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

在我们这个以信息为基础的社会中，信息的准确传输在许多应用领域都是至关重要的。但信息可能会在许多方面被扭曲，到达时往往会删除部分信息。在这种情况下，大多数基于帧的恢复消息的方法的目标要么是快速重建消息(但允许近似误差)，要么是完美重建(但通常具有昂贵的计算成本)。在这个项目中，调查员的目标是快速准确地从报文中的已知或未知位置恢复丢失的数据。这在信息技术、生物技术和通信中具有直接应用，特别是在多路传输中，其中共享昂贵的信道，并且多路传输的数据用于增强编码安全性。学生将在这个项目的过程中接受培训。主要研究人员和他的学生致力于关于使用帧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恢复信号的问题。研究内容包括理论分析和新算法。主要目的是描述和识别当擦除发生在可能未知的位置时确保对几乎所有输入信号的擦除进行快速和完美恢复的编码帧。(Ii)开发用于信道共享应用中的最佳超帧和擦除恢复的理论和算法。多路传输将多个消息组合成单个消息，然后必须对其进行解码--也就是说，必须提取其单独的组成部分消息。进行多路传输通常是为了共享昂贵的资源，例如通道，或者增强组件消息的安全性。调查者研究了超帧的条件，该条件允许从擦除中完美地恢复，同时在不同分量消息的接收者之间保持信息安全。(Iii)研究频谱最优帧。这里的主要目的是刻画谱k-一致或接近k-一致的框架。此外，研究人员继续研究与框架膨胀理论相关的问题及其与相似性问题的联系。

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.