arXiv subjects

Charles Soussen

Publications and source records attributed to Charles Soussen.

9 recordsLinked to original sources

arXiv2020

When does OMP achieve exact recovery with continuous dictionaries?

This paper presents new theoretical results on sparse recovery guarantees for a greedy algorithm, Orthogonal Matching Pursuit (OMP), in the context of continuous parametric dictionaries. Here, the continuous setting means that the dictionary is made up of an infinite uncountable number of atoms. In this work, we rely on the Hilbert structure of the observation space to express our recovery results as a property of the kernel defined by the inner product between two atoms. Using a continuous extension of Tropp's Exact Recovery Condition, we identify key assumptions allowing to analyze OMP in the continuous setting. Under these assumptions, OMP unambiguously identifies in exactly $k$ steps the atom parameters from any observed linear combination of $k$ atoms. These parameters play the role of the so-called support of a sparse representation in traditional sparse recovery. In our paper, any kernel and set of parameters that satisfy these conditions are said to be admissible. In the one-dimensional setting, we exhibit a family of kernels relying on completely monotone functions for which admissibility holds for any set of atom parameters. For higher dimensional parameter spaces, the analysis turns out to be more subtle. An additional assumption, so-called axis admissibility, is imposed to ensure a form of delayed recovery (in at most $k^D$ steps, where $D$ is the dimension of the parameter space). Furthermore, guarantees for recovery in exactly $k$ steps are derived under an additional algebraic condition involving a finite subset of atoms (built as an extension of the set of atoms to be recovered). We show that the latter technical conditions simplify in the case of Laplacian kernels, allowing us to derive simple conditions for $k$-step exact recovery, and to carry out a coherence-based analysis in terms of a minimum separation assumption between the atoms to be recovered.

cs.IT↗

arXiv2019

ROHSA: Regularized Optimization for Hyper-Spectral Analysis - Application to phase separation of 21 cm data

Extracting the multiphase structure of the neutral interstellar medium (ISM) is key to understand the star formation in galaxies. The radiative condensation of the diffuse warm neutral medium producing a thermally unstable lukewarm medium and a dense cold medium is closely related to the initial step which leads the HI-to-H2 transition and the formation of molecular clouds. Up to now the mapping of these phases out of 21 cm emission hyper-spectral cubes has remained elusive mostly due to the velocity blending of individual cold structures present on a given line of sight. To address this problem, we developed a new Gaussian decomposition algorithm named ROHSA (Regularized Optimization for Hyper-Spectral Analysis) based on a multi-resolution process from coarse to fine grid. ROHSA uses a regularized non-linear least-square criterion to take into account simultaneously the spatial coherence of the emission and the multiphase nature of the gas. In order to obtain a solution with spatially smooth parameters, the optimization is performed on the whole data cube at once. The performances of ROHSA were tested on a synthetic observation computed from numerical simulations of thermally bi-stable turbulence. An application on a 21 cm observation of a high Galactic latitude region from the GHIGLS survey is presented. The evaluation of ROHSA on synthetic 21 cm observations shows that it is able to recover the multiphase nature of the HI and physically meaningful information about the underlying three-dimensional fields (density, velocity and temperature). The application on a real 21 cm observation of a high Galactic latitude field produces a picture of the multiphase HI, with isolated, filamentary and narrow structures and wider, diffuse and space filling components. The test-case field used here contains significant intermediate-velocity clouds that were well mapped out by the algorithm.

astro-ph.GA↗

arXiv2016

Construction of extended 3D field of views of the internal bladder wall surface: a proof of concept

3D extended field of views (FOVs) of the internal bladder wall facilitate lesion diagnosis, patient follow-up and treatment traceability. In this paper, we propose a 3D image mosaicing algorithm guided by 2D cystoscopic video-image registration for obtaining textured FOV mosaics. In this feasibility study, the registration makes use of data from a 3D cystoscope prototype providing, in addition to each small FOV image, some 3D points located on the surface. This proof of concept shows that textured surfaces can be constructed with minimally modified cystoscopes. The potential of the method is demonstrated on numerical and real phantoms reproducing various surface shapes. Pig and human bladder textures are superimposed on phantoms with known shape and dimensions. These data allow for quantitative assessment of the 3D mosaicing algorithm based on the registration of images simulating bladder textures.

cs.CV↗

arXiv2015

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 $μ<1/(2k-1)$ (where $μ$ 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 $μ<1/k$ for strongly decaying sparse signals. Finally, in the noiseless setting, we prove that the proposed conditions, in particular the bound $μ<1/k$, are the tightest achievable guarantees based on mutual coherence.

cs.IT↗

arXiv2015

Comparative study of image registration techniques for bladder video-endoscopy

Bladder cancer is widely spread in the world. Many adequate diagnosis techniques exist. Video-endoscopy remains the standard clinical procedure for visual exploration of the bladder internal surface. However, video-endoscopy presents the limit that the imaged area for each image is about nearly 1cm2. And, lesions are, typically, spread over several images. The aim of this contribution is to assess the performance of two mosaicing algorithms leading to the construction of panoramic maps (one unique image) of bladder walls. The quantitative comparison study is performed on a set of real endoscopic exam data and on simulated data relative to bladder phantom.

cs.CV↗

arXiv2014

Homotopy based algorithms for $\ell_0$-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.

math.NA↗

arXiv2012

Joint k-step analysis of Orthogonal Matching Pursuit and Orthogonal Least Squares

Tropp's analysis of Orthogonal Matching Pursuit (OMP) using the Exact Recovery Condition (ERC) is extended to a first exact recovery analysis of Orthogonal Least Squares (OLS). We show that when the ERC is met, OLS is guaranteed to exactly recover the unknown support in at most k iterations. Moreover, we provide a closer look at the analysis of both OMP and OLS when the ERC is not fulfilled. The existence of dictionaries for which some subsets are never recovered by OMP is proved. This phenomenon also appears with basis pursuit where support recovery depends on the sign patterns, but it does not occur for OLS. Finally, numerical experiments show that none of the considered algorithms is uniformly better than the other but for correlated dictionaries, guaranteed exact recovery may be obtained after fewer iterations for OLS than for OMP.

stat.CO↗

arXiv2012

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 μof the dictionary and relaxes Tropp's well-known condition μ<1/(2k-1) to the case where OMP/OLS have a partial knowledge of the support.

cs.IT↗

arXiv2011

A sufficient condition on monotonic increase of the number of nonzero entry in the optimizer of L1 norm penalized least-square problem

The $\ell$-1 norm based optimization is widely used in signal processing, especially in recent compressed sensing theory. This paper studies the solution path of the $\ell$-1 norm penalized least-square problem, whose constrained form is known as Least Absolute Shrinkage and Selection Operator (LASSO). A solution path is the set of all the optimizers with respect to the evolution of the hyperparameter (Lagrange multiplier). The study of the solution path is of great significance in viewing and understanding the profile of the tradeoff between the approximation and regularization terms. If the solution path of a given problem is known, it can help us to find the optimal hyperparameter under a given criterion such as the Akaike Information Criterion. In this paper we present a sufficient condition on $\ell$-1 norm penalized least-square problem. Under this sufficient condition, the number of nonzero entries in the optimizer or solution vector increases monotonically when the hyperparameter decreases. We also generalize the result to the often used total variation case, where the $\ell$-1 norm is taken over the first order derivative of the solution vector. We prove that the proposed condition has intrinsic connections with the condition given by Donoho, et al \cite{Donoho08} and the positive cone condition by Efron {\it el al} \cite{Efron04}. However, the proposed condition does not need to assume the sparsity level of the signal as required by Donoho et al's condition, and is easier to verify than Efron, et al's positive cone condition when being used for practical applications.

stat.ML↗