Carole J. Twining

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.

Groupwise Diffeomorphic Non-rigid Registration for Automatic Model Building

We describe a framework for registering a group of images together using a set of non-linear diffeomorphic warps. The result of the groupwise registration is an implicit definition of dense correspondences between all of the images in a set, which can be used to construct statistical models of shape change across the set, avoiding the need for manual annotation of training images. We give examples on two datasets (brains and faces) and show the resulting models of shape and appearance variation. We show results of experiments demonstrating that the groupwise approach gives a more reliable correspondence than pairwise matching alone.

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.

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.

Kernel pricipal component analysis and the construction of non-linear active shape models

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.

Groupwise Non-rigid Registration Using Polyharmonic Clamped-Plate Splines

This paper introduces a novel groupwise data-driven algorithm for non-rigid registration. The motivation behind the algorithm is to enable the analysis of groups of registered images; to this end, the algorithm automatically constructs a low-dimensional, common representation of the warp fields. We demonstrate the algorithm on an example set of 2D medical images, and show that we can obtain good registration across the set, with automatic detection and correction of misaligned examples, whilst still maintaining a low-dimensional representation.

Groupwise Construction of Appearance Models using Piece-wise Affine Deformations

We describe an algorithm for obtaining correspondences across a group of imagesof deformable objects. The approach is to construct a statistical modelof appearance which can encode the training images as compactly as possible(a Minimum Description Length framework). Correspondences are defined bypiece-wise linear interpolation between a set of control points defined oneach image. Given such points a model can be constructed, which can approximateevery 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 descriptionlength, leading to good correspondence. We describe the algorithm in detailand give examples of its application to MR brain images and to faces. We alsodescribe experiments which use a recently-introduced specificity measureto evaluate the performance of different components of the algorithm.

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.

The use of kernel principal component analysis to model data distributions

Abstract We describe the use of kernel principal component analysis (KPCA) to model data distributions in high-dimensional spaces. We show that a previous approach to representing non-linear data constraints using KPCA is not generally valid, and introduce a new ‘proximity to data’ measure that behaves correctly. We investigate the relation between this measure and the actual density for various low-dimensional data distributions. We demonstrate the effectiveness of the method by applying it to the higher-dimensional case of modelling an ensemble of images of handwritten digits, showing how it can be used to extract the digit information from noisy input images.