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

信号和图像分析算法在科学和技术的各种应用中发挥着重要作用。例如，生物医学科学家通常使用成像和其他传感设备来表征正常与患病的生理学。信号和图像分析还用于许多日常生活应用，包括面部识别，机器人（例如对象识别，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，并且许多工具不一定非常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），该算法不仅使用图像（信号）的强度，而且还使用它们各自的位置进行比较。本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。该方法在很大程度上依赖于最佳运输和相关的数学技术。该研究包括可逆非线性信号变换的发展，其计算的有效算法，并测试其在一些信号建模和歧视任务（癌症检测，患病细胞表型的表征，信号数据库中变化的可视化，低分辨率图像的识别等）中的应用。 研究的主要重点是拉格朗日变换的发展，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.