Non-linear signal representations: theory, algorithms and applications

AldroubiDMS-0807464 研究人员开发了一个数学框架，以及用于非线性信号表示的快速计算方案，这些非线性信号表示“最佳”与已验证数据兼容（例如，一组信号、图像或视频）。 数据可以是有限的或无限的，模型可以存在于有限维或无限维的希尔伯特或巴拿赫空间中。 该框架可以被看作是标准最小二乘非线性设置的扩展。 调查员使用这个框架tunify，扩展，并补充一些技术中使用的压缩抽样理论，广义principal componentsanalysis，和字典设计问题。 他将结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、大脑形态学、DNA序列比较、运动跟踪）和信号建模（例如，对于具有有限新息的信号）。 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、医学成像设备、音频设备等。这些技术产生了越来越多的数据，产生了用于获取、处理、分析和传输数据的新的非线性技术和范例。 为了充分利用这些新技术，彻底理解底层信号和数据的结构是至关重要的。 这个项目开发了从一组观察中学习非线性数据结构的技术。 它在数据挖掘、数据分类、分割以及视频序列中运动目标的跟踪等方面有着直接的应用。 它在数据压缩、噪声去除和数据传输中也有直接的应用。

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.