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

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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.