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Indeed, these results suggest that the largest Fourier or wavelet coefficients of smooth signals tend to concentrate at the coarsest scales (lowest-frequencies). In Linear Approximation from Approximation , we see that linear approximations formed from just the lowest frequencyelements of the Fourier or wavelet dictionaries (i.e., the truncation of the Fourier or wavelet representation to only the lowest frequency terms) provide very accurate approximations to smooth signals. Put differently, smooth signals live near the subspace spanned by just the lowest frequency Fourier or wavelet basis functions.
Sparse signal models can be viewed as a generalization of linear models. The notion of sparsity comes from the fact that, by theproper choice of dictionary $\Psi $ , many real-world signals $x=\Psi \alpha $ have coefficient vectors $\alpha $ containing few large entries, but across different signals the locations (indicesin $\alpha $ ) of the large entries may change. We say a signal is strictly sparse (or “ $K$ -sparse”) if all but $K$ entries of $\alpha $ are zero.
Some examples of real-world signals for which sparse models have been proposed include neural spike trains (in time), music andother audio recordings (in time and frequency), natural images (in the wavelet or curveletdictionaries [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] ), video sequences (in a 3-D waveletdictionary [link] , [link] ), and sonar or radar pulses (in a chirplet dictionary [link] ). In each of these cases, the relevant information in a sparse representation of asignal is encoded in both the locations (indices) of the significant coefficients and the values to which they are assigned. This type of uncertainty is an appropriate model formany natural signals with punctuated phenomena.
Sparsity is a nonlinear model. In particular, let ${\Sigma}_{K}$ denote the set of all $K$ -sparse signals for a given dictionary. It is easy to see that the set ${\Sigma}_{K}$ is not closed under addition. (In fact, ${\Sigma}_{K}+{\Sigma}_{K}={\Sigma}_{2K}$ .) From a geometric perspective, the set of all $K$ -sparse signals from the dictionary $\Psi $ forms not a hyperplane but rather a union of $K$ -dimensional hyperplanes, each spanned by $K$ vectors of $\Psi $ (see [link] (b)). For a dictionary $\Psi $ with $Z$ entries, there are $\left(\genfrac{}{}{0pt}{}{Z}{K}\right)$ such hyperplanes. (The geometry of sparse signal collections has alsobeen described in terms of orthosymmetric sets; see [link] .)
Signals that are not strictly sparse but rather have a few “large” and many “small” coefficients are known as compressible signals. The notion of compressibility can be made more precise by considering the rate at which the sorted magnitudes of the coefficients $\alpha $ decay, and this decay rate can in turn be related to the ${\ell}_{p}$ norm of the coefficient vector $\alpha $ . Letting $\tilde{\alpha}$ denote a rearrangement of the vector $\alpha $ with the coefficients ordered in terms of decreasing magnitude, then the reordered coefficientssatisfy [link]
We recall from [link] that for a smooth signal $f$ , the largest Fourier and wavelet coefficients tend to cluster at coarse scales (low frequencies). Suppose, however, thatthe function $f$ is piecewise smooth; i.e., it is ${\mathcal{C}}^{H}$ at every point $t\in \mathbb{R}$ except for one point ${t}_{0}$ , at which it is discontinuous. Naturally, this phenomenon will be reflected in the transform coefficients.In the Fourier domain, this discontinuity will have a global effect, as the overall smoothness of the function $f$ has been reduced dramatically from $H$ to 0. Wavelet coefficients, however, depend only on local signalproperties, and so the wavelet basis functions whose supports do not include ${t}_{0}$ will be unaffected by the discontinuity. Coefficients surrounding the singularity will decay only as ${2}^{-j/2}$ , but there are relatively few such coefficients. Indeed, at each scale there are only $O\left(1\right)$ wavelets that include ${t}_{0}$ in their supports, but these locations are highly signal-dependent. (For modeling purposes, these significantcoefficients will persist through scale down the parent-child tree structure.) After reordering by magnitude, the waveletcoefficients of piecewise smooth signals will have the same general decay rate as those of smooth signals. In Nonlinear Approximation from Approximation , we see that the quality of nonlinear approximations offered by wavelets for smooth 1-D signals is nothampered by the addition of a finite number of discontinuities.
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