Charles Soussen

Automated Force Volume Image Processing for Biological Samples

Atomic force microscopy (AFM) has now become a powerful technique for investigating on a molecular level, surface forces, nanomechanical properties of deformable particles, biomolecular interactions, kinetics, and dynamic processes. This paper specifically focuses on the analysis of AFM force curves collected on biological systems, in particular, bacteria. The goal is to provide fully automated tools to achieve theoretical interpretation of force curves on the basis of adequate, available physical models. In this respect, we propose two algorithms, one for the processing of approach force curves and another for the quantitative analysis of retraction force curves. In the former, electrostatic interactions prior to contact between AFM probe and bacterium are accounted for and mechanical interactions operating after contact are described in terms of Hertz-Hooke formalism. Retraction force curves are analyzed on the basis of the Freely Jointed Chain model. For both algorithms, the quantitative reconstruction of force curves is based on the robust detection of critical points (jumps, changes of slope or changes of curvature) which mark the transitions between the various relevant interactions taking place between the AFM tip and the studied sample during approach and retraction. Once the key regions of separation distance and indentation are detected, the physical parameters describing the relevant interactions operating in these regions are extracted making use of regression procedure for fitting experiments to theory. The flexibility, accuracy and strength of the algorithms are illustrated with the processing of two force-volume images, which collect a large set of approach and retraction curves measured on a single biological surface. For each force-volume image, several maps are generated, representing the spatial distribution of the searched physical parameters as estimated for each pixel of the force-volume image.

From Bernoulli–Gaussian Deconvolution to Sparse Signal Restoration

Formulated as a least square problem under an l0 constraint, sparse signal restoration is a discrete optimization problem, known to be NP complete. Classical algorithms include, by increasing cost and efficiency, matching pursuit (MP), orthogonal matching pursuit (OMP), orthogonal least squares (OLS), stepwise regression algorithms and the exhaustive search. We revisit the single most likely replacement (SMLR) algorithm, developed in the mid-1980s for Bernoulli-Gaussian signal restoration. We show that the formulation of sparse signal restoration as a limit case of Bernoulli-Gaussian signal restoration leads to an l0-penalized least square minimization problem, to which SMLR can be straightforwardly adapted. The resulting algorithm, called single best replacement (SBR), can be interpreted as a forward-backward extension of OLS sharing similarities with stepwise regression algorithms. Some structural properties of SBR are put forward. A fast and stable implementation is proposed. The approach is illustrated on two inverse problems involving highly correlated dictionaries. We show that SBR is very competitive with popular sparse algorithms in terms of tradeoff between accuracy and computation time.

Polygonal and polyhedral contour reconstruction in computed tomography

This paper is about three-dimensional (3-D) reconstruction of a binary image from its X-ray tomographic data. We study the special case of a compact uniform polyhedron totally included in a uniform background and directly perform the polyhedral surface estimation. We formulate this problem as a nonlinear inverse problem using the Bayesian framework. Vertice estimation is done without using a voxel approximation of the 3-D image. It is based on the construction and optimization of a regularized criterion that accounts for surface smoothness. We investigate original deterministic local algorithms, based on the exact computation of the line projections, their update, and their derivatives with respect to the vertice coordinates. Results are first derived in the two-dimensional (2-D) case, which consists of reconstructing a 2-D object of deformable polygonal contour from its tomographic data. Then, we investigate the 3-D extension that requires technical adaptations. Simulation results illustrate the performance of polygonal and polyhedral reconstruction algorithms in terms of quality and computation time.

Joint K-Step Analysis of Orthogonal Matching Pursuit and Orthogonal Least Squares

Tropps 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 where k denotes the support cardinality. 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.

Exact Recovery Conditions for Sparse Representations With Partial Support Information

We address the exact recovery of a k-sparse vector in the noiseless setting when some partial information on the support is available. This partial information takes the form of either a subset of the true support or an approximate subset including wrong atoms as well. We derive a new sufficient and worst-case necessary (in some sense) condition for the success of some procedures based on ℓp-relaxation, orthogonal matching pursuit (OMP), and orthogonal least squares (OLS). Our result is based on the coherence μ of the dictionary and relaxes the well-known condition μ <; 1/2k - 1) ensuring the recovery of any k-sparse vector in the noninformed setup. It reads μ <; 1/(2k - g + b - 1) when the informed support is composed of g good atoms and b wrong atoms. We emphasize that our condition is complementary to some restricted-isometry-based conditions by showing that none of them implies the other. Because this mutual coherence condition is common to all procedures, we carry out a finer analysis based on the null space property (NSP) and the exact recovery condition (ERC). Connections are established regarding the characterization of ℓp-relaxation procedures and OMP in the informed setup. First, we emphasize that the truncated NSP enjoys an ordering property when p is decreased. Second, the partial ERC for OMP (ERC-OMP) implies in turn the truncated NSP for the informed ℓ1 problem, and the truncated NSP for p <; 1 .

Flexible calibration of structured-light systems projecting point patterns

Structured-light systems (SLSs) are widely used in active stereo vision to perform 3D modelling of a surface of interest. We propose a flexible method to calibrate SLSs projecting point patterns. The method is flexible in two respects. First, the calibration is independent of the number of points and their spatial distribution inside the pattern. Second, no positioning device is required since the projector geometry is determined in the camera coordinate system based on unknown positions of the calibration board. The projector optical center is estimated together with the 3D rays originating from the projector using a numerical optimization procedure. We study the 3D point reconstruction accuracy for two SLSs involving a laser based projector and a pico-projector, respectively, and for three point patterns. We finally illustrate the potential of our active vision system for a medical endoscopy application where a 3D cartography of the inspected organ (a large field of view surface also including image textures) can be reconstructed from a video acquisition using the laser based SLS.

Sparse multidimensional modal analysis using a multigrid dictionary refinement

We address the problem of multidimensional modal estimation using sparse estimation techniques coupled with an efficient multigrid approach. Modal dictionaries are obtained by discretizing modal functions (damped complex exponentials). To get a good resolution, it is necessary to choose a fine discretization grid resulting in intractable computational problems due to the huge size of the dictionaries. The idea behind the multigrid approach amounts to refine the dictionary over several levels of resolution. The algorithm starts from a coarse grid and adaptively improves the resolution in dependence of the active set provided by sparse approximation methods. The proposed method is quite general in the sense that it allows one to process in the same way mono-and multidimensional signals. We show through simulations that, as compared to high-resolution modal estimation methods, the proposed sparse modal method can greatly enhance the estimation accuracy for noisy signals and shows good robustness with respect to the choice of the number of components.

Fast Positive Deconvolution of Hyperspectral Images

In this brief, we provide an efficient scheme for performing deconvolution of large hyperspectral images under a positivity constraint, while accounting for spatial and spectral smoothness of the data.

Homotopy based algorithms for L0-regularized least-squares

Sparse signal restoration is usually formulated as the minimization of a quadratic cost function |y-Ax ||2² where \mbi A is a dictionary and \mbi x is an unknown sparse vector. It is well-known that imposing an ℓ₀ constraint leads to an NP-hard minimization problem. The convex relaxation approach has received considerable attention, where the ℓ₀-norm is replaced by the ℓ₁-norm. Among the many effective ℓ₁ solvers, the homotopy algorithm minimizes ||y-Ax ||2²+λ||x||₁ with respect to x for a continuum of λs. It is inspired by the piecewise regularity of the ℓ₁-regularization path, also referred to as the homotopy path. In this paper, we address the minimization problem ||y-Ax||2²+λ||x||₀ for a continuum of λs and propose two heuristic search algorithms for ℓ₀-homotopy. Continuation Single Best Replacement is a forward-backward greedy strategy extending the Single Best Replacement algorithm, previously proposed for ℓ₀-minimization at a given λ. The adaptive search of the λ-values is inspired by ℓ₁-homotopy. ℓ₀ Regularization Path Descent is a more complex algorithm exploiting the structural properties of the ℓ₀-regularization path, which is piecewise constant with respect to λ. 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.

Ultrasonic non destructive testing based on sparse deconvolution

The acoustic modality yields non destructive testing techniques of choice for in-depth investigation. Given a precise model of acoustic wave propagation in materials of possibly complex structures, acoustical imaging amounts to the so-called acoustic wave inversion. A less ambitious approach consists in processing pulse-echo data (typically, A- or B-scans) to detect localised echoes with the maximum temporal (and lateral) precision. This is a resolution enhancement problem, and more precisely a sparse deconvolution problem which is naturally addressed in the inversion framework. The paper focuses on the main sparse deconvolution methods and algorithms, with a view to apply them to ultrasonic non-destructive testing.