Frame-Based Kernel Analysis and Algorithms for Fast Recovery of Erasures and Multiplexing

信息的准确传递在我们这个信息化社会的许多应用领域都是至关重要的。但是，信息可以在很多方面被扭曲，通常会伴随着部分信息的删除。在这种情况下，大多数基于帧的消息恢复方法要么旨在快速重建消息（但允许近似误差），要么旨在完美重建消息（但通常具有昂贵的计算成本）。在这个项目中，研究者的目标是与从消息中已知或未知位置快速准确地恢复丢失的数据有关的问题。这在信息技术、生物技术和通信中有直接的应用，特别是在多路复用中，昂贵的信道被共享，多路复用的数据被用来增强编码的安全性。学生将在这个项目的过程中得到训练。首席研究员和他的学生们研究了关于使用帧来快速准确地恢复带有擦除的信号的三个基本问题：(i)开发新的方法来解决在已知或未知位置存在擦除时从帧系数中恢复信号的问题。研究包括理论分析和新的算法。主要目标是表征和识别编码帧，以确保当擦除发生在可能未知的位置时，几乎所有输入信号的擦除都能快速完美地恢复。（ii）在信道共享应用程序中发展最优超帧和擦除恢复的理论和算法。多路复用将多个消息合并为单个消息，然后必须对其进行解码——也就是说，必须提取其单独的组件消息。多路复用通常用于共享昂贵的资源（如通道）或增强组件消息的安全性。调查员研究了允许从擦除中完美恢复的超帧条件，同时在不同组件消息的接收器之间保持信息安全。（iii）研究频谱最优帧。这里的主要目的是表征光谱k均匀或近k均匀帧。此外，研究者继续研究与框架膨胀理论相关的问题及其与相似问题的联系。

Accurate transmission of messages is critical in many areas of application in our information-based society. But a message can be distorted in many ways, often arriving with erasures in parts of the message. In such cases, most frame-based methods to restore the message have been aimed at either fast reconstruction of the message (but allowing approximation errors) or perfect reconstruction (but usually with expensive computational cost). In this project the investigator targets problems related to both fast and accurate recovery of the lost data from either known or unknown locations in the message. This has direct applications in information technology, biotechnology, and communications, particularly in multiplexing where expensive channels are shared and the multiplexed data are used to enhance coding security. Students will be trained in the course of this project.The principal investigator and his students work on three fundamental questions concerning the use of frames for fast and accurate recovery of signals with erasures: (i) Develop new approaches to the problem of recovering signals from the frame coefficients when there are erasures at either known or unknown locations. The investigation includes both theoretical analysis and new algorithms. The main objective is to characterize and identify the encoding frames that ensure fast and perfect recovery of the erasures for almost all input signals when erasures occur at possibly unknown locations. (ii) Develop theory and algorithms for optimal superframes and erasure recovery in channel-sharing applications. Multiplexing combines several messages into a single one, which must then be decoded -- that is, its separate component messages must be extracted. Multiplexing usually is done to share an expensive resource, such as a channel, or to enhance the security of the component messages. The investigator studies conditions on superframes that allow perfect recovery from erasures while maintaining information security among the receivers of the different component messages. (iii) Study spectrally optimal frames. The chief aim here is to characterize spectrally k-uniform or near k-uniform frames. Additionally, the investigator continues to work on problems related to the theory of frame dilations and its connections to the Similarity Problem.