Nonparametric estimation of integral curves and surfaces

积分曲线是脑成像和大气科学中出现的各种自然结构的合理模型。脑成像中的自然结构包括连接神经元的轴突纤维。大气科学中的自然结构包括气压等线、风暴锋和急流。通常，数据是沿着曲线的点，并且被噪声破坏。本研究的目的是对这些曲线进行统计估计。这些研究人员将研究与统计积分曲线估计相关的详细问题清单，这将加深和扩大我们对自然结构的理解。pi将通过分析患病和健康大脑的图像来验证他们的统计方法。纤维的可靠统计估计将包括用于诸如阿尔茨海默病、多发性硬化症和脑肿瘤等疾病诊断程序的图像的不确定度评估。它还将增强规划成像引导神经外科手术所需的成像工具。在气象学中，对这些曲线的正确估计可以增强现有的天气图。最后，pi将利用建模曲线和建模曲面之间的相似性，并研究曲面的统计估计，这是脑成像中轴突纤维束的自然模型，也是许多科学中使用的数字化图像中的等面。积分曲线是微分方程的解，其中控制向量或张量场是直接或间接观察到的，并受到噪声的干扰。他们计划了统计分析的几个方向，包括一个完全非参数扩散函数模型，积分曲线的同步置信带，自适应估计，通过不同程序获得的同一大脑图像的比较，基于张量阶的模型简化，以及流形的非参数估计的统一方法，其中积分曲线和曲面作为特殊情况。具体来说，pi计划为最终用户提供增强图像，这些图像不仅包含估计的积分曲线，而且同时包含以统一的方式显示估计质量的置信带，而且所有的调优参数都将仅从数据中计算出来。在完成建议的研究后，从业者将配备一个系统比较不同轨迹成像算法的统计特性的程序。最后，pi计划将他们的方法从1D曲线扩展到2D曲面。本研究的基础是经验过程理论、高斯过程理论、常微分方程理论、张量摄动理论、数值分析、计算机视觉和计算统计学的协同作用。

Integral curves are reasonable models for a variety of natural structures that arise in brain imaging and in atmospheric sciences. Natural structures in brain imaging include the axonal fibers connecting neurons. Natural structures in atmospheric sciences include isolines of barometric pressure, storm fronts, and jet streams. Generally, the data are points along the curves and are corrupted by noise. The goal of this research is statistical estimation of these curves. These researchers will investigate a detailed list of issues pertinent to statistical integral curve estimation, which will deepen and widen our understanding of the natural structures. The PIs will validate their statistical methodology by analyzing images on both diseased and healthy brains. The reliable statistical estimation of fibers will include the assessment of uncertainty in the images used for diagnostic procedures in diseases such as Alzheimer's disease, multiple sclerosis, and brain tumors. It will also enhance the imaging tools needed for planning imaging-guided neurosurgeries. In meteorology the correct estimation of these curves can enhance existing weather maps. Finally, the PIs will capitalize on the similarities between modeling curves and modeling surfaces and investigate the statistical estimation of surfaces, which serve as natural models for axonal fiber bundles in brain imaging and iso-surfaces in digitized images used by many sciences. Integral curves are solutions of differential equations where the governing vector or tensor fields are observed directly or indirectly and perturbed by noise. Several directions of their statistical analysis are planned, including a fully nonparametric diffusion function model, simultaneous confidence bands for integral curves, adaptive estimation, comparison of images of the same brain obtained via different procedures, model reduction based on tensor's order, and a unified approach to nonparametric estimation of manifolds, where integral curves and surfaces serve as particular cases. Specifically, the PIs plan to provide an end-user with enhanced images that contain not only estimated integral curves, but also simultaneous confidence bands that show the quality of the estimation in a uniform way, and moreover all the tuning parameters would be calculated from the data only. Upon completion of the proposed research, a practitioner would be armed with a procedure of systematic comparison of statistical properties of different tractography algorithms. Finally, the PIs plan to extend their methodology from 1D curves to 2D surfaces. The basis of this research is a synergy of empirical processes theory, Gaussian processes theory, the theory of ordinary differential equations, perturbation theory for tensors, numerical analysis, computer vision, and computational statistics.