Integral Curve Estimation: New Methodology and Applications to Diffusion Tensor Imaging

该项目的重点是建模和估计由未知矢量场驱动的积分曲线，该矢量场无法直接观测到。相反，相关空间场的噪声数据是可用的。提出了积分曲线估计量，并研究了其相合性和渐近分布。利用概率方法来解决这些非参数推理问题。这项工作的动机是扩散张量成像（一种特殊类型的磁共振成像），其中越来越需要统计方法。给定与水扩散张量相关的信号强度的观测值，可以找到与张量的最大特征值对应的向量场，它显示了扩散的主要方向。那么这个矢量场对应的积分曲线就是一条纤维，因为水分子在一些组织中，比如大脑中的白质，主要沿着纤维扩散。其中一个重要的问题是，从某一点开始的光纤是否能到达某一区域。所提出的方法为这种连通性问题提供了答案，更重要的是，它评估了这些答案的统计质量。总之，研究者开发了统计方法，补充了扩散张量成像技术，以帮助研究软组织（如大脑或肌肉）的结构。本提案概述了基于观察到的信号强度的噪声数据来估计光纤的新模型的基础上的数学和统计问题。该模型为扩散张量磁共振成像（DT-MRI，一种脑成像技术）中的问题提供了答案，这反过来又有助于提高对活体脑的物理和生物学的理解，并推进脑疾病和障碍的诊断方法。提出的研究增加了统计在DT-MRI中的作用。它将数学和统计学与物理和医学相结合。作为一种自然的推广，所提出的曲线（纤维）估计方法不仅基于矢量或张量数据，而且基于某些空间场，这本身就很有趣。它的分析表明概率方法在非参数统计推理问题中是非常宝贵的。研究人员还看到了这项工作在气象学等其他科学领域的潜在应用。此外，该项目为吸引和培养有兴趣和技能的研究生从事跨学科研究奠定了基础。

The focus of the project is to model and to estimate the integral curves that are driven by an unknown vector field which is not observed directly. Instead, a noisy data is available for a related spatial field. The investigator proposes integral curve estimators and studies their consistency and asymptotic distribution. Probabilistic methods are utilized to address these non-parametric inference problems. This work is motivated by Diffusion Tensor Imaging (a special type of Magnetic Resonance Imaging) where there is a growing need for statistical methods. Given observations of signal intensities that are related to water diffusion tensor, one can find the vector-field corresponding to the maximal eigenvalue of the tensor, which shows the main direction of diffusion. Then the integral curve corresponding to this vector-field is a fiber, since water molecules diffuse mostly along fibers in some tissues such as white matter in brain. One of important questions is whether a fiber starting at a specific point reaches a certain region. The proposed methodology provides answers to this connectivity problem and more importantly it assesses statistical quality of those answers. To summarize, the investigator develops statistical methods that complement Diffusion Tensor Imaging technology to help study architecture of soft tissues such as brain or muscle.This proposal outlines mathematical and statistical issues underpinning the development of a novel model for estimation of fibers based on observed noisy data of signal intensities. This model provides answers to questions in Diffusion Tensor Magnetic Resonance Imaging (DT-MRI, a brain imaging technique), which in turn could help to improve understanding of physics and biology of live brain and to advance diagnostical methods for brain diseases and disorders. The proposed research increases the role of statistics in DT-MRI. It integrates mathematics and statistics with physics and medicine. As a natural generalization, the proposed methodology for curve (fiber) estimation, based on not only vector or tensor data but some spatial field, is of interest by itself. Its analysis shows how probabilistic methods are invaluable for problems in non-parametric statistical inference. The investigator also sees potential applications of this work to other fields of science such as meteorology. Furthermore, this project builds a base for attracting and training graduate students with interest and skills to work in interdisciplinary research.