Cheng Jian, Ghosh Aurobrata, Jiang Tianzi, Deriche Rachid
Center for Computational Medicine, LIAMA, Institute of Automation, Chinese Academy of Sciences, China.
Med Image Comput Comput Assist Interv. 2011;14(Pt 2):98-106. doi: 10.1007/978-3-642-23629-7_13.
In Diffusion Tensor Imaging (DTI), Riemannian framework based on Information Geometry theory has been proposed for processing tensors on estimation, interpolation, smoothing, regularization, segmentation, statistical test and so on. Recently Riemannian framework has been generalized to Orientation Distribution Function (ODF) and it is applicable to any Probability Density Function (PDF) under orthonormal basis representation. Spherical Polar Fourier Imaging (SPFI) was proposed for ODF and Ensemble Average Propagator (EAP) estimation from arbitrary sampled signals without any assumption.
Tensors only can represent Gaussian EAP and ODF is the radial integration of EAP, while EAP has full information for diffusion process. To our knowledge, so far there is no work on how to process EAP data. In this paper, we present a Riemannian framework as a mathematical tool for such task.
We propose a state-of-the-art Riemannian framework for EAPs by representing the square root of EAP, called wavefunction based on quantum mechanics, with the Fourier dual Spherical Polar Fourier (dSPF) basis. In this framework, the exponential map, logarithmic map and geodesic have closed forms, and weighted Riemannian mean and median uniquely exist. We analyze theoretically the similarities and differences between Riemannian frameworks for EAPs and for ODFs and tensors. The Riemannian metric for EAPs is diffeomorphism invariant, which is the natural extension of the affine-invariant metric for tensors. We propose Log-Euclidean framework to fast process EAPs, and Geodesic Anisotropy (GA) to measure the anisotropy of EAPs. With this framework, many important data processing operations, such as interpolation, smoothing, atlas estimation, Principal Geodesic Analysis (PGA), can be performed on EAP data.
The proposed Riemannian framework was validated in synthetic data for interpolation, smoothing, PGA and in real data for GA and atlas estimation. Riemannian median is much robust for atlas estimation.
在扩散张量成像（DTI）中，基于信息几何理论的黎曼框架已被提出用于处理张量的估计、插值、平滑、正则化、分割、统计检验等。最近，黎曼框架已被推广到方向分布函数（ODF），并且适用于正交基表示下的任何概率密度函数（PDF）。球形极坐标傅里叶成像（SPFI）被提出用于从任意采样信号估计ODF和总体平均传播子（EAP），无需任何假设。
张量仅能表示高斯EAP，而ODF是EAP的径向积分，而EAP具有扩散过程的完整信息。据我们所知，到目前为止还没有关于如何处理EAP数据的工作。在本文中，我们提出一个黎曼框架作为完成此类任务的数学工具。
我们通过用傅里叶对偶球形极坐标傅里叶（dSPF）基表示EAP的平方根（称为基于量子力学的波函数），提出了一种用于EAP的先进黎曼框架。在此框架中，指数映射、对数映射和测地线具有封闭形式，并且加权黎曼均值和中位数唯一存在。我们从理论上分析了用于EAP、ODF和张量的黎曼框架之间的异同。用于EAP的黎曼度量是微分同胚不变的，这是张量的仿射不变度量的自然扩展。我们提出对数欧几里得框架来快速处理EAP，并提出测地线各向异性（GA）来测量EAP的各向异性。利用这个框架，可以对EAP数据执行许多重要的数据处理操作，如插值、平滑、图谱估计、主测地线分析（PGA）。
所提出的黎曼框架在合成数据的插值、平滑、PGA以及真实数据的GA和图谱估计中得到了验证。黎曼中位数在图谱估计中更稳健。
