Kovacevic - Chebira - IEEE-SPL- 2007-04-86

Kovacevic - Chebira - IEEE-SPL- 2007-04-86 - [Jelena...

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IEEE SIGNAL PROCESSING MAGAZINE [ 86 ] JULY 2007 1053-5888/07/$25.00©2007IEEE R edundancy is a com- mon tool in our daily lives. Before we leave the house, we double- and triple-check that we turned off gas and lights, took our keys, and have money (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 anoth- er system, typically a basis, where its characteristics are more readily apparent in the transform coefficients. However, these representations are typically nonredundant, and thus corruption or loss of transform coefficients can be serious. In comes redundancy; we build a safety net into our representation so that we can avoid those dis- asters. The redundant counterpart of a basis is called a frame [no one seems to know why they are called frames, perhaps because of the bounds in (25)?]. It is generally acknowledged (at least in the signal processing and harmonic analysis communities) that frames were born in 1952 in the paper by Duffin and Schaeffer [32]. 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 [29]. Frame-like ideas, that is, building redundancy into a signal expansion, can be found in pyramid Life Beyond Bases: The Advent of Frames (Part I) [ Jelena Kovac ˇevic ´ and Amina Chebira ] © PH O TO D IS C Redundant: To Be or Not to Be?
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IEEE SIGNAL PROCESSING MAGAZINE [ 87 ] JULY 2007 coding [14]; source coding [7], [8], [23], [27], [28], [37], [38], [53]; denoising [20], [31], [35], [46], [68]; robust transmission [9]–[12], [17], [36], [45], [54], [58], [63]; CDMA systems [52], [59], [66], [67]; multiantenna code design [40], [44]; segmenta- tion [30], [50], [60]; classification [18], [50], [60]; prediction of epileptic seizures [5], [6]; restoration and enhancement [47]; motion estimation [51]; signal reconstruction [2]; coding theory [41], [55]; operator theory [1]; and quantum theory and computing [33], [57]. 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 rep- resent signals in a given space with a certain amount of redundancy. The simplest frame, appro- priately named Mercedes-Benz (MB), is given in “The Mercedes-Benz Frame”; just have a peek at the sidebar now as we will go into more details later. The question now is this: Why and where would one use frames? The answer is obvious: anywhere where redundancy is a must. The host of the applica- tions mentioned above and discussed in Part II of this article [48] illustrate that richly.
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.

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Kovacevic - Chebira - IEEE-SPL- 2007-04-86 - [Jelena...

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