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，该矢量场不是直接观测的。相反，噪声数据可用于相关的空间场。提出了积分曲线估计量，并研究了它们的相合性和渐近分布。概率方法被用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)，其中对统计方法的需求越来越大。给出与水扩散张量有关的信号强度的观测，可以找到与张量的最大本征值相对应的矢量场，它显示了扩散的主要方向。那么与这个矢量场对应的积分曲线就是纤维，因为在某些组织中，水分子主要是沿着纤维扩散的，比如大脑中的白质。一个重要的问题是，从特定点开始的纤维是否到达了特定区域。建议的方法为这一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，研究人员开发了补充扩散张量成像技术的统计方法，以帮助研究脑或肌肉等软组织的结构。这项建议概述了支持开发基于观察到的信号强度噪声数据的纤维估计的新模型的数学和统计学问题。这个模型回答了扩散张量磁共振成像(DT-MRI，一种脑成像技术)中的问题，这反过来又有助于提高对活体大脑的物理和生物学的理解，并改进脑部疾病和障碍的诊断方法。这项拟议的研究增加了统计学在DT-MRI中的作用。它将数学和统计学与物理学和医学结合在一起。作为一种自然的推广，所提出的不仅基于矢量或张量数据，而且基于某些空间场的曲线(纤维)估计方法本身就很有意义。它的分析表明，概率方法对于非参数统计推断中的问题是多么宝贵。这位研究人员还看到了这项工作在其他科学领域的潜在应用，如气象学。此外，该项目还为吸引和培训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.