Chen Ting, Rangarajan Anand, Eisenschenk Stephan J, Vemuri Baba C
Department of CISE, University of Florida, Gainesville, FL 32611, USA.
Med Image Comput Comput Assist Interv. 2010;13(Pt 3):65-72. doi: 10.1007/978-3-642-15711-0_9.
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.
本文提出了一种利用信息几何框架构建神经解剖形状复合体图谱的新技术。形状复合体是局部邻域内形状的集合。我们使用距离函数S(x)的零水平集来表示整个形状复合体的边界。构成形状复合体的不同解剖结构之间的空间关系通过距离变换来捕捉。然后，我们利用薛定谔方程-h2nabla2 psi + psi = 0的稳态波函数psi(x)与任何距离函数S(x)满足的程函方程//nablaS// = 1之间的熟知关系。这导致了psi(x)与S(x)之间的一一映射，并允许通过明确的数学关系从psi(x)恢复S(x)。由于波函数可被视为平方根密度函数，我们能够利用这种联系并将形状复合体距离变换转换为概率密度函数。此外，平方根密度函数可被视为单位超球面上的点，其黎曼结构是完全已知的。通过首先计算波函数的卡尔彻均值psi(x)，然后将估计均值反向映射回距离变换空间以实现图谱，从而构建形状复合体图谱。我们通过对一组脑部MRI扫描图像的实验来演示形状复合体图谱的计算。我们还展示了计算得到的对照人群图谱的变异模式，以证明形状复合体的变异性。
