Kovačević and Chebira: 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 “just once more” 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 acknowledged1 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 pioneers— Daubechies, Grossman and Meyer [49]. Frame-like ideas, that is, building redundancy into a signal expansion, can be seen in pyramid coding [26], quantization [47], [99], [15], [69], [43], [48], [72], [14], denoising [128], [36], [56], [84], [65], robust transmission [68], [79], [113], [16], [17], [18], [119], [31], [19], [101], CDMA systems [98], [122], [123], [115], multiantenna code design [74], [78], segmentation [116], [92], [50], classification [116], [92], [33], prediction of epileptic seizures [12], [13], restoration and enhancement [85], motion estimation [96], signal reconstruction [6], coding theory [75], [103], operator theory [2] and quantum theory and computing [59], [110]. While frames are often associated with wavelet frames, it is important to remember that frames are more general than that. Wavelet frames possess structure; frames are redundant representations that only need to represent signals in a given space with a certain amount of redundancy. The simplest frame, appropriately named Mercedes-Benz (MB), is given in Box I; just have a peek now, we will go into more details later. The question now is: Why and where would one use frames? The answer is obvious: anywhere where redundancy is a must.