0.5 Sparsity in redundant dictionaries

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This collection comprises Chapter 1 of the book A Wavelet Tour of Signal Processing, The Sparse Way(third edition, 2009) by Stéphane Mallat. The book's website at Academic Press ishttp://www.elsevier.com/wps/find/bookdescription.cws_home/714561/description#description The book's complementary materials are available athttp://wavelet-tour.com

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.

Frame analysis and synthesis

Suppose that a sparse family of vectors ${\left\{{\phi }_{p}\right\}}_{p\in \Lambda }$ has been selected to approximate a signal $\phantom{\rule{0.166667em}{0ex}}f$ . 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.

1. In a dual-synthesis problem, the orthogonal projection $\phantom{\rule{0.166667em}{0ex}}{f}_{\lambda }$ of $\phantom{\rule{0.166667em}{0ex}}f$ in V λ must be computed from dictionary coefficients, ${\left\{⟨\phantom{\rule{0.166667em}{0ex}}f,{\phi }_{p}⟩\right\}}_{p\in \lambda }$ , provided by an analysis operator.  This is the case when a signal transform ${\left\{⟨\phantom{\rule{0.166667em}{0ex}}f,{\phi }_{p}⟩\right\}}_{p\in \Gamma }$ is calculated in some large dictionary and a subset of inner products are selected. Such inner products may correspond tocoefficients above a threshold or local maxima values.
2. In a dual-analysis problem, the decomposition coefficients of $\phantom{\rule{0.166667em}{0ex}}{f}_{\lambda }$ must be computed on a family of selected vectors ${\left\{{\phi }_{p}\right\}}_{p\in \Lambda }$ . This problem appears when sparse representation algorithmsselect vectors as opposed to inner products. This is the case forpursuit algorithms, which compute approximation supports in highly redundant dictionaries.

The frame theory gives energy equivalence conditions to solve both problems with stable operators.A family ${\left\{{\phi }_{p}\right\}}_{p\in \Lambda }$ is a frame of the space V it generatesif there exists $B\ge A>0$ such that

$\forall h\in {\mathbf{V},\phantom{\rule{5.0pt}{0ex}}A\parallel h\parallel }^{2}\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}\sum _{m\in \lambda }|⟨h,{\phi }_{p}{⟩|}^{2}\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}B{\parallel h\parallel }^{2}.$

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 ${\left\{{\stackrel{˜}{\phi }}_{p}\right\}}_{p\in \Lambda }$ that solves both the dual-synthesis and dual-analysisproblems:

$\begin{array}{c}\hfill {f}_{\lambda }=\sum _{p\in \lambda }⟨\phantom{\rule{0.166667em}{0ex}}f,{\phi }_{p}⟩\phantom{\rule{0.166667em}{0ex}}{\stackrel{˜}{\phi }}_{p}=\sum _{p\in \lambda }⟨\phantom{\rule{0.166667em}{0ex}}f,{\stackrel{˜}{\phi }}_{p}⟩\phantom{\rule{0.166667em}{0ex}}{\phi }_{p}.\end{array}$

Algorithms are provided to calculate these decompositions.The dual frame is also stable:

$\forall f\phantom{\rule{0.166667em}{0ex}}\in \phantom{\rule{0.166667em}{0ex}}\mathbf{V},\phantom{\rule{5.0pt}{0ex}}{B}^{-1}{\parallel \phantom{\rule{0.166667em}{0ex}}f\parallel }^{2}\le \sum _{m\in \gamma }|⟨\phantom{\rule{0.166667em}{0ex}}f,{\stackrel{˜}{\phi }}_{p}{⟩|}^{2}\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}{B}^{-1}{\parallel \phantom{\rule{0.166667em}{0ex}}f\parallel }^{2}.$

The frame bounds A and B are redundancy factors. If the vectors ${\left\{{\phi }_{p}\right\}}_{p\in \Gamma }$ are normalized and linearly independent, then $A\le 1\le B$ . Such a dictionary is called a Riesz basis of V and the dual frame is biorthogonal:

$\forall \left(\phantom{\rule{0.166667em}{0ex}}p,p\text{'}\right)\in {\lambda }^{2},\phantom{\rule{3.33333pt}{0ex}}⟨{\phi }_{p},{\stackrel{˜}{\phi }}_{p\text{'}}⟩=\delta \left[\phantom{\rule{0.166667em}{0ex}}p-p\text{'}\right].$

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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