Statistical Shape Analysis of Surfaces in Medical Images Applied to the Tetralogy of Fallot Heart - Modeling in Computational Biology and Biomedicine - page 162

5.1.2 Chapter Overview

Public

This chapter is targeting people interested in statistics of surfaces, the surface being
considered as the random variable itself. The reader is assumed to have good notions
of multivariate statistics (expectation, moments, PCA, regression, hypothesis tests)
and some notions of analysis in infinite dimensional vector spaces, in particular
with reproducing kernel Hilbert spaces (RKHS). This last notion is borrowed from
machine learning theory and is applied here to the characterization of geometric
shapes through geometric integration theory. A few notions from Riemannian
geometry are used but no specific knowledge is required.

Outline

In the following sections, we will first review in Sect. 5.2 statistical shape analysis
methods with a particular focus on the statistical analysis of surfaces. We will
introduce the formalism of deformations which is at the center of most of the
current works in computational anatomy [ 21 ]. Then, we will detail the framework
of currents to represent surfaces, and show how this can be turned into an effective
shape analysis technique. Section 5.3 will apply this methodology to the shape of the
heart of 13 patients with repaired tetralogy of Fallot. Correlating shape with clinical
variables will illustrate how we can extract some insight about the relationship
between morphology and physiology. Last but not least, we will exemplify how
the lack of longitudinal measurements can be bypassed by building a statistical
generative growth model from cross-sectional data which summarizes the heart
shape remodeling at the population level.

5.2 Statistical Shape Analysis

5.2.1 Shapes, Forms and Deformations

There is generally no physical model that can faithfully relate the shape of organs
in different patients. Thus, to analyze their variability in a population, one usually
extracts some anatomically representative landmarks (or more generally geometric
features), and models their statistical distribution across the population, via a
mean shape and covariance structure analysis after a group-wise matching for
instance. One of the earliest methods [ 2 , 3 ] consists in studying the variability of
anatomical landmark positions among a population: after a global pose (position