The rigid group-wise point set registration methods proposed in this chapter, namely, TMM and mrTMM, were shown to outperform state-of-the-art GMM- based techniques in terms of registration accuracy, using the synthetic data set. Their performance with the clinical data sets however, was comparable to gCPD. The proposed methods and gCPD consistently outperformed the other GMM- based approaches investigated, across all clinical data experiments. Although mrTMM offered significant improvement in registration accuracy over TMM on
synthetic data, their performance was comparable in most experiments involving clinical data. However, mrTMM also offered significant improvement in compu- tational efficiency, achieving substantial reduction in execution times when reg- istering all data sets, relative to TMM. Quicker convergence, without loss in reg- istration accuracy is especially attractive for the analysis of large data sets.
The TMM-based non-rigid group-wise registration approach (TMM-NR) per- formed comparably to, or better than, a group-wise variant of non-rigid CPD, and pair-wise non-rigid CPD in terms of registration accuracy, using the synthetic 3D face data set. Using clinical data, comprising ventricular structures generated from AD, MCI and healthy cohorts, the proposed non-rigid group-wise regis- tration approach (TMM-NR) performed comparably to non-rigid gCPD in most cases and consistently outperformed TMM-rigid, in terms of registration accu- racy. Improvements in registration accuracy offered by both rigid and non-rigid formulations of the proposed framework in some cases, and their comparable performance in others, relative to the state-of-the-art, is promising for their appli- cation in a variety of computer vision and medical image analysis tasks, requiring automatic robustness to outliers.
Chapter 4
Statistical Shape Models
4.1
Introduction
‘Shape’ as defined by (Kendall,1989), is the geometric information that remains once an object has been normalized with respect to rotation, scaling and transla- tion. Various methods to represent this information and analyse the statistics of their variation across an ensemble of similar shapes have been proposed, some of which include point- or mesh-based discretisation (Cootes et al., 1995), im- plicit functions (signed distance maps) (Leventon, Grimson, and Faugeras,2000), spherical harmonics (SPHARM) based parametrisation (Brechbühler, Gerig, and Kübler, 1995), (Gerig et al., 2001b) and medial shape representation (Pizer et al.,
2003), (Styner et al., 2003). Among these, point-based representations of shape are the most prevalent for training SSMs using Principal Component Analysis (PCA), due to their simplicity and independence to topology. The latter property in particular is a desirable trait for anatomical structures, not afforded by some techniques such as SPHARM for example, which only permit shapes of spheri- cal topology. Medial models are ’skeleton-like’ representations which yield more compact shape descriptions than landmark-based approaches but utilise surface boundaries parametrised by SPHARM and consequently have identical topolog- ical constraints. Choice of shape representation thus plays a central role in, de- termining the variety of shapes that may be analysed, and in the selection of an appropriate registration technique for correcting global differences in pose and establishing correspondences.
Past approaches to automatic SSM generation have included: (1) a pair-wise, template-to-training set (or one-to-many) registration strategy where an atlas is non-rigidly registered to each training shape, thereby propagating the landmarks
used to represent the atlas shape across the training set and establishing corre- spondences (Lorenz and Krahnstöver,2000), (Frangi et al.,2002); (2) population- based techniques based on minimum description length (MDL) (Davies et al.,
2002), (Davies et al., 2010) or entropy (equivalent to MDL) (Cates et al., 2007), which automatically estimate correspondences across training shapes by opti- mizing an objective function dependent on model quality; and (3) group-wise point set registration methods for jointly aligning a group of shapes and estab- lishing correspondences across them (Hufnagel et al.,2008), (Gooya, Davatzikos, and Frangi,2015). A thorough review of various correspondence estimation ap- proaches for training SSMs is provided in (Heimann and Meinzer,2009). As dis- cussed in the previous chapter, of particular interest in this thesis are the third group of techniques, due to their inherent flexibility and ability to establish cor- respondences in an unbiased manner. In contrast, population based techniques often perform the registration (for global pose corrections) and correspondence estimation steps, separately. While pair-wise registration approaches result in a bias towards the chosen atlas, registered to each sample in the training set.
Based on these criteria, point-based representations of shapes and the TMM- based group-wise registration framework discussed in the previous chapter are employed for training SSMs, as the objective is to design a framework indepen- dent of topology, automatic and robust to the presence of outliers. Adopting a probabilistic view of correspondences, also helps account for uncertainties in the registration process, not afforded by techniques relying on one-to-one mapping. This is particularly crucial when automatic segmentation and point set generation techniques are employed, to generate the requisite training shapes, as done in this study. Additionally, point-based representation of shapes are computation- ally less expensive than their implicit (signed distance functions) counterparts for example, in relation to SSM construction and application.
Statistical shape models have found widespread use in a variety of medical image analysis applications in recent years such as segmentation (Patenaude et al.,2011), (Castro-Mateos et al.,2015), shape-based prediction of tissue anisotropy (Lekadir et al.,2014), quantitative shape analysis and classification for computer- aided-diagnosis (Styner et al., 2004), (Shen et al., 2012), (Gooya et al., 2015), to name a few. Their primary challenge has persistently been the availability of training sets of sufficient size, necessary to adequately describe anatomical shape variability observed across different demographic and diagnostic populations. A training set of segmentations delineating the structure of interest in medical im- ages, is typically generated manually or semi-automatically, which is laborious
and prohibitive when analysing 3D structures from large cohorts. In the past, var- ious solutions have been proposed, such as merging pre-existing SSMs trained from different cohorts (Pereañez et al., 2014), generating synthetic variations in shape using deformable transformations (Koikkalainen et al.,2008) to enrich the data set with a higher degree of variability, and employing automatic techniques to generate the required training set of segmentations, which is of particular in- terest in this study. The major challenges with this approach are the potential in- clusion of outliers and the presence of missing information in the segmentations, as a result of variable image resolution and quality, motion artefacts, pathology- induced intensity inhomogeneities, among others. Consequently, in order to fa- cilitate large-scale statistical shape analysis of anatomical structures using auto- mated techniques for generating training shapes, a robust framework capable of rigidly aligning such instances and establishing anatomically valid correspon- dences across the same, is imperative. The TMM-based methods presented in the previous chapter address these issues and are therefore used here to construct SSMs by PCA. The quality of the resulting SSMs are quantitatively evaluated and compared with the state-of-the-art, in this chapter.