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)医疗诊断(MRI、CT扫描、大脑DTI)的数字图像，(2)来自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。对这些的正确的几何描述需要现代微分几何的工具。经典的统计方法不足以进行分析；假定基础数据以有限数目的未知参数为模的分布形式的参数模型经常被错误地指定。由PI和其他人开发的独立于模型的方法已被证明在分析这些数据方面非常有效。本项目旨在极大地扩大这一方法的应用范围。提出的方法的一个基本组成部分是度量空间上概率Q的Fre‘chet均值的概念，它最小化到一点的期望平方距离。度量空间一般是微分流形，通常具有自然黎曼度量。但它也可能是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合在一起组成的分层空间。要使方法论发挥作用，必须确定(A)Fre‘chet极小值的唯一性和(B)样本Fre’chet均值的渐近分布。本项目的目标之一是显著扩展这方面的早期理论，为许多新的应用开辟道路。另一个重要目标是将密度估计、分类和回归的非参数贝叶斯理论推广到这类空间。这里的一个特别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优于核密度估计，而且当数据从参数模型模拟时，其最大似然估计也远远优于最大似然估计！对这一点的理解有望导致非参数贝叶斯方法的更广泛和更有效的使用。最后，PI建议开发一种用于机器人物体视觉的图形方法，与其他方法相比，计算复杂性要低得多。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.