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

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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.