Non-linear signal representations: theory, algorithms and applications

研究者开发了一个数学框架，以及与观测数据（例如，一组信号，图像或视频）“最佳”兼容的非线性信号表示的快速计算方案。数据可以是有限或无限的，模型可以存在于有限或无限维的希尔伯特或巴拿赫空间中。该框架可以看作是标准最小二乘对非线性设置的扩展。研究者使用这个框架来统一、扩展和补充压缩抽样理论、广义主成分分析和字典设计问题中使用的一些技术。他将其成果应用于工程和生物医学中的几个问题，包括数据分类和分割（例如，面部识别，脑形态学，DNA序列比较，运动跟踪）和信号建模（例如，具有创新的有限信号）。许多新技术在其功能中使用数字信号处理技术，例如电话和计算机网络、医疗成像设备、音频设备等。这些技术产生的数据量越来越大，产生了新的非线性技术和范式，用于获取、处理、分析和传输数据。为了充分利用这些新技术，对潜在信号和数据的结构有一个彻底的了解是至关重要的。该项目开发了从一组观测数据中学习非线性数据结构的技术。它直接应用于数据挖掘、数据分类、分割和跟踪视频序列中的运动物体。它在数据压缩、噪声去除和数据传输方面也有直接的应用。

AldroubiDMS-0807464 The investigator develops a mathematical framework togetherwith fast computational schemes for nonlinear signalrepresentations that are "optimally" compatible with observeddata (e.g., a set of signals, images, or videos). The data canbe finite or infinite and the models can live in finite- orinfinite-dimensional Hilbert or Banach spaces. The framework canbe viewed as an extension of the standard least squares to anonlinear setting. The investigator uses this framework tounify, extend, and complement some of the techniques used incompressed sampling theory, the generalized principal componentsanalysis, and the dictionary design problem. He applies theresults to several problems in engineering and biomedicineincluding data classification and segmentation (e.g., facerecognition, brain morphology, DNA sequence comparison, movementtracking), and signal modeling (e.g., for signals with finiterate of innovation). Many of the new technologies use digital signal processingtechniques in their functioning, e.g., telephone and computernetworks, medical imaging devices, audio equipment, etc. Thesetechnologies, which are producing greater and greater volumes ofdata, gave rise to new nonlinear techniques and paradigms foracquiring, processing, analyzing, and transmitting the data. Inorder to take advantage of these novel techniques, a thoroughunderstanding of the structure of the underlying signals and datais fundamental. This project develops techniques for learningthe nonlinear data structure from a set of observations. It hasdirect applications in data mining, data classification,segmentation, and tracking moving objects in video sequences. Italso has direct applications in data compression, noise removal,and data transmission.