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该领域。大多数使用的多分辨率技术是非冗余的，也就是说，使用的基础数学结构是基础。然而，今天的许多应用需要系统中的一些冗余。冗余的要求需要比基更一般的数学结构，称为帧。虽然框架的最初工作可以追溯到20世纪50年代，但框架只是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。在框架上已经做了相当多的工作;然而，它们的成熟程度远不及小波。这种情况即将改变，因为大量的应用程序需要帧提供冗余;理论需要快速跟进。这项研究解决了当前知识中的空白，并解决了帧中的一些悬而未决的问题。这些都涉及到某些类的特性，以及在生物成像，生物识别和强大的传输问题的动机框架工具箱的建设。发展框架理论到这个程度，使框架达到小波的成熟程度，并显着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.