Frame Toolbox for Bioimaging, Biometrics and Robust Transmission

在过去的几十年里，信号和图像处理中的多分辨率技术已经彻底改变了这个领域。大多数使用的多分辨率技术都是非冗余的，也就是说，使用的底层数学结构是基。然而，今天的许多应用程序都需要系统中的一些冗余。冗余的要求需要一种比基更通用的数学结构，称为框架。虽然关于帧的最初工作可以追溯到20世纪50年代，但帧直到最近才变得更加流行，主要是由于新兴的应用程序需要提供冗余的工具。相当数量的工作已经在框架上完成了；然而，它们的成熟程度远不及小波。随着大量应用程序需要帧提供冗余，这种情况即将改变；理论需要快速跟进。本研究解决了现有知识的空白，并解决了框架中的一些开放问题。这些与某些类别的特征以及由生物成像，生物识别和鲁棒传输问题驱动的框架工具箱的构建有关。将框架理论发展到这种程度，使框架达到了小波的成熟水平，并大大扩展了多分辨率工具箱。

In the past couple of decades, multiresolution techniques in signal and image processing have revolutionized the field. Most of the multiresolution techniques in use are nonredundant, that is, the underlying mathematical structures used are bases. However, many of today's applications require some redundancy in the system.The requirement of redundancy requires a mathematical structure more general than bases, termed frames. Although the initial work on frames dates back to the 1950s, frames have become more popular only recently, mostly due to emerging applications requiring tools which provide redundancy. A fair amount of work has already been done on frames; however, their level of maturity is nowhere near that of wavelets. This is about to change as a host of applications requires redundancy offered by frames; the theory needs to follow fast.This research addresses gaps in the current knowledge and solves some of the open questions in frames. These are related to the characterization of certain classes as well as the construction of a frame toolbox motivated by problems in bioimaging, biometrics and robust transmission. Developing the frame theory to this extent brings frames to the level of maturity of wavelets and significantly expands the multiresolution toolbox.