Rhodri H. Davies

Evaluation of 3D Correspondence Methods for Model Building

The correspondence problem is of high relevance in the construction and use of statistical models. Statistical models are used for a variety of medical application, e.g. segmentation, registration and shape analysis. In this paper, we present comparative studies in three anatomical structures of four different correspondence establishing methods. The goal in all of the presented studies is a model-based application. We have analyzed both the direct correspondence via manually selected landmarks as well as the properties of the model implied by the correspondences, in regard to compactness, generalization and specificity. The studied methods include a manually initialized subdivision surface (MSS) method and three automatic methods that optimize the object parameterization: SPHARM, MDL and the covariance determinant (DetCov) method. In all studies, DetCov and MDL showed very similar results. The model properties of DetCov and MDL were better than SPHARM and MSS. The results suggest that for modeling purposes the best of the studied correspondence method are MDL and DetCov.

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.

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.

Building optimal 2D statistical shape models

Statistical shape models are used widely as a basis for segmenting and interpreting images. A major drawback of the approach is the need, during training, to establish a dense correspondence across a training set of segmented shapes. We show that model construction can be treated as an optimisation problem, automating the process and guaranteeing the effectiveness of the resulting models. This is achieved by optimising an objective function with respect to the correspondence. We use an information theoretic objective function that directly promotes desirable features of the model. This is coupled with an effective method of manipulating correspondence, based on reparameterising each training shape, to build optimal statistical shape models. The method is evaluated on several training sets of shapes, showing that it constructs better models than alternative approaches. q 2003 Elsevier B.V. All rights reserved.

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.

Corpus Callosum Analysis using MDL-based Sequential Models of Shape and Appearance

This paper describes a method for automatically analysing and segmenting the corpus callosum from magnetic resonance images of the brain based on the widely used Active Appearance Models (AAMs) by Cootes et al. Extensions of the original method, which are designed to improve this specific case are proposed, but all remain applicable to other domain problems. The well-known multi-resolution AAM optimisation is extended to include sequential relaxations on texture resolution, model coverage and model parameter constraints. Fully unsupervised analysis is obtained by exploiting model parameter convergence limits and a maximum likelihood estimate of shape and pose. Further, the important problem of modelling object neighbourhood is addressed. Finally, we describe how correspondence across images is achieved by selecting the minimum description length (MDL) landmarks from a set of training boundaries using the recently proposed method of Davies et al. This MDL-approach ensures a unique parameterisation of corpus callosum contour variation, which is crucial for neurological studies that compare reference areas such as rostrum, splenium, et cetera. We present quantitative and qualitative results that show that the method produces accurate, robust and rapid segmentations in a cross sectional study of 17 subjects, establishing its feasibility as a fully automated clinical tool for analysis and segmentation.

An Efficient Method for Constructing Optimal Statistical Shape Models

Statistical shape models show considerable promise as a basis for segmenting and interpreting images. A major drawback of the approach is the need to establish a dense correspondence 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 a correspondence across a set of shapes. As the correspondence does not use an explicit ordering constraint, it generalises to 3D shapes. Results are given for several different training sets of 2D boundaries, showing the automatic method constructs better models than ones built by hand.

Consistent spherical parameterisation for statistical shape modelling

We have described previously a method of automatically constructing statistical models of shape. The method treats model-building as an optimisation problem by re-parameterising each shape so as to minimise the description length of the training set. The approach requires an explicit parameterisation of each shape, which is straightforward in 2D, but non-trivial in 3D. It is necessary to provide some parameterisation of the training set to initialise the optimisation. An inappropriate initial parameterisation can cause the optimisation to converge at a slower rate or stop it from converging to a satisfactory solution. In this paper we describe a method of producing a consistent parameterisation for a given set of surfaces. The consistent parameterisations were used to initialise the model-building algorithm and produced results that were significantly better than alternative approaches

Non-parametric surface-based regularisation for building statistical shape models

Determining groupwise correspondence across a set of unlabelled examples of either shapes or images, by the use of an optimisation procedure, is a well-established technique that has been shown to produce quantitatively better models than other approaches. However, the computational cost of the optimisation is high, leading to long convergence times. In this paper, we show how topologically non-trivial shapes can be mapped to regular grids (called shape images). This leads to an initial reduction in computational complexity. By also considering the question of regularisation, we show that a non-parametric fluid regulariser can be applied in a principled manner, the fluid flowing on the shape surface itself, whilst not loosing the computational gain made by the use of shape images. We show that this non-parametric regularisation leads to a further considerable gain, when compared to previous parametric regularisation methods. Quantitative evaluation is performed on biological datasets, and shown to yield a substantial decrease in convergence time, with no loss of model quality.

GROUP-WISE CORRESPONDENCE OF SURFACES USING NON-PARAMETRIC REGULARISATION AND SHAPE IMAGES

Previous automatic methods for constructing statistical shape models have located the correspondence across a training set of shapes by minimising an objective function based on the minimum description length principle (MDL). Although this method has been shown to produce quantitatively better results than other methods, the computational cost is high, leading to long convergence times. In this paper, we show how topologically non-trivial training shapes can be mapped to regular grids (called shape images). This leads to a significant reduction in computational complexity, and effectively re-casts shape model-building in a similar framework to that of groupwise image registration. A further reduction in computational cost is achieved by using a non-parametric fluid-based regulariser to manipulate the correspondence between shape images. Quantitative evaluation of using shape images and fluid-based regularisation demonstrates reduced convergence time (without loss of model quality) when compared to previous methods.