Relaxed Recovery Conditions for OMP/OLS by Exploiting both Coherence and Decay

We propose extended coherence-based conditions for exact sparse support recovery using orthogonal matching pursuit (OMP) and orthogonal least squares (OLS). Unlike standard uniform guarantees, we embed some information about the decay of the sparse vector coefficients in our conditions. As a result, the standard condition $\mu<1/(2k-1)$ (where $\mu$ denotes the mutual coherence and $k$ the sparsity level) can be weakened as soon as the non-zero coefficients obey some decay, both in the noiseless and the bounded-noise scenarios. Furthermore, the resulting condition is approaching $\mu<1/k$ for strongly decaying sparse signals. Finally, in the noiseless setting, we prove that the proposed conditions, in particular the bound $\mu<1/k$, are the tightest achievable guarantees based on mutual coherence

Coherence-based Partial Exact Recovery Condition for OMP/OLS

We address the exact recovery of the support of a k-sparse vector with Orthogonal Matching Pursuit (OMP) and Orthogonal Least Squares (OLS) in a noiseless setting. We consider the scenario where OMP/OLS have selected good atoms during the first l iterations (l<k) and derive a new sufficient and worst-case necessary condition for their success in k steps. Our result is based on the coherence \mu of the dictionary and relaxes Tropp's well-known condition \mu<1/(2k-1) to the case where OMP/OLS have a partial knowledge of the support

Homotopy based algorithms for ℓ0\ell_0-regularized least-squares

Sparse signal restoration is usually formulated as the minimization of a quadratic cost function $\|y-Ax\|_2^2$, where A is a dictionary and x is an unknown sparse vector. It is well-known that imposing an $\ell_0$ constraint leads to an NP-hard minimization problem. The convex relaxation approach has received considerable attention, where the $\ell_0$-norm is replaced by the $\ell_1$-norm. Among the many efficient $\ell_1$ solvers, the homotopy algorithm minimizes $\|y-Ax\|_2^2+\lambda\|x\|_1$ with respect to x for a continuum of $\lambda$'s. It is inspired by the piecewise regularity of the $\ell_1$-regularization path, also referred to as the homotopy path. In this paper, we address the minimization problem $\|y-Ax\|_2^2+\lambda\|x\|_0$ for a continuum of $\lambda$'s and propose two heuristic search algorithms for $\ell_0$-homotopy. Continuation Single Best Replacement is a forward-backward greedy strategy extending the Single Best Replacement algorithm, previously proposed for $\ell_0$-minimization at a given $\lambda$. The adaptive search of the $\lambda$-values is inspired by $\ell_1$-homotopy. $\ell_0$ Regularization Path Descent is a more complex algorithm exploiting the structural properties of the $\ell_0$-regularization path, which is piecewise constant with respect to $\lambda$. Both algorithms are empirically evaluated for difficult inverse problems involving ill-conditioned dictionaries. Finally, we show that they can be easily coupled with usual methods of model order selection.Comment: 38 page

On the properties of the solution path of the constrained and penalized L2-L0 problems

12 pagesTechnical report on the properties of the L0-constrained least-square minimization problem and the L0-penalized least-square minimization problem: domain of optimization, notion of solution path, properties of the "penalized" solution path..

Approche multirésolution pour la reconstruction 3D de défaut en tomographie X

- L'objet de ce travail est la reconstruction de la forme d'un défaut compact dans un milieu homogène en tomographie X pour des applications en contrôle non destructif. Nous modélisons la forme du défaut par un polyèdre et estimons les coordonnées de ses sommets directement à partir des projections. L'estimateur choisi est le maximum a posteriori et l'algorithme proposé est de type recuit simulé ou descente itérative par coordonnées. L'initialisation se fait à l'aide d'une méthode qui estime les moments géométriques de l'objet à partir des projections. Le caractère multirésolution intervient au niveau algorithmique : nous utilisons la solution obtenue à une résolution donnée comme initialisation à la résolution plus fine suivante

Source separation approach for the analysis of spatially resolved multiply excited autofluorescence spectra during optical clearing of ex vivo skin

Spatially resolved multiply excited autofluorescence spectroscopy is a valuable optical biopsy technique to investigate skin UV-visible optical properties in vivo in clinics. However, it provides bulk fluorescence signals from which the individual endogenous fluorophore contributions need to be disentangled. Skin optical clearing allows for increasing tissue transparency, thus providing access to more accurate in-depth information. The aim of the present contribution was to study the time changes in skin spatially resolved and multiply excited autofluorescence spectra during skin optical clearing. The latter spectra were acquired on an ex vivo human skin strip lying on a fluorescent gel substrate during 37 minutes of the optical clearing process of a topically applied sucrose-based solution. A Non Negative Matrix Factorization-based blind source separation approach was proposed to unmix skin tissue intrinsic fluorophore contributions and to analyze the time evolution of this mixing throughout the optical clearing process. This spectral unmixing exploited the multidimensionality of the acquired data, i.e., spectra resolved in five excitation wavelengths, four source-to-detector separations, and eight measurement times. Best fitting results between experimental and estimated spectra were obtained for optimal numbers of 3 and 4 sources. These estimated spectral sources exhibited common identifiable shapes of fluorescence emission spectra related to the fluorescent gel substrate and to known skin intrinsic fluorophores matching namely dermis collagen/elastin and epidermis flavins. The time analysis of the fluorophore contributions allowed us to highlight how the clearing process towards the deepest skin layers impacts skin autofluorescence through time, namely with a strongest contribution to the bulk autofluorescence signal of dermis collagen (respectively epidermis flavins) fluorescence at shortest (respectively longest) excitation wavelengths and longest (respectively shortest) source-to-detector separations