Accurate digital representation and recovery for redundant frames

冗余帧表示在信号处理应用中起着核心作用。通过使用过完备的系统来表示信号，帧提供了更高的设计灵活性，从而能够在许多情况下提供对噪声和数据丢失的稳健性。研究人员研究了用数字表示冗余有限帧展开的数学方法，重点研究了两个关键步骤：系数量化和信号重构。量化(编码步骤)的工作集中在Sigma-Delta算法的类，并研究了严格的逼近误差界，提高性能的双帧方法，以及多描述编码和正交频分复用的新算法的设计。Sigma-Delta算法非常适合于利用帧系数的冗余集合中固有的相关性，并且在实践中是合乎需要的，因为它们可以使用非常粗略的例如一位标量量化器来稳健地实现。在信号恢复部分(译码部分)，研究了在脉冲编码调制和Sigma-Delta量化的情况下，模数转换的非线性一致性重构新算法。研究人员还研究了如何将噪声整形和一致性重建方法扩展到基于融合框架的分布式处理应用中。数字数据在现代技术中普遍存在。该项目解决了如何在噪声环境中准确处理、传输和恢复数字信号的一般问题。许多数字数据集的巨大规模推动了对能够提供高效和健壮的数据处理算法的新的数学技术的需求。融合帧的量化工作适用于在远程环境中部署密集的低分辨率传感器集合的情况；该项目研究通过传感器网络进行最佳通信和提取数字信息的方法。有限帧扩展的量化工作适用于物理限制或其他干扰导致传输信息丢失的擦除信道上的通信；该项目研究减轻数据擦除影响的程序。

Redundant frame representations play a central role in signal processing applications. By using overcomplete systems to represent signals, frames offer increased design flexibility and are thereby able to provide robustness against noise and data loss in many settings. The investigator studies the mathematics of digitally representing redundant finite frame expansions, with an emphasis on two key steps: coefficient quantization and signal reconstruction. The work on quantization (the encoding step) focuses on the class of Sigma-Delta algorithms and studies rigorous approximation error bounds, dual frame methods for boosting performance, and the design of new algorithms for multiple description coding and orthogonal frequency division multiplexing. Sigma-Delta algorithms are well suited for utilizing the correlations inherent in redundant collections of frame coefficients, and are desirable in practice since they can be robustly implemented using very coarse, for example one bit, scalar quantizers. In the work on signal recovery (the decoding step), the investigator studies new nonlinear consistent reconstruction algorithms for analog-to-digital conversion in the settings of Pulse Code Modulation and Sigma-Delta quantization. The investigator also studies how to extend noise-shaping and consistent reconstruction methods to distributed processing applications based on fusion frames.Digital data is ubiquitous in modern technology. The project addresses the general problem of how to accurately process, transmit, and recover digital signals in noisy environments. The massive size of many digital data sets drives a need for new mathematical techniques that are capable of providing efficient and robust data processing algorithms. The work on quantization of fusion frames applies to scenarios where a dense collection of low resolution sensors is deployed in a remote environment; the project studies methods for optimally communicating and extracting digital information through the sensor network. The work on quantization of finite frame expansions is applicable to communication over erasure channels where physical constraints or other interference result in a loss of transmitted information; the project studies procedures for mitigating the effect of data erasures.