Numerical Methods for Digital Signal Reconstruction

研究者开发有效的数值方法，公式维信号重建使用工具从谐波分析，如局部三角变换和小波。所考虑的重建问题范围从离散数据的非平稳信号的逼近到数字图像序列中缺失数据的恢复。这些重建问题的成功解决需要一个关于信号的优先级信息，否则这些问题是不适定的。通常这种先验知识可以从过程的物理特性中得到，例如信号的“局部频率行为”。局部三角变换和小波变换已成为数字信号分析、压缩和去噪的有力方法。研究者为这些概念在信号和图像重建中的应用推导了一个数学框架（超出了去噪的良好穿越区域）。这些构造方法的设计伴随着多层次正则化技术的发展，以便为结合不同类型的信号先验知识提供灵活的框架。研究人员将新的数值方法应用于来自分散数据的地球物理信号的近似和接收编码比特流的纠错。该项目的动机是在勘探、地球物理、医学成像和电信等不同领域出现的一些多维信号重建问题。为了优化压缩、传输或分析不完整或失真的信号，有必要首先从可用数据中重建原始信号。这种重建过程包括恢复丢失的数据以及从分散的噪声测量中恢复信号。这里开发的数学框架允许设计灵活和鲁棒的算法来重建多维信号。由于数据量巨大，并且可能需要实时处理，因此重建算法也必须快速有效。推导出的数值方法对于数据传输中的可靠误差校正、地球物理信号的有效重建以及其他应用具有重要意义。

The investigator develops efficient numerical methods formulti-dimensional signal reconstruction using tools from harmonicanalysis such as local trigonometric transforms and wavelets.The reconstruction problems that are considered range fromapproximation of nonstationary signals from scattered data torecovery of missing data in digital image sequences. Successfulsolution of these reconstruction problems requires a prioriinformation about the signal, otherwise these problems areill-posed. Often this a priori knowledge can be derived fromphysical properties of the process, for instance in terms of the"local frequency behavior" of the signal. Local trigonometrictransforms and wavelets have become powerful methods foranalysis, compression and denoising of digital signals. Theinvestigator derives a mathematical framework for the applicationof these concepts to signal and image reconstruction (beyond thewell-traversed area of denoising). The design of thesereconstruction methods is accompanied by the development ofmulti-level regularization techniques in order to provide aflexible framework for the incorporation of different kinds of apriori knowledge about the signal. The investigator applies thenew numerical methods to the approximation of geophysical signalsfrom scattered data and to error correction in received coded bitstreams. This project is motivated by a number of multi-dimensionalsignal reconstruction problems arising in areas as diverse asexploration geophysics, medical imaging, and telecommunication.In order to optimally compress, transmit or analyze an incompleteor distorted signal, it is necessary to first reconstruct theoriginal signal from the available data. This reconstructionprocess includes the restoration of lost data as well as therecovery of a signal from scattered noisy measurements. Themathematical framework being developed here allows the design offlexible and robust algorithms for the reconstruction ofmulti-dimensional signals. Due to huge amount of data andpossibly required real-time processing the reconstructionalgorithms must also be fast and efficient. The derivednumerical methods are important for reliable error correction indata transmission, for efficient reconstruction of geophysicalsignals, and for other applications.