Chen T, Rangarajan A, Eisenschenk SJ, Vemuri BC.
Abstract
This paper proposes a novel technique for constructing a neuroanatomical shape complex atlas using an information geometry framework. A shape complex is a collection of shapes in a local neighborhood. We represent the boundary of the entire shape complex using the zero level set of a distance function S(x). The spatial relations between the different anatomical structures constituting the shape complex are captured via the distance transform. We then leverage the well known relationship between the stationary state wave function psi(x) of the Schrödinger equation -h2nabla2 psi + psi = 0 and the eikonal equation //nablaS// = 1 satisfied by any distance function S(x). This leads to a one-to-one map between psi(x) and S(x) and allows for recovery of S(x) from psi(x) through an explicit mathematical relationship. Since the wave function can be regarded as a square-root density function, we are able to exploit this connection and convert shape complex distance transforms into probability density functions. Furthermore, square-root density functions can be seen as points on a unit hypersphere whose Riemannian structure is fully known. A shape complex atlas is constructed by first computing the Karcher mean psi(x) of the wave functions, followed by an inverse mapping of the estimated mean back to the space of distance transforms in order to realize the atlas. We demonstrate the shape complex atlas computation via a set of experiments on a population of brain MRI scans. We also present modes of variation from the computed atlas for the control population to demonstrate the shape complex variability.
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