Nonlinear sampling theory for signals with finite rate of innovation

SUNDMS-1109063从理论上和实际实现上发展了一种新的有限新息率信号的非线性采样和恢复的数学框架。研究人员致力于几个基本问题，涉及从有噪声的采样数据中恢复信号，探索存在大量噪声的有效算法方法，以及将新的采样方法应用于工程问题。这个跨学科的项目是基于这样的观察，即许多工程问题中的信号具有有限的新息率，并且可以用稀疏表示的信号来逼近，并且采样过程具有很强的邻域相关性。抽样理论是数学科学和工程科学中最基本、最引人入胜的课题之一。在全球定位系统(GPS)、通信中的超宽带(UWB)测距系统和医疗诊断中的质谱学等工程应用中，需要获得含有噪声的采样数据，实时恢复是首选的，非常准确的恢复对于有意义的证明至关重要。标准傅立叶方法和常规采样技术不适用于这类问题。挑战在于对快速、准确和强健恢复的要求。研究人员将微积分应用于无限矩阵，并引入了新的压缩技术来解决基本采样问题。该项目的成功在数学上和技术上都可能是重要的，并对信息技术和生物技术等战略领域产生潜在影响。

SunDMS-1109063 The investigator develops a new mathematical framework for nonlinear sampling and recovery of signals with finite rate of innovation from theoretical viewpoint and in practical realization. The investigator works on several fundamental problems concerning the recovery of a signal from its noisy sampled data, the exploration of efficient algorithmic methods in the presence of substantial noise, and the application of the novel sampling methodology to engineering problems. This interdisciplinary project is based on the observations that signals in many engineering problems have finite rate of innovation and could be approximated by signals with sparse representations, and that the sampling process has strong neighbor dependency. Sampling theory is one of the most basic and fascinating topics in mathematical science and in engineering sciences. In engineering applications, such as Global Positioning Systems (GPS), Ultra-wideband (UWB) ranging systems in communication, and mass spectrometry in medical diagnosis, noisy sampled data are obtained, real-time recovery is preferred, and very accurate restoration is crucial for meaningful justification. Standard Fourier approaches and conventional sampling techniques are inapplicable in such problems. The challenge resides in the requirement of rapid, accurate, and robust recovery. The investigator applies calculus for infinite matrices and introduces novel compressive techniques to tackle fundamental sampling problems. Success in the project could be both mathematically fundamental and technologically important, and has potential impact in the strategic areas of information technology and biotechnology.