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.
