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Frame Compatibility: Discrete Versus Continuous Redundant Expansions, Strategies for Narrowing the Digital-Analog Gap

框架兼容性：离散扩展与连续冗余扩展、缩小数模差距的策略

基本信息

批准号：
1715735
负责人：
Bernhard Bodmann
金额：
$ 25.7万
依托单位：
University of Houston
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1715735&HistoricalAwards=false
关键词：
Frame Compatibility Discrete Versus Continuous

项目摘要

The theory of compressed sensing promises to revolutionize remote sensing such as radar, biomedical imaging, and perhaps even digital photography. The main insight from this theory is that a compressible signal can be acquired with much less effort than a signal with a high information content. However, these results are commonly based on mathematical models for signals that are already digitized and for sensors that measure randomly, which makes them somewhat disconnected from realistic physical signals and apparatuses. This work explores recent trends in narrowing the gap between theory and practice. Instead of digital signals, models for analog signals are used to define compressibility. The signal space includes the possibility of continuous changes without producing artifacts in the signal recovery procedure. This idea will be applied to radar, X-ray crystallography, and other sensing systems. The work is also anticipated to have application to neural networks that form the basis for modern machine learning algorithms. Although the application to machine learning is entirely concerned with digital data, the use of continuous models ensures that encoded information can be retrieved accurately. Redundant, stable expansions with frames have become central to many applications of mathematics in data science, signal acquisition, and communications, from remote sensing to packet-based, wireless, fiber optical, or quantum communications and recently in compressed sensing and super-resolution. Despite the successes of the frame-based expansion and acquisition of signals, there is often a mismatch between the stylized mathematical signal and measurement models that are assumed and the corresponding physical models in the analog domain. For example, signal acquisition is typically described by specific linear functionals, not randomly chosen, unstructured ones. This research project addresses the need to improve compatibility between continuous and discrete representation spaces on a fundamental level. A typical model for analog signals is given by infinite-dimensional Hilbert spaces with a reproducing kernel and an associated expansion with respect to a continuous, highly coherent family of vectors. A natural measure of sparsity of a signal is in this setting the minimal number of kernel functions needed in its expansion. Signal acquisition is usually based on sampling from a group-invariant, discrete family of functionals. The expected outcomes of the project include: (1) accurate recovery for signals that are sparsely synthesized in a finite- or infinite-dimensional reproducing kernel space and measured with physically relevant sensing models, using a sparsity-inducing norm that is stable with respect to continuous deformations; (2) phase retrieval, signal recovery based on magnitudes of frame coefficients, in reproducing kernel Hilbert spaces such as multivariate Paley-Wiener spaces, which will be done using sparsity to demonstrate injectivity of measurements, stability, and feasibility of recovery algorithms in a general class of kernel spaces; and (3) a version of Mallat's scattering transform in a redundant representation with approximate invertibility based on phase retrieval and sparsity. The scattering transform extracts nonlinear features from data that are powerful descriptors in classification problems. It is designed from a viewpoint of desirable properties in the analog domain, but its application is mostly to digitized data of limited size, for which the claims need to be properly adapted. The investigators will use phase retrieval and sparsity to demonstrate the approximate invertibility of the transform, which is needed to verify the faithful encoding of data.

压缩感知理论有望彻底改变雷达、生物医学成像、甚至数字摄影等遥感技术。该理论的主要见解是，与具有高信息内容的信号相比，获取可压缩信号要花费更少的精力。然而，这些结果通常基于已经数字化的信号和随机测量的传感器的数学模型，这使得它们在某种程度上与现实的物理信号和设备脱节。这项工作探讨了缩小理论与实践之间差距的最新趋势。使用模拟信号模型而不是数字信号来定义可压缩性。信号空间包括连续变化的可能性，而不会在信号恢复过程中产生伪影。这一想法将应用于雷达、X 射线晶体学和其他传感系统。这项工作还有望应用于构成现代机器学习算法基础的神经网络。尽管机器学习的应用完全涉及数字数据，但连续模型的使用确保了可以准确地检索编码信息。冗余、稳定的帧扩展已成为数据科学、信号采集和通信领域许多数学应用的核心，从遥感到基于数据包、无线、光纤或量子通信，以及最近的压缩传感和超分辨率。尽管基于帧的信号扩展和采集取得了成功，但假设的程式化数学信号和测量模型与模拟域中相应的物理模型之间经常存在不匹配。例如，信号采集通常由特定的线性泛函描述，而不是随机选择的非结构化泛函。该研究项目解决了从根本上提高连续和离散表示空间之间兼容性的需要。模拟信号的典型模型由无限维希尔伯特空间给出，该空间具有再现核以及关于连续的、高度相干的向量族的相关扩展。在这种情况下，信号稀疏性的自然衡量标准是其扩展所需的最小核函数数量。信号采集通常基于对群不变、离散泛函族进行采样。该项目的预期成果包括：（1）使用对连续变形稳定的稀疏诱导范数，准确恢复在有限或无限维再现核空间中稀疏合成并通过物理相关传感模型测量的信号； (2) 相位检索，基于帧系数大小的信号恢复，在再现内核希尔伯特空间（例如多元佩利-维纳空间）中，这将使用稀疏性来完成，以证明测量的单射性、稳定性以及一般类内核空间中恢复算法的可行性； (3) Mallat 散射变换的冗余表示形式，具有基于相位检索和稀疏性的近似可逆性。散射变换从数据中提取非线性特征，这些特征是分类问题中强大的描述符。它是从模拟域中所需特性的角度出发设计的，但其应用主要是有限大小的数字化数据，因此需要适当调整权利要求。研究人员将使用相位检索和稀疏性来证明变换的近似可逆性，这是验证数据的忠实编码所必需的。

项目成果

期刊论文数量（4）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Binary Parseval frames from group orbits

来自群轨道的二元帕塞瓦尔框架

DOI：
10.1016/j.laa.2018.07.016
发表时间：
2018
期刊：
Linear Algebra and its Applications
影响因子：
1.1
作者：
Mendez, Robert P.;Bodmann, Bernhard G.;Baker, Zachery J.;Bullock, Micah G.;McLaney, Jacob E.
通讯作者：
McLaney, Jacob E.

On the minimum of the mean-squared error in 2-means clustering

关于2均值聚类中均方误差的最小值