Let f=∑m∈Γa[m]gmf=∑m∈Γa[m]gm be the representation of ff in an orthonormal basis B={gm}m∈ΓB={gm}m∈Γ. An approximation must be recovered from

A basis BB of singular vectors diagonalizes U*UU*U. Then U transforms a subset of Q vectors {gm}m∈ΓQ{gm}m∈ΓQ of BB into an orthogonal basis {Ugm}m∈ΓQ{Ugm}m∈ΓQ of ImU and sets all other vectors to zero. A singular value decomposition estimates the coefficients a[m]a[m] of ff by projecting Y on this singular basis and by renormalizing the resultingcoefficients

where h_m² 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 BB is the Fourier basis and a singular value estimation implements a regularized inverseconvolution.

The basis that diagonalizes U*UU*U rarely provides a sparse signal representation. For example, a Fourier basis that diagonalizes convolution operators does not efficiently approximate signals including singularities.

Donoho (Donoho:95) introduced more flexibility by looking for a basis BB providing a sparse signal representation, where a subset of Q vectors {gm}m∈ΓQ{gm}m∈ΓQ are transformed by U in a Riesz basis {Ugm}m∈ΓQ{Ugm}m∈ΓQ of ImU, while the others are set to zero. With an appropriate renormalization, {λ˜m-1Ugm}m∈ΓQ{λ˜m-1Ugm}m∈ΓQ has a biorthogonal basis {φ˜m}m∈ΓQ{φ˜m}m∈ΓQ that is normalized ∥φ˜m∥=1∥φ˜m∥=1. The sparse coefficients of ff in BB can then be estimated with a thresholding

for thresholds T_m appropriately defined.

For classes of signals that are sparse in BB, such thresholding estimators may yield 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 and potentially optimal in cases where super-resolution is not possible.

Let wλ=f-fλwλ=f-fλ be the approximation error of a sparse representation fλ=∑p∈λa[p]gpfλ=∑p∈λa[p]gp of ff. The observed signal can be written as

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

then we can recover a super-resolution estimation of ff

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(λ)<1ERC(λ)<1 and if {Ugp}p∈Λ{Ugp}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-resolution zooming, and inpainting illustrate these results.

Candès and Tao (candes-near-optimal), and Donoho (donoho-cs) proved that stable super-resolution is possible for any sufficiently sparse signal ff if U is an operator with random coefficients. Compressive sensing then becomes possible by recovering a close approximation of f∈CNf∈CN from Q≪NQ≪N linear measurements (candes-cs-review).

A recovery is stable for a sparse approximation set |λ|≤M|λ|≤M only if the corresponding dictionary family {Ugm}m∈Λ{Ugm}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⊂γλ⊂γ with |λ|≤M|λ|≤M:

This is a strong incoherence condition on the P vectors of {Ugm}m∈Γ{Ugm}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={gm}m∈ΓD={gm}m∈Γ, this is possible for M≤CQ(logN)-1M≤CQ(logN)-1 if U is a matrix with independent Gaussian random coefficients. A pursuit algorithm then provides a stable approximation of any f∈CNf∈CN having a sparse approximation from vectors in DD.

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 nonlinear algorithm, 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.

Sparsity in redundant dictionaries also provides efficient strategies to separate a family of signals {fs}0≤s<S{fs}0≤s<S that are linearly mixed in K≤SK≤S observed signals with noise:

From a stereo recording, separating the sounds of S musical instruments is an example of source separation with k=2k=2. Most often the mixing matrix U={uk,s}0≤k<K,0≤s<SU={uk,s}0≤k<K,0≤s<S is unknown. Source separation is a super-resolution problem since SNSN data values must be recovered from Q=KN≤SNQ=KN≤SN measurements. Not knowing the operator U makes it even more complicated.

If each source fsfs has a sparse approximation support λ_s in a dictionary DD, with ∑s=0S-1|λs|≪N∑s=0S-1|λs|≪N, then it is likely that the sets {λs}0≤s<s{λs}0≤s<s are nearly disjoint. In this case, the operator U, the supports λ_s, and the sources fsfs are approximated by computing sparse approximations of the observed data Y_k in DD. The distribution of these coefficients identifies the coefficients of the mixing matrix U and the nearly disjoint source supports. Time-frequency separation of sounds illustrate these results.