Collaborative Research: Multidimensional Curve Estimation for Diffusion MRI

积分曲线是各种生物现象的自然模型，从大脑成像数据中的神经纤维到大气数据中的急流。传统上，它们被建模为在3D域中用噪声观测到的方向向量场上定义的微分方程组的解。但成像技术的进步现在提供了更复杂的方向信息--在3D球体上定义的函数--在域中的每个位置。从这种增强的方向性数据跟踪的积分曲线有可能极大地增加我们对脑连接等生物现象的理解，但对于这些前沿数据的积分曲线估计器的统计特性还没有很好地理解。因此，在这个项目中，研究人员将为复杂方向数据的3D领域的积分曲线估计提供坚实的理论基础，并将其应用于正在进行的科学研究中的大量真实数据集。主要计划是使用高阶超对称张量对定向数据进行局部建模，并根据其伪本征矢量场上定义的常微分方程组进行积分曲线估计。研究人员将证明所提出的积分曲线估计在极小极大意义下具有最优收敛速度，并证明伪本征矢量场的气球估计将导致更好的收敛。然后，积分曲线估值器将被链接到伴随的随机过程，以允许在曲线的点估计周围构造统一的置信带；并将探索这些置信带的自适应估计，以使其实用。然后，研究人员将研究是否可以通过选择任意的3D测量位置，可能使用增强的成像技术来进一步改善估计。最后，将构建积分曲线分支的测试，例如在轴突纤维分叉或交叉的位置。拟议的工作可能会极大地提高扩散磁共振成像(MRI)数据的有用性，这是一项具有巨大潜力的技术，可以在活人中探测大脑的“线路图”--它的连接性。目前，大脑连通性测量被广泛认为是脆弱的、复杂的和难以验证的。对于每个接受扩散磁共振扫描的人，研究人员将估计描述轴突纤维轨迹的曲线，轴突纤维是大脑的电线。这些纤维将大脑区域连接成分布式网络，从而产生思维；这种大脑线路的进化对正常发育、基因表达、衰老、疾病、药物和环境因素的反应是广泛的神经科学的主要兴趣所在。简单地向科学终端用户提供他们是否应该相信由计算机程序提供给他们的估计的纤维轨迹的感觉，将极大地增强他们对这些轨迹与其他科学数据之间的关系做出有信心的决定的能力。此外，所提出的方法在气象学上也是相关的。在那里，天气数据中的等值线、锋面、急流和压力槽可以用类似的曲线轨迹建模，这些轨迹可以用来增强现有的天气图。最后，这项提议具有令人兴奋的教育影响。研究人员是一名统计学家和一名接受过神经科学培训的计算机科学家，他们设想建立一个由统计学和神经成像领域有前途的年轻研究人员组成的跨学科团队，他们通过参加小组会议、研究生课程以及与理论和应用相关的网络资源，接触曲线估计的数学和神经科学方面。这种独特的交叉授粉将使受训者做好准备，为在科学界方兴未艾的跨学科研究团队做出贡献。

Integral curves are natural models for a variety of biological phenomena, from neuron fibers in brain imaging data to jet streams in atmospheric data. Traditionally they have been modeled as solutions to differential equations defined on fields of direction vectors that are observed with noise in a 3D domain. But advances in imaging technology now provide much more complex directional information--functions defined on the 3D sphere-at each location in the domain. Integral curves traced from this enhanced directional data have the potential to dramatically increase our understanding of biological phenomena such as brain connectivity, but the statistical properties of integral curve estimators for this cutting-edge data are not well understood. Therefore in this project the investigators will provide a solid theoretical foundation for integral curve estimation in 3D fields of complex directional data and apply it to large corpuses of real data sets from ongoing scientific studies. The primary plan will be to model directional data locally using high-order supersymmetric tensors, and pose integral curve estimation in terms of ODEs defined on the field of their pseudo-eigenvectors. The investigators will show that the proposed integral curve estimators enjoy optimal convergence rates in a minimax sense, and prove that balloon estimators of the pseudo-eigenvector fields will lead to improved convergence. Then integral curve estimators will be linked to accompanying random processes to allow construction of uniform confidence bands around point estimates for curves; and adaptive estimation of these confidence bands will be explored to make them practically useful. The investigators will then study whether estimation may be improved further by selecting arbitrary 3D measurement locations, possibly using enhanced imaging techniques. Finally, a test for branching of integral curves will be constructed, for example at locations where axon fibers diverge or cross.The proposed work has the potential to dramatically increase the usefulness of diffusion magnetic resonance imaging (MRI) data, a technology with tremendous potential to probe the "wiring diagram" of the brain-- its connectivity-- in living people. Currently, brain connectivity measurements are widely regarded as brittle, complicated, and difficult to validate. For each individual receiving a diffusion MRI scan, the investigators will estimate curves describing the trajectories of axon fibers, the electrical "wires" of the brain. These fibers connect brain regions into distributed networks that give rise to thought; the evolution of this brain wiring in response to normal development, gene expression, aging, disease, drugs, and environmental factors is of primary interest to a broad swath of neuroscience. Simply providing scientific end-users with a sense of whether or not they should believe the estimated fiber trajectories provided to them by computer programs will greatly enhance their ability to make confident decisions about relations between such trajectories and other scientific data. In addition, the proposed methodology is also relevant in meteorology. There, isolines, fronts, jetstreams, and pressure troughs in weather data can be modeled by similar curve trajectories that can be used to enhance existing weather maps. Finally, this proposal has an exciting educational impact. The investigators, a statistician and a computer scientist with neuroscience training, envision building an interdisciplinary team of promising young researchers in statistics and neuroimaging who gain exposure to both the mathematical and neuroscience aspects of curve estimation through joined group meetings, graduate courses, and web resources related to theory and applications. This unique cross-pollination will prepare the trainees to contribute to the broadly interdisciplinary research teams that are ascendant in the sciences.