CIF: Small: Transport and other Lagrangian transforms for signal analysis and discrimination

信号和图像分析算法在科学技术中的各种应用中扮演着重要的角色。例如，生物医学科学家经常使用成像和其他感官设备来表征正常和疾病的生理学。信号和图像分析还用于许多日常生活应用，包括人脸识别、机器人(例如，物体识别、自动导航)和语音识别。目前为信号和图像分析开发的大多数数学工具(例如傅立叶小波变换)都是为了解决通信和相关学科中的问题而设计的，其中许多工具不一定很适合与信号检测和分类相关的现代问题。研究人员研究新的信号分析和合成算法(即变换)，这些算法不仅基于图像(信号)的强度，而且还基于图像(信号)的各自位置。该项目的目标是开发一个利用拉格朗日观点的信号和图像分析框架。这种方法在很大程度上依赖于最优运输和相关的数学技术。这项研究包括开发可逆的非线性信号变换，计算它们的有效算法，并测试它们在一些信号建模和识别任务中的应用(癌症检测、疾病细胞表型表征、信号数据库中变异的可视化、低分辨率图像识别等)。研究的重点是拉格朗日变换的发展，它在信号识别任务中具有具体的理论和实践优势。

Signal and image analysis algorithms play an important role in a variety of applications in science and technology. Biomedical scientists, for example, routinely use imaging and other sensory devices to characterize normal versus diseased physiology. Signal and image analysis is also used for numerous daily life applications including face identification, robotics (e.g. object identification, automated navigation), and voice recognition. Most of the mathematical tools currently developed for signal and image analysis (e.g. Fourier & wavelet transforms) were designed to address problems in communications and related disciplines, and many are not necessarily well suited for modern problems related to signal detection and classification. The investigators study new signal analysis and synthesis algorithms (i.e. transforms) derived based on comparing images (signals) using on not only their intensities, but also their respective locations.The goal of this project is to develop a framework for signal and image analysis that utilizes a `Lagrangian' point of view. The approach relies heavily on optimal transport and related mathematical techniques. The study includes the development of invertible nonlinear signal transforms, efficient algorithms for their computation, and testing their application in a number of signal modeling and discrimination tasks (cancer detection, characterization of diseased cell phenotypes, visualization of variations in signal databases, recognition from low resolution images, and others). The main emphasis of the study is the development of Lagrangian transforms that have concrete theoretical and practical advantages in signal discrimination tasks.