0.1 Signal dictionaries and representations

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This collection reviews fundamental concepts underlying the use of concise models for signal processing. Topics are presented from a geometric perspective and include low-dimensional linear, sparse, and manifold-based signal models, approximation, compression, dimensionality reduction, and Compressed Sensing.

For a wide variety of signal processing applications (including analysis, compression, noise removal, and so on) it is useful toconsider the representation of a signal in terms of some dictionary  [link] . In general, a dictionary $\phantom{\rule{2pt}{0ex}}\Psi \phantom{\rule{2pt}{0ex}}$ issimply a collection of elements drawn from the signal space whose linear combinations can be used to represent or approximatesignals.

Considering, for example, signals in ${\mathbb{R}}^{N}$ , we may collect and represent the elements of the dictionary $\Psi$ as an $N×Z$ matrix, which we also denote as $\Psi$ . From this dictionary, a signal $x\in {\mathbb{R}}^{N}$ can be constructed as a linear combination of the elements (columns) of $\Psi$ . We write

$x=\Psi \alpha$
for some $\alpha \in {\mathbb{R}}^{Z}$ . (For much of our notation in this section, we concentrate on signals in ${\mathbb{R}}^{N}$ , though the basic concepts translate to other vector spaces.)

Dictionaries appear in a variety of settings. The most common may be the basis, in which case $\Psi$ has exactly $N$ linearly independent columns, and each signal $x$ has a unique set of expansion coefficients $\alpha ={\Psi }^{-1}x$ . The orthonormal basis (where the columns are normalized and orthogonal) is also ofparticular interest, as the unique set of expansion coefficients $\alpha ={\Psi }^{-1}x={\Psi }^{{}^{T}}x$ can be obtained as the inner products of $x$ against the columns of $\Psi$ . That is, $\alpha \left(i\right)=〈x,,,{\psi }_{i}〉,i=1,2,\cdots ,N$ , which gives us the expansion

$x=\sum _{i=1}^{N}〈x,,,{\psi }_{i}〉{\psi }_{i}.$

We also have that ${∥x∥}_{2}^{2}={\sum }_{i=1}^{N}{〈x,,,{\psi }_{i}〉}^{2}$ .

Frames are another special type of dictionary  [link] . A dictionary $\Psi$ is a frame if there exist numbers $A$ and $B$ , $0 such that, for any signal $x$

$A{∥x∥}_{2}^{2}\le \sum _{z}{〈x,,,{\psi }_{z}〉}^{2}\le B{∥x∥}_{2}^{2}.$
The elements of a frame may be linearly dependent in general (see [link] ), and so there may exist many ways to express a particular signal among the dictionary elements.However, frames do have a useful analysis/synthesis duality: for any frame $\Psi$ there exists a dual frame $\stackrel{˜}{\Psi }$ such that
$x=\sum _{z}〈x,,,{\psi }_{z}〉{\stackrel{˜}{\psi }}_{z}=\sum _{z}〈x,,,{\stackrel{˜}{\psi }}_{z}〉{\psi }_{z}.$
In the case where the frame vectors are represented as columns of the $N$ x $Z$ matrix $\Psi$ , the matrix $\stackrel{˜}{\Psi }$ containing the dual frame elements is simply the transpose of the pseudoinverse of $\Psi$ . A frame is called tight if the frame bounds $A$ and $B$ are equal. Tight frames have the special properties of (i) being theirown dual frames (after a rescaling by $1/A$ ) and (ii) preserving norms, i.e., ${\sum }_{i=1}^{N}{〈x,,,{\psi }_{i}〉}^{2}=A{∥x∥}_{2}^{2}$ . The remainder of this section discusses several importantdictionaries.

The canonical basis

The standard basis for representing a signal is the canonical (or “spike”) basis. In ${\mathbb{R}}^{N}$ , this corresponds to a dictionary $\Psi ={I}_{N}$ (the $N×N$ identity matrix). When expressed in the canonical basis, signals are often said tobe in the “time domain.”

Fourier dictionaries

The frequency domain provides one alternative representation to the time domain. The Fourier series and discrete Fourier transformare obtained by letting $\Psi$ contain complex exponentials and allowing the expansion coefficients $\alpha$ to be complex as well. (Such a dictionary can be used to represent real or complexsignals.) A related “harmonic” transform to express signals in ${\mathbb{R}}^{N}$ is the discrete cosine transform (DCT), in which $\Psi$ contains real-valued, approximately sinusoidal functions and the coefficients $\alpha$ are real-valued as well.

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