The investigator develops a mathematical framework together with fast computational schemes for nonlinear signal representations that are "optimally" compatible with observed data (e.g., a set of signals, images, or videos). The data can be finite or infinite and the models can live in finite- or infinite-dimensional Hilbert or Banach spaces. The framework can be viewed as an extension of the standard least squares to a nonlinear setting. The investigator uses this framework to unify, extend, and complement some of the techniques used in compressed sampling theory, the generalized principal components analysis, and the dictionary design problem. He applies the results to several problems in engineering and biomedicine including data classification and segmentation (e.g., face recognition, brain morphology, DNA sequence comparison, movement tracking), and signal modeling (e.g., for signals with finite rate of innovation).
Many of the new technologies use digital signal processing techniques in their functioning, e.g., telephone and computer networks, medical imaging devices, audio equipment, etc. These technologies, which are producing greater and greater volumes of data, gave rise to new nonlinear techniques and paradigms for acquiring, processing, analyzing, and transmitting the data. In order to take advantage of these novel techniques, a thorough understanding of the structure of the underlying signals and data is fundamental. This project develops techniques for learning the nonlinear data structure from a set of observations. It has direct applications in data mining, data classification, segmentation, and tracking moving objects in video sequences. It also has direct applications in data compression, noise removal, and data transmission.
