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In natural languages, large dictionaries are needed to refine ideas with short sentences, and they evolve with usage.Eskimos have eight different words to describe snow quality , whereas a single word is typically sufficient in a Parisian dictionary.Similarly, large signal dictionaries of vectors are needed to constructsparse representations of complex signals. However, computing and optimizing a signal approximation by choosingthe best M dictionary vectors is much more difficult.
Suppose that a sparse family of vectors has been selected to approximate a signal . An approximation can be recoveredas an orthogonal projection in the space V λ generated by these vectors. We then face one of the following twoproblems.
The frame theory gives energy equivalence conditions to solve both problems with stable operators.A family is a frame of the space V it generatesif there exists such that
The representation is stable since any perturbation of frame coefficientsimplies a modification of similar magnitude on h . Chapter 5 proves that the existence of a dual frame that solves both the dual-synthesis and dual-analysisproblems:
Algorithms are provided to calculate these decompositions.The dual frame is also stable:
The frame bounds A and B are redundancy factors. If the vectors are normalized and linearly independent, then . Such a dictionary is called a Riesz basis of V and the dual frame is biorthogonal:
When the basis is orthonormal, then both bases are equal. Analysis and synthesis problems are then identical.
The frame theory is also used to construct redundant dictionaries that define complete, stable, and redundant signal representations,where V is then the whole signal space. The frame bounds measure the redundancy of such dictionaries.Chapter 5 studies the construction of windowed Fourier and wavelet frame dictionaries by samplingtheir time, frequency, and scaling parameters, while controlling frame bounds.In two dimensions, directional wavelet frames include wavelets sensitive to directional image structures such as textures or edges.
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