Frames as codes and classifiers

模拟信号的冗余线性表示的数学描述被称为框架理论，它是许多应用数学领域开放问题的核心，如近似理论和数值以及统计分析。简而言之，一个坐标系是一个（过）完备的向量集合，它给出稳定的展开式。以前的工作已经测量了框架的误差校正能力与手段推导自几何或泛函解析公式。然而，为了信号通信和统计分析的目的，信息论应该为优化设计提供指导原则。该项目的目标是通过结合框架和信息理论来缩小这一差距。将信息论的最优性标准应用于框架设计，预计将对所有涉及在存在噪声的情况下传输或解释模拟信号的任务产生重大影响。研究者试图为两个特定的应用领域描述和构建线性信号编码和解码图：寻找具有相关编码和解码图的帧，以实现(1)数字流媒体的纠错和(2)噪声显微镜图像中的信号分类的信息论优化。数字信号传输已经彻底改变了我们的日常生活，从移动电话和互联网语音协议电话到高清电视和其他流媒体。数字化的使用有助于抑制模拟设备的典型失真，并通过合并冗余（即重复信息）实现看似无故障的通信。然而，有时错误抑制的数字特性会导致一些伪影，这些伪影并不像我们从模拟传输中回想起来的优雅退化。造成这种缺陷的原因是通用数字通信协议不能很好地适应编码模拟信号的特性。研究人员的目标是恢复通过互联网发送的流媒体或不可靠环境中的无线通信的优雅退化的可能性。模拟-数字编码的优化设计对模拟信号的解码具有类似的含义，例如从生物学中的显微镜图像中提取信息。显微镜图像中细胞成分的可靠监测和分类对于测量细胞的生理反应（如细胞周期的阶段）和检测由于疾病而发生的异常行为至关重要。对细胞行为进行分类与高通量测定的药物评估相关，包括癌细胞生长抑制剂的开发。

Bodmann0807399 The mathematical description of redundant linearrepresentations of analog signals is called frame theory, and itis central to open problems in many fields of applied mathematicssuch as approximation theory and numerical as well as statisticalanalysis. In short, a frame is an (over)complete set of vectorsthat gives stable expansions. Previous works have measured theerror correction capabilities of frames with means derived fromgeometric or functional analytic formulations. However, forpurposes of signal communication and statistical analysis,information theory should provide the guiding principles foroptimal design. The goal of the project is to close this gap bycombining frame and information theory. Applyinginformation-theoretic optimality criteria to frame design isexpected to have a significant impact on all tasks that involveanalog signals being transmitted or interpreted in the presenceof noise. The investigator seeks to characterize and constructlinear signal encoding and decoding maps for two specific areasof applications: Find frames with associated encoding anddecoding maps that achieve the information-theoretic optimum for(1) error correction of digital streaming media and (2)classification of signals in noisy microscopy images. Digital signal transmissions have revolutionized our dailylives, from cellular phones and Voice-over-Internet-Protocoltelephony to high-definition television and other streamingmedia. The use of digitization helps suppress distortionstypical for analog devices and allows seemingly faultlesscommunication by incorporating redundancy, that is, repetetiveinformation. However, at times the digital nature of errorsuppression leads to artifacts that do not resemble the gracefuldegradation we recall from analog transmissions. The reason forthis deficiency is that the general-purpose digital communicationprotocols are not well-adapted to the properties of the encodedanalog signal. The investigator aims to restore the possibilityof graceful degradation to streaming media sent across theinternet, or to wireless communications in unreliableenvironments. The optimal design of analog-digital encoding hassimilar implications for the decoding of analog signals, such asthe extraction of information from microscopy images in biology. The reliable monitoring and classification of cell constituentsin microscopy images is essential for measuring physiologicresponses in cells (such as the stages in the cell cycle) and fordetecting anomalous behavior that occurs as a result of disease. Classifying cell behavior is relevant for the evaluation of drugswith high-throughput assays, including the development ofcancer-cell growth inhibitors.