GAMEs: Growing and adaptive meshes for fully automatic shape modeling and analysis

Abstract

This paper presents a new framework for shape modeling and analysis, rooted in the pattern recognition theory and based on artificial neural networks. Growing and adaptive meshes (GAMEs) are introduced: GAMEs combine the self-organizing networks which grow when require (SONGWR) algorithm and the Kohonen’s self-organizing maps (SOMs) in order to build a mesh representation of a given shape and adapt it to instances of similar shapes. The modeling of a surface is seen as an unsupervised clustering problem, and tackled by using SONGWR (topology-learning phase). The point correspondence between point distribution models is granted by adapting the original model to other instances: the adaptation is seen as a classification task and performed accordingly to SOMs (topology-preserving phase). We thoroughly evaluated our method on challenging synthetic datasets, with different levels of noise and shape variations. Finally, we describe its application to the analysis of a challenging medical dataset. Our method proved to be reproducible, robust to noise, and capable of capturing real variations within and between groups of shapes.

Introduction

Statistical shape modeling and analysis have been increasingly used during the last decade as a basis for image segmentation and interpretation. Many successful 2D-applications have been described in the literature (Cootes et al., 1995). Building a statistical model requires often the establishing of correspondence between shape surfaces over a set of training examples. Defining corresponding points on different shapes is not trivial: in some 2D applications, manual landmark definition might be possible but it becomes unpractical when 3D/4D shapes are considered.

Different techniques have been proposed in the literature to address this problem. In Gerig et al. (2001), the authors successfully applied a shape representation based on spherical harmonics (SPHARM) to the analysis of brain ventricles. The SPHARM is a multi-scale approach, which allows for smooth shape representation, even at a very fine scale. The two major limitations of this method are (1) the need of pre-processing steps to generate a spherical-topology object, and (2) the non-intuitive nature of parameters, which do not allow for an easy interpretation when significant shape differences are found. In Pizer et al. (2003), the authors introduced deformable medial representations (M-reps) for segmentation of 3D medical structures: they successfully applied their method to the analysis of kidney and hippocampus (see also Styner et al., 2004). The M-reps are multi-scale and can represent objects as composition of multiple figures related with each other. A key step to build up M-reps is the construction of medial models, which can either be created manually (time consuming and error prone), or generated automatically as suggested in Styner et al. (2003a). Nevertheless, the first step described in Styner et al. (2003a) requires a boundary parametrization using SPHARM: thus, the spherical-topology constraint still remains. Moreover, to identify medial models in thin elongated structures is not a trivial task. Another solution, rooted in an information theory framework, is the Minimum Description Length (MDL) approach described in Davies et al. (2001): the authors suggest to (1) use a descriptive function to describe corresponding points on different shapes, (2) build up a first model, and (3) evaluate its performances through an objective function. New models are generated with new parameters which are tuned in order to optimize the objective function. The process continues until convergence. The MDL has been successfully applied to the modeling of different structures (Davies et al., 2003, Styner et al., 2003b). Nevertheless, due to the highly non-linear objective function, genetic algorithms are used for the optimization: the computation load is therefore high, and a general global optimum is not granted. A completely different approach is the one suggested by Rueckert et al. (2003): instead of modeling shapes, the authors suggest to model deformation fields obtained by non-rigid registration. This approach has several advantages when the whole anatomy of a certain organ is studied: it does not require prior segmentation and it provides inter-object relationships. On the other hand, the authors consider shape-based approaches to be a better choice whenever the inter-object relationship is not important, or confounds the modeling process. All the modeling approaches described before can be divided into two groups: groupwise and pairwise. Groupwise analyses aim to optimize an objective function over the whole dataset while creating the statistical model (Davies et al., 2001), while pairwise solutions start with a representative shape of the dataset and build up a model rooted on it. Many advantages of groupwise analysis have already been highlighted in the literature (Davies et al., 2003); nevertheless, in a recent study on non-rigid registration and segmentation, Crum et al. (2005) compared pairwise and groupwise approaches, showing that the simpler framework of pairwise methods performs systematically better.

The method presented in this work is a pairwise approach to shape modeling. The shape-modeling problem can be summarized as follows: given a set of shapes, one needs to generate Point Distribution Models (PDMs) which describe them. Since shapes need to be compared, a correspondence between points in the PDMs has to be established. Three questions rise: How many nodes are needed? Where should they be located on the surface? How can one define correspondence across shapes? In this work, we introduce a pattern recognition framework to address these questions. In a first phase, a topology-learning unsupervised clustering algorithm is used to select the optimal number of nodes (clusters) and their locations in the input space (3-dimensional space of surface points). In a second phase, the correspondence problem among models is tackled as a classification task: the generalization property of a classifier is used to match unseen cases to similar previously seen points, preserving the topology learned through unsupervised clustering. Growing and adaptive meshes (GAMEs) are introduced to implement both the unsupervised clustering and the adapting algorithms. The growing phase of GAMEs (unsupervised topology-learning clustering) is based on self-organizing networks which grow when required (SONGWR), introduced by Marsland et al. (2002) (see also Fritzke, 1992, Fritzke, 1994 for more details on growing cell structures): SONGWR proved to be (1) more data-driven while growing, and (2) faster in learning input representation, when compared with previous models. The adaptation phase of GAMEs is based on self-organizing maps (SOM), which have been proved to be perfectly topology-preserving (Kohonen, 1990). To the best of our knowledge, this is the first work showing how a combination of growing structures and self-organizing maps can be used to address the issues of shape modeling and analysis.

In this paper, we provide a thorough evaluation of our method. Working with challenging synthetic shapes, we tested GAMEs’ robustness, reproducibility, and accuracy in detecting landmarks. We used the outcome of GAMEs to build statistical models and tested its ability of representing real variations within and between groups of shapes. We also successfully applied GAMEs to the analysis of shape variation in populations of brain ventricles, both for healthy elderly individuals and for patients subject to Alzheimer’s disease.

The rest of the paper is organized as follows. In Section 2 we provide an overview of our method. In Section 3 we describe the methods we used to evaluate and compare different PDMs, reporting the results for synthetic shapes. Section 4 shows the successful modeling of the brain ventricles in Magnetic Resonance Imaging (MRI). We finally give a detailed discussion of the proposed method, highlighting our main contributions, and provide a general conclusion.