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. We consider two approaches: discretizing the tensor calculus, and a discrete mesh version of differential forms. While these two approaches are equivalent in the continuum, we show that this is not true in the discrete case. Nevertheless, we show that it is possible to construct mesh versions of the Levi-Civita connection (and hence the tensorial covariant derivative and the associated covariant exterior derivative), the torsion, and the curvature. We show how discrete analogs of the usual vector integral theorems are constructed in such a way that the appropriate conservation laws hold exactly on the mesh, rather than only as approximations to the continuum limit. We demonstrate the success of our method by constructing a mesh version of classical electromagnetism and discuss how our formalism could be used to deal with other physical models, such as fluids.

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. We consider the large-numbers limit of the specificity, and derive expressions which show that it can be considered as an estimator of the divergence between the unknown pdf from which the training data was drawn and the model pdf built from the training data. Experiments using artificial data enable us to show that these limiting large-number relations enable us to obtain good quantitative and qualitative predictions of the behavior of the measured specificity, even for small numbers of training examples and in some extreme cases. We demonstrate that specificity can provide a more sensitive measure of difference between various modeling methods than some previous graph-based techniques. Key points are illustrated using real data sets. We thus establish a proper theoretical basis for the previously ad hoc concept of specificity, and obtain useful insights into the application of specificity in the analysis of real data.

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. Given the complexity of the task, the final accuracy is significantly affected by the choices made for each component. Here, we present a groupwise registration algorithm which can take advantage of the statistics of both the image intensities and the range of shapes across the group to achieve accurate matching. By testing on large sets of images (in both 2D and 3D), we explore the effects of using different image representations and different statistical shape constraints. We demonstrate that careful choice of such representations can lead to significant improvements in overall performance.

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. In 2-D, a correspondence is often defined using a set of manually placed 'landmarks' and linear interpolation to sample the shape in between. Such annotation is, however, time-consuming and subjective, particularly when extended to 3-D. In this paper, we show that it is possible to establish a dense correspondence across the whole training set automatically by treating correspondence as an optimization problem. The objective function we use for the optimization is based on the minimum description length principle, which we argue is a criterion that leads to models with good compactness, specificity, and generalization ability. We manipulate correspondence by reparameterizing each training shape. We describe an explicit representation of reparameterization for surfaces in 3-D that makes it impossible to generate an illegal (i.e., not one-to-one) correspondence. We also describe several large-scale optimization strategies for model building, and perform a detailed analysis of each approach. Finally, we derive quantitative measures of model quality, allowing meaningful comparison between models built using different methods. Results are given for several different training sets of 3-D shapes, which show that the minimum description length models perform significantly better than other approaches.

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). This leads to an initial reduction in computational complexity. We also consider the question of regularization, and show that by borrowing ideas from image registration, it is possible to build a non-parametric, fluid regularizer for shapes, without losing the computational gain made by the use of shape images. We show that this non-parametric regularization leads to a further considerable gain, when compared to parametric regularization methods. Quantitative evaluation is performed on biological datasets, and shown to yield a substantial decrease in convergence time, with no loss of model quality.

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. The performance of this algorithm is evaluated by extending the concepts of generalisability and specificity from shape models to image models. This provides an independent metric for comparing registration algorithms of groups of images. Experimental results on MR data of brains for various pairwise and groupwise registration algorithms is presented, and demonstrates the feasibility of the combined registration/modelling framework, as well as providing quantitative evidence for the superiority of groupwise approaches to registration.

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. This paper presents methods for constructing a general class of multi-dimensional diffeomorphic representations of deformations. We demonstrate, for the particular case of the polyharmonic clamped-plate splines, that these representations are suitable for the description of deformations of medical images in both two and three dimensions, using a set of two-dimensional annotated MRI brain slices and a set of three-dimensional segmented hippocampi with optimized correspondences. The class of diffeomorphic representations also defines a non-Euclidean metric on the space of patterns, and, for the case of compactly supported deformations, on the corresponding diffeomorphism group. In an experimental study, we show that this non-Euclidean metric is superior to the usual ad hoc Euclidean metrics in that it enables more accurate classification of legal and illegal variations.

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. We compare the results with those obtained using another previously published method for building 3D models, and show that our approach results in a model that is significantly more specific, general, and compact. The two models are used to investigate the hypothesis that there are differences in hippocampal shape between age-matched schizophrenic and normal control subgroups within the training set. Linear discriminant analysis is used to find the combination of shape parameters that best separates the two subgroups. We perform an unbiased test that shows there is a statistically significant shape difference using either shape model, but that the difference is more significant using the model built using our approach. We show also that the difference between the two subgroups can be visualised as a mode of shape variation.

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. Often this is achieved by locating a set of "landmarks" manually on each training image, which is time consuming and subjective in two dimensions and almost impossible in three dimensions. We describe how shape models can be built automatically by posing the correspondence problem as one of finding the parameterization for each shape in the training set. We select the set of parameterizations that build the "best" model. We define "best" as that which minimizes the description length of the training set, arguing that this leads to models with good compactness, specificity and generalization ability. We show how a set of shape parameterizations can be represented and manipulated in order to build a minimum description length model. Results are given for several different training sets of two-dimensional boundaries, showing that the proposed method constructs better models than other approaches including manual landmarking-the current gold standard. We also show that the method can be extended straightforwardly to three dimensions.