Nonlinear sampling theory for signals with finite rate of innovation

SunDMS-1109063 研究者从理论上和实际实现上提出了一个新的数学框架，用于非线性采样和恢复信号的有限新息率。 调查工作的几个基本问题，有关恢复的信号从其嘈杂的采样数据，有效的算法方法在大量噪声的存在下的探索，和应用新的采样方法的工程问题。 这个跨学科的项目是基于观察，在许多工程问题中的信号具有有限的创新率，可以近似的信号与稀疏表示，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本、最令人感兴趣的课题之一。 在工程应用中，例如全球定位系统（GPS）、通信中的超宽带（UWB）测距系统和医疗诊断中的质谱分析，获得噪声采样数据，优选实时恢复，并且非常准确的恢复对于有意义的调整是至关重要的。 标准的傅立叶方法和传统的采样技术是不适用于这样的问题。 挑战在于需要快速、准确和强大的恢复。 研究人员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。 该项目的成功可能具有数学基础和技术重要性，并可能对信息技术和生物技术的战略领域产生影响。

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.