Non-uniform sampling and reconstruction:Theory and algorithms

Aldroubi0103104 研究者和他的同事们开发了一个数学框架和快速计算方案，用于从噪声中重建函数，信号或图像，非常大的采样数据集，通过非理想的采集设备在非均匀网格上采集。 非均匀采样和reconstructionis处理的上下文中的移位不变的子空间，Besovspaces，在任意尺寸的问题。 本文发展了由加权平均得到的样本的理论，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，根据采样密度、平均泛函和噪声统计量，导出了重建信号与原始信号之间误差的界。 该理论框架和计算方案的发展需要一套新的技术和思想，涉及到小波理论、框架理论、泛函分析和调和分析等多个领域。 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，都会遇到从一组非均匀样本中重建信号或图像的问题。 例如，数据包在通过互联网或卫星传输过程中的丢失可以看作是一个非均匀采样和重构问题。 在地球物理勘探中，地球磁场的测量是由空中快速移动的采集设备和分散的固定设备组合而成的，这导致了高度不均匀的采样模式和巨大的数据集。 目标是重建磁场并利用它来揭示地质特征。 事实上，现代数字信号或图像处理总是使用原始模拟信号或图像的采样版本。 然而，这些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，并被噪声破坏。 然后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）量化条件下，它是可能的，以重建一个信号准确地从不同的集合的非均匀平均样本; 2）使用这些分析结果，以发展明确的，计算效率高的重建计划;和3）分析的算法在不利条件下的性能，或当数据是不完整的或被噪声破坏。 理论和算法的发展，在严格和现实的情况下表现良好，将有助于分析，处理和管理非常大的数据集数字获得的新的采集方式，并发送或接收的通信网络，如互联网，手机，和其他分布式通信系统。

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.