# 0.6 Inverse problems  (Page 2/2)

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## Singular value decompositions

Let $\phantom{\rule{0.166667em}{0ex}}f={\sum }_{m\in \Gamma }a\left[m\right]\phantom{\rule{0.166667em}{0ex}}{g}_{m}$ be the representation of $\phantom{\rule{0.166667em}{0ex}}f$ in an orthonormal basis $\mathcal{B}={\left\{{g}_{m}\right\}}_{m\in \Gamma }$ . An approximation must be recovered from

$Y=\sum _{m\in \Gamma }a\left[m\right]\phantom{\rule{0.166667em}{0ex}}U{g}_{m}+W.$

A basis $\mathcal{B}$ of singular vectors diagonalizes ${U}^{*}U$ . Then U transforms a subset of Q vectors ${\left\{{g}_{m}\right\}}_{m\in {\Gamma }_{Q}}$ of $\mathcal{B}$ into an orthogonal basis ${\left\{U{g}_{m}\right\}}_{m\in {\Gamma }_{Q}}$ of ImU and sets all other vectors to zero. A singular value decomposition estimates the coefficients $a\left[m\right]$ of $\phantom{\rule{0.166667em}{0ex}}f$ by projecting Y on this singular basis and by renormalizing the resultingcoefficients

$\forall m\in \gamma ,\phantom{\rule{1.em}{0ex}}\stackrel{˜}{a}\left[m\right]=\frac{⟨Y,U{g}_{m}⟩}{\parallel U{g}_{m}{\parallel }^{2}+{h}_{m}^{2}},$

where h m 2 are regularization parameters.

Such estimators recover nonzero coefficients in a space of dimension Q and thus bring no super-resolution. If U is a convolution operator, then $\mathcal{B}$ is the Fourier basis and a singular value estimationimplements a regularized inverseconvolution.

## Diagonal thresholding estimation

The basis that diagonalizes ${U}^{*}U$ rarely provides a sparse signal representation.For example, a Fourier basis that diagonalizes convolution operators does notefficiently approximate signals including singularities.

Donoho (Donoho:95) introduced more flexibility by looking for a basis $\mathcal{B}$ providing a sparse signal representation, where a subset of Q vectors ${\left\{{g}_{m}\right\}}_{m\in {\Gamma }_{Q}}$ are transformed by U in a Riesz basis ${\left\{U{g}_{m}\right\}}_{m\in {\Gamma }_{Q}}$ of ImU , while the others are set to zero. With an appropriate renormalization, ${\left\{{\stackrel{˜}{\lambda }}_{m}^{-1}\phantom{\rule{0.166667em}{0ex}}U{g}_{m}\right\}}_{m\in {\Gamma }_{Q}}$ has a biorthogonal basis ${\left\{{\stackrel{˜}{\phi }}_{m}\right\}}_{m\in {\Gamma }_{Q}}$ that is normalized $\parallel {\stackrel{˜}{\phi }}_{m}\parallel =1$ . The sparse coefficients of $\phantom{\rule{0.166667em}{0ex}}f$ in $\mathcal{B}$ can then be estimated with a thresholding

$\forall m\in {\gamma }_{Q},\phantom{\rule{1.em}{0ex}}\stackrel{˜}{a}\left[m\right]={\rho }_{{T}_{m}}\left({\stackrel{˜}{\lambda }}_{m}^{-1}⟨Y\phantom{\rule{-0.166667em}{0ex}},{\stackrel{˜}{\phi }}_{m}⟩\right)\phantom{\rule{1.em}{0ex}}\text{with}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\rho }_{T}\left(x\right)=x\phantom{\rule{0.166667em}{0ex}}{1}_{|x|>T},$

for thresholds T m appropriately defined.

For classes of signals that are sparse in $\mathcal{B}$ , such thresholding estimators mayyield a nearly minimax risk, but they provide no super-resolution since this nonlinear projector remains in a space of dimension Q . This result applies to classes of convolution operators U in wavelet or wavelet packet bases. Diagonal inverse estimators are computationally efficient andpotentially optimal in cases where super-resolution is not possible.

## Super-resolution and compressive sensing

Suppose that $\phantom{\rule{0.166667em}{0ex}}f$ has a sparse representation in some dictionary $\mathcal{D}={\left\{{g}_{p}\right\}}_{p\in \Gamma }$ of P normalized vectors. The P vectors of the transformed dictionary ${\mathcal{D}}_{U}=U\mathcal{D}={\left\{U{g}_{p}\right\}}_{p\in \Gamma }$ belong to the space ImU of dimension $Q and thus define a redundant dictionary. Vectors in the approximation support λ of $\phantom{\rule{0.166667em}{0ex}}f$ are not restricted a priori to a particular subspace of ${\mathbb{C}}^{N}$ . Super-resolution is possible if the approximation support λ of $\phantom{\rule{0.166667em}{0ex}}f$ in $\mathcal{D}$ can be estimated by decomposing the noisy data Y over D U . It dependson the properties of the approximation support λ of $\phantom{\rule{0.166667em}{0ex}}f$ in γ .

## Geometric conditions for super-resolution

Let ${w}_{\lambda }=f-{f}_{\lambda }$ be the approximation error of a sparse representation $\phantom{\rule{0.166667em}{0ex}}{f}_{\lambda }={\sum }_{p\in \lambda }a\left[\phantom{\rule{0.166667em}{0ex}}p\right]\phantom{\rule{0.166667em}{0ex}}{g}_{p}$ of $\phantom{\rule{0.166667em}{0ex}}f$ . The observed signal can be written as

$Y=Uf+W=\sum _{p\in \lambda }a\left[\phantom{\rule{0.166667em}{0ex}}p\right]\phantom{\rule{0.166667em}{0ex}}U{g}_{p}+U{w}_{\lambda }+W.$

If the support λ can be identified by finding a sparse approximation of Y in D U

${Y}_{\lambda }=\sum _{p\in \lambda }\stackrel{˜}{a}\left[\phantom{\rule{0.166667em}{0ex}}p\right]\phantom{\rule{0.166667em}{0ex}}U{g}_{p},$

then we can recover a super-resolution estimation of $\phantom{\rule{0.166667em}{0ex}}f$

$\stackrel{˜}{F}=\sum _{p\in \lambda }\stackrel{˜}{a}\left[\phantom{\rule{0.166667em}{0ex}}p\right]\phantom{\rule{0.166667em}{0ex}}{g}_{p}.$

This shows that super-resolution is possible if the approximation support λ can be identified by decomposing Y in the redundant transformed dictionary D U . If the exact recovery criteria is satisfy $ERC\left(\lambda \right)<1$ and if ${\left\{U{g}_{p}\right\}}_{p\in \Lambda }$ is a Riesz basis, then λ can be recovered using pursuit algorithms with controlled error bounds.

For most operator U , not all sparse approximation sets can be recovered. It is necessary to impose some further geometric conditions on λ in γ , which makes super-resolution difficult and often unstable. Numerical applications to sparse spike deconvolution, tomography, super-resolutionzooming, and inpainting illustrate these results.

## Compressive sensing with randomness

Candès and Tao (candes-near-optimal), and Donoho (donoho-cs) proved that stable super-resolution is possible for anysufficiently sparse signal $\phantom{\rule{0.166667em}{0ex}}f$ if U is an operator with random coefficients. Compressive sensing then becomespossible by recovering a close approximation of $\phantom{\rule{0.166667em}{0ex}}f\in {\mathbb{C}}^{N}$ from $Q\ll N$ linear measurements (candes-cs-review).

A recovery is stable for a sparse approximation set $|\lambda |\le M$ only if the corresponding dictionary family ${\left\{U{g}_{m}\right\}}_{m\in \Lambda }$ is a Riesz basis of the space it generates. The M-restricted isometry conditions of Candès, Tao, and Donoho (donoho-cs) imposes uniform Riesz bounds for all sets $\lambda \subset \gamma$ with $|\lambda |\le M$ :

$\begin{array}{c}\hfill \forall c\in {\mathbb{C}}^{|\lambda |},\phantom{\rule{5.0pt}{0ex}}\left(1-{\delta }_{M}\right){\phantom{\rule{0.166667em}{0ex}}\parallel c\parallel }^{2}\le \parallel \sum _{m\in \lambda }c\left[\phantom{\rule{0.166667em}{0ex}}p\right]\phantom{\rule{0.166667em}{0ex}}U{g}_{p}{\parallel }^{2}\le \left(1+{\delta }_{M}\right)\phantom{\rule{0.166667em}{0ex}}{\parallel c\parallel }^{2}.\end{array}$

This is a strong incoherence condition on the P vectors of ${\left\{U{g}_{m}\right\}}_{m\in \Gamma }$ , which supposes that any subset of less than M vectors is nearly uniformly distributed on the unit sphere of ImU .

For an orthogonal basis $\mathcal{D}={\left\{{g}_{m}\right\}}_{m\in \Gamma }$ , this is possible for $M\le C\phantom{\rule{0.166667em}{0ex}}Q{\left(logN\right)}^{-1}$ if U is a matrix with independent Gaussian random coefficients. A pursuit algorithm thenprovides a stable approximation of any $\phantom{\rule{0.166667em}{0ex}}f\in {C}^{N}$ having a sparse approximation from vectors in $\mathcal{D}$ .

These results open a new compressive-sensing approach to signal acquisition and representation.Instead of first discretizing linearly the signal at a high-resolution N and then computing a nonlinear representation over M coefficients in some dictionary, compressive-sensing measures directly M randomized linear coefficients. A reconstructed signal is then recovered by a nonlinearalgorithm, producing an error that can be of the same order of magnitude as the error obtained by the more classic two-step approximation process,with a more economic acquisiton process. These results remain valid for several types of random matrices U . Examples of applications to single-pixel cameras,video super-resolution, new analog-to-digital converters, and MRI imaging are described.

## Blind source separation

Sparsity in redundant dictionaries also provides efficient strategies to separate a family of signals ${\left\{\phantom{\rule{0.166667em}{0ex}}{f}_{s}\right\}}_{0\le s that are linearly mixed in $K\le S$ observed signals with noise:

${Y}_{k}\left[n\right]=\sum _{s=0}^{S-1}{u}_{k,s}\phantom{\rule{0.166667em}{0ex}}{f}_{s}\left[n\right]+{W}_{k}\left[n\right]\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{5.0pt}{0ex}}0\le n\phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.166667em}{0ex}}N\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{5.0pt}{0ex}}0\le k

From a stereo recording, separating the sounds of S musical instruments is an example of source separation with $k=2$ . Most often the mixing matrix $U={\left\{{u}_{k,s}\right\}}_{0\le k is unknown. Source separation is a super-resolution problem since $S\phantom{\rule{0.166667em}{0ex}}N$ data values must be recovered from $Q=K\phantom{\rule{0.166667em}{0ex}}N\le S\phantom{\rule{0.166667em}{0ex}}N$ measurements. Not knowing the operator U makes it even more complicated.

If each source $\phantom{\rule{0.166667em}{0ex}}{f}_{s}$ has a sparse approximation support λ s in a dictionary $\mathcal{D}$ , with ${\sum }_{s=0}^{S-1}|{\lambda }_{s}|\ll N$ , then it is likely that the sets ${\left\{{\lambda }_{s}\right\}}_{0\le s are nearly disjoint. In this case,the operator U , the supports λ s , and the sources $\phantom{\rule{0.166667em}{0ex}}{f}_{s}$ are approximated by computing sparse approximations of the observed data Y k in $\mathcal{D}$ . The distribution of these coefficients identifies the coefficients of the mixingmatrix U and the nearly disjoint source supports. Time-frequency separation of sounds illustrate these results.

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