Statistics of Shape: Simple Statistics on Interesting Spaces

Sarang Joshi
University of Utah
Radiation Oncology, Biomedical Engineering and Computer Science

A primary goal of Computational Anatomy is the statistical analysis of anatomical variability. A natural question that arises is how dose one define the image of an “Average Anatomy”. Such an “average” must represent the intrinsic geometric anatomical variability present. Large Deformation Diffeomorphic transformations have been shown to accommodate the geometric variability but performing statistics of Diffeomorphic transformations remains a challenge. In this lecture I will further extend this notion of averaging for studying change of anatomy on average from a cross sectional study of growth. Regression analysis is a powerful tool for the study of changes in a dependent variable as a function of an independent repressor variable, and in particular it is applicable to the study of anatomical growth and shape change. When the underlying process can be modeled by parameters in a Euclidean space, classical regression techniques~\cite{hardle90,wand95} are applicable and have been studied extensively. However, recent work suggests that attempts to describe anatomical shapes using {\em flat Euclidean spaces} undermines our ability to represent natural biological variability. In this lecture I will develop a method for regression analysis of general, manifold-valued data. Specifically, we extend Nadaraya-Watson kernel regression by recasting the regression problem in terms of Fr\'echet expectation. Although this method is quite general, our driving problem is the study anatomical shape change as a function of age from random design image data. I will demonstrate our method by analyzing shape change in the brain from a random design dataset of MR images of 97 healthy adults ranging in age from 20 to 79 years. To study the small scale changes in anatomy, we use the infinite dimensional manifold of diffeomorphic transformations, with an associated metric. We regress a representative anatomical shape, as a function of age, from this population.

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