Discrete differential geometry: the nonplanar quadrilateral mesh.

We consider the problem of constructing a discrete differential geometry defined on nonplanar quadrilateral meshes. Physical models on discrete nonflat spaces are of inherent interest, as well as being used in applications such as computation for electromagnetism, fluid mechanics, and image analysis. However, the majority of analysis has focused on triangulated meshes.

Specificity: A Graph-Based Estimator of Divergence.

In statistical modeling, there are various techniques used to build models from training data. Quantitative comparison of modeling techniques requires a method for evaluating the quality of the fit between the model probability density function (pdf) and the training data. One graph-based measure that has been used for this purpose is the specificity.

Computing accurate correspondences across groups of images.

Groupwise image registration algorithms seek to establish dense correspondences between sets of images. Typically, they involve iteratively improving the registration between each image and an evolving mean. A variety of methods have been proposed, which differ in their choice of objective function, representation of deformation field, and optimization methods.

Building 3-D statistical shape models by direct optimization.

Statistical shape models are powerful tools for image interpretation and shape analysis. A simple, yet effective, way of building such models is to capture the statistics of sampled point coordinates over a training set of example shapes. However, a major drawback of this approach is the need to establish a correspondence across the training set.

Groupwise surface correspondence by optimization: representation and regularization.

Groupwise optimization of correspondence across a set of unlabelled examples of shapes or images is a well-established technique that has been shown to produce quantitatively better models than other approaches. However, the computational cost of the optimization is high, leading to long convergence times. In this paper, we show how topologically non-trivial shapes can be mapped to regular grids, hence represented in terms of vector-valued functions defined on these grids (the shape image representation).

A unified information-theoretic approach to groupwise non-rigid registration and model building.

The non-rigid registration of a group of images shares a common feature with building a model of a group of images: a dense, consistent correspondence across the group. Image registration aims to find the correspondence, while modelling requires it. This paper presents the theoretical framework required to unify these two areas, providing a groupwise registration algorithm, where the inherently groupwise model of the image data becomes an integral part of the registration process.

Shape discrimination in the hippocampus using an MDL model.

We extend recent work on building 3D statistical shape models, automatically, from sets of training shapes and describe an application in shape analysis. Using an existing measure of model quality, based on a minimum description length criterion, and an existing method of surface re-parameterisation, we introduce a new approach to model optimisation that is scalable, more accurate, and involves fewer parameters than previous methods. We use the new approach to build a model of the right hippocampus, using a training set of 82 shapes, manually segmented from 3D MR images of the brain.

Constructing diffeomorphic representations for the groupwise analysis of nonrigid registrations of medical images.

Groupwise nonrigid registrations of medical images define dense correspondences across a set of images, defined by a continuous deformation field that relates each target image in the group to some reference image. These registrations can be automatic, or based on the interpolation of a set of user-defined landmarks, but in both cases, quantifying the normal and abnormal structural variation across the group of imaged structures implies analysis of the set of deformation fields. We contend that the choice of representation of the deformation fields is an integral part of this analysis.

A minimum description length approach to statistical shape modeling.

We describe a method for automatically building statistical shape models from a training set of example boundaries/surfaces. These models show considerable promise as a basis for segmenting and interpreting images. One of the drawbacks of the approach is, however, the need to establish a set of dense correspondences between all members of a set of training shapes.