Life Beyond Bases: The Advent of Frames

Redundancy is a common tool in our daily lives. We doubleand triple-check that we turned off gas and lights, took our keys, money, etc. (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming â\u80\u9cjust once more â\u80\u9d they are on top of it. Of course, the reason we are doing that is to avoid a disaster by missing or forgetting something, not to drive our loved ones crazy. The same idea of removing doubt is present in signal representations. Given a signal, we represent it in another system, typically a basis, where its characteristics are more readily apparent in the transform coefficients (for example, wavelet-based compression). However, these representations are typically nonredundant, and thus corruption or loss of transform coefficients can be fatal. In comes redundancy; we build a safety net into our representation so that we can avoid those fatal disasters. The redundant counterpart of a basis is called a frame (no one seems to know why they are called frames, perhaps because they are bounded from two sides as in (15)?). It is generally acknowledged 1 that frames were born in 1952 in the paper by Duffin and Schaeffer [57]. Despite being over half a century old, frames gained popularity only in the last decade, due mostly to the work of the three wavelet pioneersâ\u80\u9