Non-uniform sampling and reconstruction:Theory and algorithms

Aldroubi0103104 研究人员和他的同事开发了一个数学框架和快速计算方案，用于从非理想采集设备在非均匀网格上采集的噪声很大的采样数据集中重建函数、信号或图像。 非均匀采样和重建问题在平移不变子空间、贝索夫空间和任意维度的背景下进行处理。 该理论是针对从加权平均值获得样本的情况而发展的。建立了精确重建的密度条件。 当数据有噪声、不完整或不满足精确重建所需的假设时，可以根据采样密度、平均泛函和噪声统计推导出重建信号和原始信号之间的误差界限。 数学框架和计算方案的发展需要一套新的技术和思想，涉及小波理论、框架论、泛函分析和调和分析等多个数学领域。 该项目的动机是数据传输、地球物理勘探、天文学、光谱学和生物医学成像中出现的问题。 在信号或图像处理的许多应用中都会遇到从一组不均匀样本重建信号或图像的问题。 例如，通过互联网或卫星传输期间数据包的丢失可以被视为非均匀采样和重建问题。 在地球物理勘探中，地球磁场是通过空中快速移动的采集设备以及分散的固定设备的组合来测量的，从而导致采样模式高度不均匀和庞大的数据集。 目标是重建磁场并用它来揭示地质特征。 事实上，现代信号或图像的数字数据处理总是使用原始模拟信号或图像的采样版本。 然而，采样设备从来都不是理想的，并且收集的数据由平均样本组成。 此外，这些数据通常非常大、不完整并且被噪声破坏。 接下来的问题是是否以及如何从数据中恢复原始信号。 因此，研究者的目标是：1）量化可以从不同的非均匀平均样本集精确重建信号的条件； 2）使用这些分析结果来开发明确的、计算高效的重建方案； 3）分析不利条件下或数据不完整或被噪声破坏时算法的性能。 在严格和现实的情况下表现良好的理论和算法的发展将有助于分析、处理和管理通过新的采集方式数字化获得的、通过互联网、蜂窝电话和其他分布式通信系统等通信网络传输或接收的超大型数据集。

Aldroubi0103104 The investigator and his colleagues develop a mathematicalframework and fast computational schemes for the reconstructionof functions, signals or images from noisy, very large sampleddata sets, acquired on nonuniform grids, by nonideal acquisitiondevices. The problem of nonuniform sampling and reconstructionis treated in the context of shift-invariant subspaces, Besovspaces, and in arbitrary dimensions. The theory is developed forthe case when the samples are obtained from weighted averages.Density conditions for exact reconstruction are established. Whenthe data are noisy, incomplete, or when the assumptions neededfor exact reconstruction are not satisfied, bounds on the errorbetween the reconstructed and original signal are derived interms of the sampling densities, the averaging functionals, andthe noise statistics. The development of the mathematicalframework and the computational schemes requires a new set oftechniques and ideas, and involves several areas of mathematicsincluding wavelet theory, frame theory, functional analysis, andharmonic analysis. The project is motivated by problems arising in datatransmission, geophysical exploration, astronomy, spectroscopy,and biomedical imaging. The problem of reconstructing a signal oran image from a set of nonuniform samples is encountered in manyapplications of signal or image processing. For example, the lossof data packets during transmission through internet or fromsatellites can be viewed as a nonuniform sampling andreconstruction problem. In geophysical exploration, the earth'smagnetic field is measured by a combination of airborn, fastmoving acquisition devices, as well as scattered stationarydevices resulting in highly nonuniform sampling patterns, and ahuge data set. The goal is to reconstruct the magnetic field anduse it to reveal geological features. In fact, modern digitaldata processing of signals or images always uses a sampledversion of the original analog signals or images. However, thesampling devices are never ideal, and the collected data consistof average samples. Moreover these data are often very large,incomplete, and corrupted by noise. The question then ariseswhether and how the original signal can be recovered from thedata. Therefore the investigator aims to 1) quantify theconditions under which it is possible to reconstruct a signalexactly from different sets of nonuniform average-samples; 2) usethese analytical results to develop explicit, and computationallyefficient reconstruction schemes; and 3) analyze the performanceof the algorithms under adverse conditions, or when the data areincomplete or corrupted by noise. The development of a theory andalgorithms that perform well under stringent and realisticsituations will help the analysis, processing and management ofvery large data sets obtained digitally by new acquisitionmodalities, and transmitted or received by communication networkssuch as the internet, cellular phones, and other distributedcommunication systems.