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Singular value decompositions

Let f = m Γ a [ m ] g m be the representation of f in an orthonormal basis B = { g m } m Γ . An approximation must be recovered from

Y = m Γ a [ m ] U g m + W .

A basis B of singular vectors diagonalizes U * U . Then U transforms a subset of Q vectors { g m } m Γ Q of B into an orthogonal basis { U g m } m Γ Q of ImU and sets all other vectors to zero. A singular value decomposition estimates the coefficients a [ m ] of f by projecting Y on this singular basis and by renormalizing the resultingcoefficients

m γ , a ˜ [ m ] = Y , U g m U g m 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 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 B providing a sparse signal representation, where a subset of Q vectors { g m } m Γ Q are transformed by U in a Riesz basis { U g m } m Γ Q of ImU , while the others are set to zero. With an appropriate renormalization, { λ ˜ m - 1 U g m } m Γ Q has a biorthogonal basis { φ ˜ m } m Γ Q that is normalized φ ˜ m = 1 . The sparse coefficients of f in B can then be estimated with a thresholding

m γ Q , a ˜ [ m ] = ρ T m ( λ ˜ m - 1 Y , φ ˜ m ) with ρ T ( x ) = x 1 | x | > T ,

for thresholds T m appropriately defined.

For classes of signals that are sparse in 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 f has a sparse representation in some dictionary D = { g p } p Γ of P normalized vectors. The P vectors of the transformed dictionary D U = U D = { U g p } p Γ belong to the space ImU of dimension Q < P and thus define a redundant dictionary. Vectors in the approximation support λ of f are not restricted a priori to a particular subspace of C N . Super-resolution is possible if the approximation support λ of f in D can be estimated by decomposing the noisy data Y over D U . It dependson the properties of the approximation support λ of f in γ .

Geometric conditions for super-resolution

Let w λ = f - f λ be the approximation error of a sparse representation f λ = p λ a [ p ] g p of f . The observed signal can be written as

Y = U f + W = p λ a [ p ] U g p + U w λ + W .

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

Y λ = p λ a ˜ [ p ] U g p ,

then we can recover a super-resolution estimation of f

F ˜ = p λ a ˜ [ p ] 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 E R C ( λ ) < 1 and if { U g p } p Λ 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 f if U is an operator with random coefficients. Compressive sensing then becomespossible by recovering a close approximation of f C N from Q N linear measurements (candes-cs-review).

A recovery is stable for a sparse approximation set | λ | M only if the corresponding dictionary family { U g m } m Λ 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 λ γ with | λ | M :

c C | λ | , ( 1 - δ M ) c 2 m λ c [ p ] U g p 2 ( 1 + δ M ) c 2 .

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

For an orthogonal basis D = { g m } m Γ , this is possible for M C Q ( log N ) - 1 if U is a matrix with independent Gaussian random coefficients. A pursuit algorithm thenprovides a stable approximation of any f C N having a sparse approximation from vectors in 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 { f s } 0 s < S that are linearly mixed in K S observed signals with noise:

Y k [ n ] = s = 0 S - 1 u k , s f s [ n ] + W k [ n ] for 0 n < N and 0 k < 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 = { u k , s } 0 k < K , 0 s < S is unknown. Source separation is a super-resolution problem since S N data values must be recovered from Q = K N S N measurements. Not knowing the operator U makes it even more complicated.

If each source f s has a sparse approximation support λ s in a dictionary D , with s = 0 S - 1 | λ s | N , then it is likely that the sets { λ s } 0 s < s are nearly disjoint. In this case,the operator U , the supports λ s , and the sources f s are approximated by computing sparse approximations of the observed data Y k in 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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Source:  OpenStax, A wavelet tour of signal processing, the sparse way. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10711/1.3
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