New Statistical Challenges Posed by Multiscale and Adaptive Representations

多尺度和自适应演示带来的新的统计挑战Donoho I. M.约翰斯通，PI的。项目期间：07/01/00 - 06/30/05该项目将解决以下具体问题：1。断层扫描中的估计：根据初步计算，曲波紧框架的表示似乎比传统的断层摄影方法实现更快的收敛速度.时频分析中的估计和测试：通过在时频平面上部署曲线波，建立了新的线性调频波紧框架。 这种新的表示方法对于噪声环境下啁啾信号的检测和估计具有重要意义.最近在计算视觉方面的工作提出了一个问题：“什么是表示自然图像的最佳基础？”有人提出，在这个主题的快速增长的文献中的许多实验结果可能会被解释和改进使用解析构造的表示：脊波和曲波.几何驱动扩散在应用图像处理中很受欢迎，但其定量性能还不清楚。建议使用最近的工具，如多尺度脊波，发展定量统计理论.基于稀疏性思想的扩展。寻找“白色噪声中的边”和“白色噪声中的子空间”的问题为推广现有的稀疏性思想提供了具有挑战性和及时的方向。测试稀疏均值。 提出了一种自适应的方法来检验一个随机向量的均值是否为非零，当该向量可能表现出未知的稀疏度时.大型协方差矩阵顶部特征值的渐近性。根据随机矩阵理论的最新进展，制定了一个程序，以发展大型数据矩阵主成分的统计理论。与“可重复研究”的原则相一致，该项目的软件和数字将在公共领域WaveLab系统的未来版本中提供。近年来，小波和时频方法的研究已经扩展到构建新的表示系统，包括为特定现象定制的系统。 例子包括小波包和余弦包，最近，系统喜欢edgelets，ridgelets，chirplets，warplets和curvelets。 与此同时，统计分析和同源领域的研究允许数据自己决定自己的最佳系统的代表设计。 主成分（即Karhunen-Loeved组合）是这种数据自适应表示的最古老的例子;最近提出了许多新的想法，如独立成分分析。 这些提议者在这两个领域都很活跃，创造了新的图像和信号表示，并发展了统计理论来支持自适应信号表示。 目前的项目将（a）追求两个机会，从最近推出的curvelets，（B）解决两个活跃的应用研究领域，计算视觉和几何驱动的扩散，和（c）攻击一些问题，被认为是在统计决策理论的新发展的核心。主题（a）可能对层析成像、图像和信号处理的应用工作产生影响，（c）可能影响主成分在气候和全球变化研究等领域的应用。

NEW STATISTICAL CHALLENGES POSED BY MULTISCALE AND ADAPTIVEREPRESENTATIONS D.L. Donoho & I.M. Johnstone, PI's.Project period: 07/01/00 -- 06/30/05The project will address the following specific topics:1. Estimation in tomography: the curvelet tight frame ofrepresentation seems, according to preliminary calculations, toachieve faster rates of convergence than traditional tomographicmethods.2. Estimation and testing in time-frequency analysis: a new tightframe of chirplets has been built by deploying curvelets in thetime-frequency plane. The new representation has promise for detection andestimation of chirps in the presence of noise.3. Recent work in computational vision asks ``what is the best basisfor representing natural images?'' It is proposed that many empiricalresults in the rapidly growing literature on this topic might beexplained and improved using analytically-constructed representations:ridgelets and curvelets.4. Geometry-driven diffusions are popular in applied image processing-- but their quantitative performance is not well-understood. It isproposed to develop a quantitative statistical theory using recenttools such as multiscale ridgelets.5. Extensions of Sparsity-Based ideas. Problems of finding `edgels inwhite noise' and `subspaces in white noise' offer challenging andtimely directions in which to generalize existing sparsity ideas.6. Testing Sparse Means. An adaptive approach is suggested fortesting if the mean of a random vector is nonzero, when the vectormight exhibit an unknown degree of sparsity.7. Asymptotics of top eigenvalues of large covariance matrices.A program is set out to develop statistical theory for principalcomponents of large data matrices based on recent progress inrandom matrix theory.Consistent with the principle of ``reproducible research'', softwareand figures from this project will be made available in futurereleases of the public domain WaveLab system.In recent years, research in wavelets and time-frequency methods hasbroadened to construct new systems of representation, includingsystems custom-tailored for specific phenomena. Examples includewavelet packets and cosine packets, and very recently, systems likeedgelets, ridgelets, chirplets, warplets, and curvelets. In parallel,research in statistical analysis and cognate fields allows datathemselves to dictate the design of their own optimal systems ofrepresentation. Principal components (i.e. Karhunen-Loevedecomposition) is the oldest example of such data-adaptiverepresentation; many newer ideas have been proposed recently, such asindependent components analysis. The proposers have beenactive in both domains, creating new image and signalrepresentations and developing statistical theory to underpin adaptivesignal representations. The current project will (a) pursue twoopportunities arising from the recent introduction of curvelets, (b)address two active applied research areas, computational vision andgeometry driven diffusions, and (c) attack some issues which areargued to be at the core of new developments in statistical decisiontheory. Topic (a) may have implications for applied work intomography, image and signal processing, and (c) may impact applieduses of principal components in domains such as climate and globalchange studies.