Frames as codes and classifiers

Bodmann0807399对模拟信号冗余线性化表示的数学描述称为框架理论，它是应用数学的许多领域，如逼近理论、数值分析和统计分析中的公开问题的核心。简而言之，框架是给出稳定展开的(超过)向量的完整集合。以前的工作是用几何或泛函解析公式得出的方法来测量帧的纠错能力。然而，为了信号交流和统计分析的目的，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将框架和信息论相结合来缩小这一差距。将信息论最优标准应用于框架设计，预计将对涉及在有噪声的情况下传输或解释模拟信号的所有任务产生重大影响。研究人员试图为两个特定的应用领域刻画和构建线性信号编码和解码映射：找到帧和相关的编码和解码映射，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最优。数字信号传输已经给我们的日常生活带来了革命性的变化，从手机和互联网语音协议电话到高清晰度电视和其他流媒体。数字化的使用有助于抑制模拟设备的典型失真，并通过加入冗余，即重复信息，实现表面上无故障的通信。然而，有时错误抑制的数字本质会导致不像我们回忆起的模拟传输中优雅的退化那样的人工制品。这一缺陷的原因是通用数字通信协议不能很好地适应编码的模拟信号的特性。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优雅降级的可能性。模拟-数字编码的优化设计对模拟信号的解码具有重要意义，例如从生物学的显微图像中提取信息。对显微图像中的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(如细胞周期的各个阶段)和检测疾病引起的异常行为至关重要。对细胞行为进行分类对于高通量药物的评估是相关的，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.