Accurate digital representation and recovery for redundant frames

冗余框架表示在信号处理应用中起着核心作用。 通过使用过完备系统来表示信号，帧提供了增加的设计灵活性，从而能够在许多设置中提供针对噪声和数据丢失的鲁棒性。 研究人员研究了数字表示冗余有限帧扩展的数学，重点是两个关键步骤：系数量化和信号重建。 量化（编码步骤）的工作集中在类的Sigma-Delta算法，并研究严格的近似误差界，双帧的方法，以提高性能，并设计新的算法，多描述编码和正交频分复用。 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，并且在实践中是期望的，因为它们可以使用非常粗略的（例如一位）标量量化器来鲁棒地实现。 在信号恢复（解码步骤）的工作中，研究者研究了在脉冲编码调制和Sigma-Delta量化的设置下用于模数转换的新的非线性一致重构算法。 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。 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鲁棒的数据处理算法的新数学技术的需求。 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的场景;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擦除信道上的通信，其中物理约束或其他干扰导致传输信息的丢失;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.