Nonparametric Statistical Image Analysis: Theory and Applications

非欧几里得数据无处不在。它们以多种形式出现，例如用于（1）医学诊断（MRI，CT扫描，大脑的DTI），（2）卫星图像的场景识别，（3）识别制造产品中的缺陷，（4）人工智能（物体的机器人识别）等的数字图像。经典的统计方法不足以进行分析;参数模型假设基础数据的分布形式以有限数量的未知参数为模，经常被错误指定。PI和其他人开发的独立于模型的方法已被证明在分析此类数据方面非常有效。本项目旨在大大扩大这一方法的应用范围。所提出的方法的一个基本组成部分是一个度量空间上的概率Q的Fre'chet平均值的概念，它最小化了从一个点的预期平方距离。度量空间通常是微分流形，通常具有自然的黎曼度量。但它也可能是一个所谓的非正曲率测地线空间，包括许多图形空间以及由不同维度的流形粘在一起组成的分层空间。为了使方法有效，必须建立（a）Frechet极小值的唯一性和（B）样本Frechet平均值的渐近分布。本项目的目标之一是在这方面显著扩展早期理论，为许多新的应用开辟道路。另一个重要的目标是扩展到这样的空间的非参数贝叶斯理论的密度估计，分类和回归。这里的一个特殊目的是探索一个有趣的现象：在中等样本量的模拟研究中，Q密度的非参数贝叶斯估计不仅远远优于核密度估计，而且当数据从参数模型模拟时，MLE也优于核密度估计！理解这一点，预计将导致更广泛和更有效地使用非参数贝叶斯方法。最后，PI建议开发一种用于机器人物体视觉的图形方法，其计算复杂度远低于其他方法。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Non-Euclidean data are ubiquitous. They arise in many forms such as digital images for (1) medical diagnostics (MRI, CT scans, DTI of the brain), (2) scene recognition from satellite images, (3) identifying defects in manufactured products, (4) artificial intelligence (robotic identification of objects), etc. Proper geometric descriptions of these require tools from modern differential geometry. Classical statistical methods are inadequate for their analysis; parametric models, which assume the form of the distribution of the underlying data modulo a finite number of unknown parameters, are often misspecified. A model-independent methodology developed by the PI and others has been shown to be very effective in analyzing such data. The present project aims at vastly broadening the scope of this methodology for applications. A basic component of the methodology proposed is the notion of the Fre'chet mean of a probability Q on a metric space, which minimizes the expected squared distance from a point. The metric space is generally a differential manifold, often provided with a natural Riemannian metric. But it may also be a so called geodesic space of non-positive curvature, including many graphical spaces as well as stratified spaces made up of manifolds of different dimensions glued together. For the methodology to work one must establish (a) the uniqueness of the Fre'chet minimizer and (b) the asymptotic distribution of the sample Fre'chet mean. It is one of the goals of the present project to significantly extend the earlier theory in this regard, opening the way to many new applications. Another important objective is to extend to such spaces the nonparametric Bayes theory of density estimation, classification and regression. One special aim here is to explore an intriguing phenomenon: in simulation studies with moderate sample sizes, the nonparametric Bayes estimator of the density of Q far outperforms not only the kernel density estimator, but also the MLE when the data are simulated from a parametric model! An understanding of this is expected to lead to a wider and more effective use of the nonparametric Bayes methodology. Finally, the PI proposes to develop a graphical method for robotic vision of objects, with much less computational complexity than that of other methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.