Nonparametric Statistical Image Analysis: Theory and Applications

非参数统计图像分析：理论与应用

基本信息

批准号：
1811317
负责人：
Rabindra Bhattacharya
金额：
$ 22.5万
依托单位：
University of Arizona
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-05-01 至 2022-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811317&HistoricalAwards=false
关键词：
Nonparametric Statistical Image Analysis Theory

项目摘要

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.

非欧盟数据无处不在。它们以多种形式出现，例如（1）医学诊断（MRI，CT扫描，大脑的DTI），（2）从卫星图像中识别场景识别，（3）识别制成品中的缺陷，（4）人工智能（机器人识别物体）等。经典的统计方法不足以分析其分析；参数模型通常被误指定，该模型假设基础数据模块的分布形式是有限数量的未知参数。 PI和其他人开发的独立于模型的方法学在分析此类数据方面非常有效。本项目旨在极大地扩大该方法的应用范围。提出的方法的一个基本组成部分是频率平均值Q的概念q，这可以最大程度地减少与点的预期平方距离。度量空间通常是一个差分歧管，通常具有天然的riemannian度量。但这也可能是一个所谓的非阳性曲率的大地测量空间，包括许多图形空间以及由粘合在一起的不同尺寸的歧管组成的分层空间。为了使方法可以建立（a）频芯最小化器的唯一性，以及（b）样品频道平均值的渐近分布。它是本项目的目标之一，可以显着扩展这方面的早期理论，从而为许多新应用开放。另一个重要的目标是扩展到此类空间，非参数贝叶斯的密度估计，分类和回归理论。这里的一个特殊目的是探索一种有趣的现象：在具有中等样本量的模拟研究中，Q的密度的非参数贝叶斯估计量不仅超过了内核密度估计器，而且当数据是从参数模型中模拟时的MLE！对此的理解有望导致非参数贝叶斯方法的更广泛，更有效的使用。最后，PI提议为物体的机器人视觉开发一种图形方法，其计算复杂性要比其他方法的复杂性要少得多。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛的影响审查标准的评估来获得支持的。