Optimal and Structured Frames with Applications

HanDMS-1106934 该项目有三个主要目标：（i）发展用于擦除和稀疏信号重建的最佳对偶（和融合）框架的理论和算法。(ii)发展分形测度的框架谱理论，可以为集中在分形区域的函数提供更快的重建（例如，与经典的香农采样方法相比）。(iii)研究群表示框架的Feichtinger框架猜想与相关问题（如Kadison-Singer问题）之间的联系。 在当今的数字世界中，各种应用对高效数据处理的要求越来越高。 主要研究谐波和功能分析中的基本问题，适用于无线通信，雷达，地震传感，图像处理，生物医学图像中功能的可靠分类以及信号数字传输中数据丢失的补偿算法等领域。 删除和稀疏的最优对偶帧设计问题直接关系到信号的冗余表示和变换的基本问题。 调查导致适用的工具，在工程和信息应用中的问题，并提供理论分析和算法建设的计算效率的框架。

HanDMS-1106934 This project has three main goals: (i) Develop the theory and algorithms for optimal dual (and fusion) frames for erasures and for sparse signal reconstruction. (ii) Develop the theory of the frame spectrum for fractal measures that can provide much faster reconstruction (for example, compared to the classical Shannon sampling method) for functions that are concentrated on fractal-like regions. (iii) Investigate the connections between the Feichtinger frame conjecture for group representation frames and related problems such as the Kadison-Singer problem. In today's digital world efficient data processing has become more demanding in all kinds of applications. The principal investigator studies fundamental questions in harmonic and functional analysis that are applicable in areas such as wireless communications, radar, seismic sensing, image processing, reliable classification of features in biomedical images, and algorithms for the compensation of data loss in digital transmission of signals. The optimal dual frame design problems for erasures and sparsity are directly related to basic issues in redundant representation and transformation of signals. The investigation leads to applicable tools for problems in engineering and information applications, and provides theoretical analysis and algorithmic construction for computationally efficient frames.