificity 90%, and AUC 94%. Based on the same dataset, the proposed approach outperforms two alternative ones that either depends on the baseline image only, or uses longitudinal information from larger brain areas. Good results were also obtained when scans at 6, 12, or 24 months were used for training the classifier. Besides the classification power, the proposed method can quantitatively compare brain regions that have a significant difference in development between the MCIc and MCIs groups.