Nested Sphere Statistics of Skeletal Models

Abstract

We seek a form of object model that exactly and completely captures the interior of most non-branching anatomic objects and simultaneously is well suited for probabilistic analysis on populations of such objects. We show that certain nearly medial, skeletal models satisfy these requirements. These models are first mathematically defined in continuous 3-space, and then discrete representations formed by a tuple of spoke vectors are derived. We describe means of fitting these skeletal models into manual or automatic segmentations of objects in a way stable enough to support statistical analysis, and we sketch means of modifying these fits to provide good correspondences of spoke vectors across a training population of objects. Understanding will be developed that these discrete skeletal models live in an abstract space made of a Cartesian product of a Euclidean space and a collection of spherical spaces. Based on this understanding and the way objects change under various rigid and nonrigid transformations, a method analogous to principal component analysis called composite principal nested spheres will be seen to apply to learning a more efficient collection of modes of object variation about a new and more representative mean object than those provided by other representations and other statistical analysis methods. The methods are illustrated by application to hippocampi.

Analysis of principal nested spheres

“…As noted in Marron et al (2010) and Pizer et al (2012), our method can be viewed as a backward generalization of principal component analysis. A conventional principal component analysis approach begins with lower dimensions and builds up, that is, finding the mean first, then the least squares line through the mean, then the second line which determines the best fitting plane and so on.…”

“…Such situations can be found in landmark shape analysis, examples of which are in § 5, and in computer-aided image analysis. For example, in investigations of shape variations of human organs including the lung, prostate, and hippocampus, applications of our method lead to succinct representations of the data compared to those from geodesic or linear principal component analysis (Jung et al, , 2011Pizer et al, 2012). The proposed method can also be seen as an extension of Jung et al (2011) …”

Analysis of Rotational Deformations From Directional Data

“…In skeletal representations (s-rep), a 3D object is modeled by skeletal positions lying inside of the object and spoke vectors pointing to the boundary of the object (Siddiqi and Pizer, 2008;Pizer et al, 2013). See Fig.…”

“…The skeletal representation (s-rep) gives a rich and efficient description of 3D objects (Siddiqi and Pizer, 2008;Pizer et al, 2013). The s-rep of human organs has been used to study structural and statistical properties and to promote precise segmentation of the organ from images.…”

“…In a number of applications, generalized rotations appear to be the major forms of deformation. For instance, the major variation of shapes of hippocampi in the human brain has been shown to be bending of the object (Joshi et al, 2002;Pizer et al, 2013); Human joint movements, such as the motion of the knee or the elbow, consist of bending and twisting about the joint (Rivest, 2001;Rivest et al, 2008;Oualkacha and Rivest, 2012). A direct modeling of such rotational deformations will promote a precise description of object variation and will be important for surgery or treatment planning.…”