Yang Xianfeng, Goh Alvina, Qiu Anqi
Division of Bioengineering, National University of Singapore, Singapore.
Inf Process Med Imaging. 2011;22:257-70. doi: 10.1007/978-3-642-22092-0_22.
In neuroimaging studies based on anatomical shapes, it is well-known that the dimensionality of the shape information is much higher than the number of subjects available. A major challenge in shape analysis is to develop a dimensionality reduction approach that is able to efficiently characterize anatomical variations in a low-dimensional space. For this, there is a need to characterize shape variations among individuals for N given subjects. Therefore, one would need to calculate (2(N)) mappings between any two shapes and obtain their distance matrix. In this paper, we propose a method that reduces the computational burden to N mappings. This is made possible by making use of the first- and second-order approximations of the metric distance between two brain structural shapes in a diffeomorphic metric space. We directly derive these approximations based on the so-called conservation law of momentum, i.e., the diffeomorphic transformation acting on anatomical shapes along the geodesic is completely determined by its velocity at the origin of a fixed template. This allows for estimating morphological variation of two shapes through the first- and second-order approximations of the initial velocity in the tangent space of the diffeomorphisms at the template. We also introduce an alternative representation of these approximations through the initial momentum, i.e., a linear transformation of the initial velocity, and provide a simple computational algorithm for the matrix of the diffeomorphic metric. We employ this algorithm to compute the distance matrix of hippocampal shapes among an aging population used in a dimensionality reduction analysis, namely, ISOMAP. Our results demonstrate that the first- and second-order approximations are sufficient to characterize shape variations when compared to the diffeomorphic metric constructed through (2(N)) mappings in ISOMAP analysis.
在基于解剖形状的神经成像研究中,众所周知,形状信息的维度远高于可用受试者的数量。形状分析中的一个主要挑战是开发一种降维方法,该方法能够在低维空间中有效地表征解剖变异。为此,需要表征(N)个给定受试者个体之间的形状变异。因此,需要计算任意两个形状之间的(C_{N}^2=\frac{N(N - 1)}{2})个映射,并获得它们的距离矩阵。在本文中,我们提出了一种将计算负担降低到(N)个映射的方法。这是通过利用微分同胚度量空间中两个脑结构形状之间度量距离的一阶和二阶近似来实现的。我们基于所谓的动量守恒定律直接推导这些近似,即沿着测地线作用于解剖形状的微分同胚变换完全由其在固定模板原点处的速度决定。这允许通过模板处微分同胚切空间中初始速度的一阶和二阶近似来估计两个形状的形态变异。我们还通过初始动量引入了这些近似的另一种表示,即初始速度的线性变换,并为微分同胚度量矩阵提供了一种简单的计算算法。我们使用该算法计算用于降维分析(即等距映射,ISOMAP)的老年人群中海马形状的距离矩阵。我们的结果表明,与ISOMAP分析中通过(C_{N}^2=\frac{N(N - 1)}{2})个映射构建的微分同胚度量相比,一阶和二阶近似足以表征形状变异。