RUI: Development of Frame Extensions and Applications, III

研究者研究帧的扩展及其在信号处理中的应用。应用程序中的一个常见问题是数据融合，即将代表同一项不同方面的数据组合在一起。例如，在试图锐化一张图片时，分别处理前景和背景可能会产生更好的整体图像。研究者研究了数据融合的框架-理论公式，强调了在多光谱图像融合中的应用以及相关的数学和计算问题。他还研究了使用融合框架的融合并行稀疏表示，研究了最优性和预期计算效率的问题。融合帧是处理一组重叠子空间的帧的扩展；目的是将来自重叠子空间的信息融合到复合总空间中。融合稀疏表示涉及以并行方式获得（重叠）子空间中信号的稀疏表示，然后进行融合操作。研究者还研究了具有最小支持或最小支持和最大近似阶的b样条riesz序列的伪框架对偶。考虑了伪框架对偶小支撑的极限行为。在这里，伪框架来源于子空间伪框架（PFFS）的概念。pffs是用于子空间线性分析和综合的类框架系统。它们与帧的不同之处在于，分析和合成序列并不一定包含在子空间中。信号处理一般是指使信号易于处理或具有更好的质量。它在现代社会随处可见。在这个项目中，引入信号处理的主要新颖之处是数据融合应用的新帧理论方法和使用融合帧概念执行数据压缩的想法。数据融合将以不同方式采样的数据或表示一个项目的不同方面，以获得更好的项目表示。研究者的目的是开发一种方法，将不同类型的相机（如红外和常规）获得的图像组合成一个具有所有信息的单一图像。特别是，他致力于提供一个更精确的组合图像，而不需要传统技术的临时程序。这方面的进展对于商业成像、地理调查、地理调查、制图和传感器阵列数据处理等信息技术领域非常重要。同样，研究人员开发了一些工具，通过将昂贵的压缩任务减少到一组并行的小任务，然后由融合框架理论控制的精确组合机制，来提高数据压缩的效率。更一般地说，这些方法可以在广泛的通信和信号处理环境中提高信息传输的速度。

Li0709384 The investigator studies extensions of frames and theirapplications in signal processing. A common problem inapplications is data fusion, the combination of data representingdifferent aspects of the same item. For example, in attemptingto sharpen a picture, treating the foreground and backgroundseparately may produce a better overall image. The investigatorstudies frame-theoretical formulations for data fusion,emphasizing applications in multi-spectral image fusion andrelated mathematical and computational issues. He also studiesfused parallel sparse representation using fusion frames,examining issues of optimality and expected computationalefficiency. Fusion frames are an extension of frames to dealwith a set of overlapping subspaces; the aim is to fuse togetherinformation from overlapping subspaces into the composite totalspace. Fused sparse representation involves ways to obtainsparse representations of a signal in (overlapping) subspaces ina parallel fashion followed by a fusion operation. Theinvestigator also studies pseudoframe duals of B-spline Rieszsequences having either minimum support or both minimum supportand maximum approximation order. Limiting behavior of the smallsupport of pseudoframe duals is considered. Here pseudoframeduals arise from a notion of pseudoframes for subspaces (PFFS). PFFSs are frame-like systems for the linear analysis andsynthesis of a subspace. They differ from frames in that bothanalysis and synthesis sequences are not necessarily contained inthe subspace. Signal processing generally refers to making the signal easyto handle or of better quality. It is everywhere seen in ourmodern society. In this project, the principal noveltiesintroduced to signal processing are a new frame-theoreticalapproach to data fusion applications and ideas of performing datacompression using a notion of fusion frames. Data fusioncombines data sampled in different ways or represnting differentaspects of an item in order to obtain a better representation ofthe item. The investigator aims at developing methods ofcombining images obtained in different type of cameras (such asinfrared and conventional) into one single image with allinformation presented. In particular, he works to provide acombined image of much greater precision without ad-hocprocedures of conventional techniques. Advances here areimportant for such areas of information technology as commercialimaging, geographic surveys, geographic survey, mapping, andsensor array data processing. Similarly, the investigatordevelops tools to improve the efficiency of data compression byreducing a costly compression task into a set of parallel smallertasks followed by a precise combination mechanism governed by thetheory of fusion frames. More generally, the methods canincrease the speed of information transmission in a broad rangeof communication and signal processing environments.