The statistical analysis of data lying on a differentiable, locally Euclidean, manifold introduces a variety of challenges because the analogous measures to standard Euclidean statistics are local, that is only defined within a neighbourhood of each datapoint. This is because the curvature of the space means that the connection of Riemannian geometry is path dependent. In this paper we transfer the problem to Weitzenbock space, which has torsion, but not curvature, meaning that parallel transport is path independent, and rather than considering geodesics, it is natural to consider autoparallels, which are `straight' in the sense that they follow the local basis vectors. We demonstrate how to generate these autoparallels in a data-driven fashion, and show that the resulting representation of the data is a useful space in which to perform further analysis.

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.

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.

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.

We describe an approach for automated analysis of deformable objects which extracts structure information from groups of images containing different examples of the object with a particular application to human imaging. The proposed analysis framework simultaneously segments and registers a set of images, incrementally constructing a model of the composition of the object. By fitting an appropriate intensity distribution model to the image we obtain a soft segmentation which allows us to explicitly model the construction of each pixel from constituent image segments, rather than its expected intensity. This effectively decouples the model from the effects of the imaging system and varying statistics in different examples. When estimating the optimal deformation field for each example, the original image is compared to a reconstruction, generated using the composition model and its intensity distribution parameters for each segment (i.e. an estimate of how the model would appear given the imaging conditions for that image). In the paper we describe the algorithm in detail and show results of applying it to two sets of medical images of different anatomies taken with different imaging modalities. We present quantitative results demonstrating that the proposed algorithm is more powerful than current state of the art methods at extracting structural information such as spatial correspondences across groups of images with varying statistics. 1

There are various methods for building statistical models of shape or shape and appearance from training data. Quantitative comparison of such methods requires a method of evaluating the quality of the fit between the model pdf and the training data. One quantitative graph-based measure which has been used for this purpose is the specificity, and the associated measure of generalisation. In this paper we consider the large-numbers limit of the specificity, and derive expressions which show that specificity can be considered as a graph-based estimator of cross-entropy. We also give explicit expressions for the various constants involved. We perform simple experiments using artificial data, and show that these limiting relations hold good even for relatively small numbers of training examples. We hence establish a proper theoretical context for the previously ad hoc concept of specificity. 1

We describe an algorithm for obtaining correspondences across a group of images of deformable objects. The approach is to construct a statistical model of appearance which can encode the training images as compactly as possible (a Minimum Description Length framework). Correspondences are defined by piece-wise linear interpolation between a set of control points defined on each image. Given such points a model can be constructed, which can approximate every image in the set. The description length encodes the cost of the model, the parameters and most importantly, the residuals not explained by the model. By modifying the positions of the control points we can optimise the description length, leading to good correspondence. We describe the algorithm in detail and give examples of its application to MR brain images and to faces. We also describe experiments which use a recently-introduced specificity measure to evaluate the performance of different components of the algorithm. 1

The principled non-rigid registration of groups of images requires a fully groupwise objective function. We consider the problem as one of finding the optimal dense correspondence between the images in the set, where optimality is defined using the Minimum Description Length (MDL) principle, that the transmission of a model of the data, together with the parameters of that model, should be as short as possible. We demonstrate that this approach provides a suitable objective function by applying it to the task of non-rigid registration of a set of 2D T1-weighted MR images of the human brain. Furthermore, we show that even in the case when substantial portions of the images are missing, the algorithm not only converges to the correct solution, but also allows meaningful integration of image data across the training set, allowing the original image to be reconstructed. 1

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.

This paper considers the problem of measuring the differences between deformations. It is intended to be applied in the context of applications such as the analysis of sets of non-rigidly registered medical images, where diffeomorphic warps between pairs of images establish a dense pixel-to-pixel correspondence. We introduce a new spline with explicit boundary conditions, and show how this can be used to generate general flows of bounded diffeomorphisms. Techniques are developed to describe an arbitrary continuous diffeomorphism and to invert and interpolate between such diffeomorphisms. We show how these constructions and the geodesic distance can be used in the analysis of training sets of general diffeomorphic warps.

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.

Statistical shape models have been used widely as a basis for segmenting and interpreting images. A major drawback of the approach is the need to establish a set of dense correspondences across a training set of segmented shapes. By posing the problem as one of minimising the description length of the model, we develop an efficient method that automatically defines correspondences across a set of shapes. Results are given for several different training sets of shapes, showing that the automatic method constructs significantly better models than those built by hand- the current gold standard. 1

The use of Kernel Principal Component Analysis (KPCA) to model data distributions in high-dimensional spaces is described. Of the many potential applications, we focus on the problem of modelling the variability in a class of shapes. We show that a previous approach to representing non-linear shape constraints using KPCA is not generally valid, and introduce a new ‘proximity to data ’ measure that behaves correctly. This measure is applied to the building of models of both synthetic and real shapes of nematode worms. It is shown that using such a model to impose shape constraints during Active Shape Model (ASM) search gives improved segmentations of worm images than those obtained using linear shape constraints. 1