Nonlinear sampling theory for signals with finite rate of innovation

研究人员从理论角度和实际实现中开发了一种新的数学框架，用于具有有限创新率的信号的非线性采样和恢复。研究者的工作涉及几个基本问题，包括从噪声采样数据中恢复信号，在存在大量噪声的情况下探索有效的算法方法，以及将新的采样方法应用于工程问题。这个跨学科项目是基于许多工程问题中的信号具有有限的创新率，并且可以用具有稀疏表示的信号来近似，并且采样过程具有很强的邻居依赖性。抽样理论是数学科学和工程科学中最基础、最引人入胜的课题之一。在工程应用中，例如全球定位系统（GPS）、通信中的超宽带（UWB）测距系统以及医疗诊断中的质谱，都会获得有噪声的采样数据，因此首选实时恢复，并且非常精确的恢复对于有意义的论证至关重要。标准的傅里叶方法和传统的采样技术不适用于这类问题。挑战在于对快速、准确和稳健的恢复的要求。研究者将微积分应用于无限矩阵，并引入新的压缩技术来解决基本的采样问题。项目的成功在数学上是基本的，在技术上也是重要的，并对信息技术和生物技术的战略领域有潜在的影响。

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.