The geometric median on Riemannian manifolds with application to robust atlas estimation.

One of the primary goals of computational anatomy is the statistical analysis of anatomical variability in large populations of images. The study of anatomical shape is inherently related to the construction of transformations of the underlying coordinate space, which map one anatomy to another. It is now well established that representing the geometry of shapes or images in Euclidian spaces undermines our ability to represent natural variability in populations. In our previous work we have extended classical statistical analysis techniques, such as averaging, principal components analysis, and regression, to Riemannian manifolds, which are more appropriate representations for describing anatomical variability. In this paper we extend the notion of robust estimation, a well established and powerful tool in traditional statistical analysis of Euclidian data, to manifold-valued representations of anatomical variability. In particular, we extend the geometric median, a classic robust estimator of centrality for data in Euclidean spaces. We formulate the geometric median of data on a Riemannian manifold as the minimizer of the sum of geodesic distances to the data points. We prove existence and uniqueness of the geometric median on manifolds with non-positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds. Generalizing the Weiszfeld procedure for finding the geometric median of Euclidean data, we present an algorithm for computing the geometric median on an arbitrary manifold. We show that this algorithm converges to the unique solution when it exists. In this paper we exemplify the robustness of the estimation technique by applying the procedure to various manifolds commonly used in the analysis of medical images. Using this approach, we also present a robust brain atlas estimation technique based on the geometric median in the space of deformable images.

VLOD: high-fidelity walkthrough of large virtual environments.

We present visibility computation and data organization algorithms that enable high-fidelity walkthroughs of large 3D geometric data sets. A novel feature of our walkthrough system is that it performs work proportional only to the required detail in visible geometry at the rendering time. To accomplish this, we use a precomputation phase that efficiently generates per cell vLOD: the geometry visible from a view-region at the right level of detail. We encode changes between neighboring cells' vLODs, which are not required to be memory resident. At the rendering time, we incrementally construct the vLOD for the current view-cell and render it. We have a small CPU and memory requirement for rendering and are able to display models with tens of millions of polygons at interactive frame rates with less than one pixel screen-space deviation and accurate visibility.