Frames as codes and classifiers

博德曼0807399 对模拟信号的冗余线性表示的数学描述称为框架理论，它是应用数学的许多领域，如近似理论、数值分析和数学分析中的开放问题的核心。 简而言之，一个框架是一个（上）完备的向量集，它给出了稳定的展开式。 以前的作品测量theerror correction capabilities的框架与手段来自几何或功能的解析公式。 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。 应用信息理论的最优性标准的帧设计预计将有一个显着的影响，所有的任务，涉及模拟信号被传输或解释存在的噪声。 研究人员试图表征和构建线性信号编码和解码映射，用于两个特定的应用领域：找到具有相关编码和解码映射的帧，这些映射实现信息理论最优，用于（1）数字流媒体的纠错和（2）噪声显微图像中的信号分类。 数字信号传输已经彻底改变了我们的日常生活，从蜂窝电话和互联网语音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，并通过引入冗余，即重复信息，允许看似无故障的通信。 然而，有时误差抑制的数字特性会导致伪影，这些伪影与我们从模拟传输中回忆到的令人不安的降级不相似。 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降级的可能性。 模拟-数字编码的最佳设计对模拟信号的解码具有类似的意义，例如从生物学中的显微图像中提取信息。在显微镜图像中对细胞成分进行可靠的监测和分类对于测量细胞的生理反应（如细胞周期中的阶段）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药物评价有关，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.