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This paper considers the microwave imaging reconstruction problem, based on additive penalization and gradient-based optimization. Each evaluation of the cost function and of its gradient requires the resolution of as many high-dimensional linear systems as the number of incident fields, which represents a large amount of computations. Since all such systems involve the same matrix, we propose a block inversion strategy, based on the block-biconjugate gradient stabilized (BiCGStab) algorithm, with efficient implementations specific to the microwave imaging context. Numerical experiments performed on synthetic data and on real measurements show that savings in computing time can reach a factor of two compared to the standard, sequential, BiCGStab implementation. Improvements brought by the block approach are even more important for the most difficult reconstruction problems, that is, with high-frequency illuminations and/or highly contrasted objects. The proposed reconstruction strategy is shown to achieve satisfactory estimates for objects of the Fresnel database, even on the most contrasted ones.

Ultrasonic inspection of coarse-grained steels is a common challenge in various industrial fields. This task is often difficult because of acoustic scattering that creates structural noise in the ultrasonic signals and images. Therefore, inspections usually use low-frequency probes, which achieve poor resolution with standard delay-and-sum (DAS) imaging techniques, such as the well-known total focusing method (TFM). The purpose of this paper is to evaluate the performance of a super-resolution ultrasonic imaging technique presented by Laroche et al. (IEEE Trans Comput Imaging 7:935–947, 2021) for the inspection of industrial coarse-grained materials. An image deconvolution problem (with spatially varying blur) is formulated, relying on a forward model that links the TFM image to the acoustic reflectivity map. The experiments consider an austenitic-ferritic stainless steel sample insonified using array probes at 3 MHz and 5 MHz placed in contact. The goal is to resolve two close reflectors corresponding to side-drilled holes (SDH) with diameter 0.4 mm spaced by 0.4 mm edge-to-edge and positioned at different depths (10, 20, 30, 40 mm). This configuration corresponds to a critical case where the distance between the two reflectors is much lower than the Rayleigh distance, that is the resolution limit of a DAS imaging system. These are typical cases where DAS images obtained from low-frequency inspections critically lack resolution, but where higher frequency probes cannot be used in practice, because a too week signal-to-noise ratio would affect the detection capability. As predicted by the Rayleigh criterion, TFM is not able to separate the reflectors. The proposed image reconstruction method successfully resolves the majority of the reflectors with a rather accurate distance estimation. In the context of coarse-grained structure inspection, subwavelength reflectors distant from each other by two times less than the resolution limit given by the Rayleigh criterion have been successfully detected and separated. This approach hence enables the use of low-frequency probes, in order to improve the signal-to-noise ratio, while keeping high resolution capability which can be particularly interesting for industrial applications. In particular, the proposed approach shows promising results for the sizing of a real crack in an industrial sample.

In 3-D microwave imaging, gradient-based optimization algorithms usually make use of the so-called stabilized version of the biconjugate gradient iterative method (BiCGStab) in order to solve multiple linear systems. We propose to use a block version of BiCGStab to jointly solve the mutiple right-hand side linear systems. Illuminations are partitioned in subgroups, which makes the method more efficient. The reconstruction process is studied on realistic simulated data and illustrates the efficiency of the method compared to BiCGStab.

Ultrasonic inspection of coarse-grained steels is a common challenge in various industrial fields. This task is often difficult because of acoustic scattering that creates structural noise in the ultrasonic signals and images. This drives inspections using low-frequency probes at the cost of a lower resolution of standard delay and sum (DAS) imaging techniques, such as the well-known total focusing method (TFM). The purpose of this paper is to present and evaluate the performances of an image reconstruction technique that aims at improving the resolution when inspecting industrial coarse-grained materials. An image deconvolution problem (with spatially varying blur) is formulated, relying on a forward model that links the TFM image to the acoustic reflectivity map. A particular attention is paid to the estimation of the PSF used for the deconvolution approach in an experimental context. The experiments are based on an austenitic-ferritic sample insonified using array probes at 3 MHz and 5 MHz placed in contact. The goal is to resolve two close reflectors corresponding to side drilled holes (SDH) with diameter 0.4 mm spaced by 0.4 mm edge to edge and positioned at different depths (10, 20, 30, 40 mm). This configuration corresponds to a critical case where the distance between the two reflectors is significantly inferior to the Rayleigh distance, that is the resolution limit of a DAS imaging system. These are typical cases where the employed frequency is actually too low and where a higher frequency probe should be used, which is not possible in practice, because it would affect the detection capability due to higher noise level. As predicted by the Rayleigh criterion, TFM is not able to separate the reflectors. The proposed image reconstruction method successfully resolves the majority of the reflectors with a rather accurate distance estimation. In the context of coarse-grained structure inspection, this approach enables the use of low-frequency probes, in order to improve the signal-to-noise ratio, while keeping high resolution capability.

In this paper, we propose a new greedy algorithm for sparse approximation, called SLS for Single L_1 Selection. SLS essentially consists of a greedy forward strategy, where the selection rule of a new component at each iteration is based on solving a least-squares optimization problem, penalized by the L_1 norm of the remaining variables. Then, the component with maximum amplitude is selected. Simulation results on difficult sparse deconvolution problems involving a highly correlated dictionary reveal the efficiency of the method, which outperforms popular greedy algorithms and Basis Pursuit Denoising when the solution is sparse.

This chapter discusses different approaches to the estimation of sinusoidal oscillations, formalized as a deconvolution problem in the Fourier domain. It formalizes the relationship between the estimation of spectral lines and sparse approximation, insisting on the specificities of the model, caused notably by the irregular nature of the sampling scheme. The chapter analyzes and compares several iterative (greedy) methods, and presents techniques based on the principle of the CLEAN algorithm, frequently used in astronomy. It examines in a thorough fashion the estimation by means of the minimization of a least squares criterion penalized by the l 1 ‐norm. Finally, the chapter focuses on the probabilistic modeling of sparsity by a Bernoulli–Gaussian (BG) model. With regard to these approaches, which represent the standards of sparse approximation methods, a more thorough exploitation of the data is proposed using Markov chain Monte‐Carlo algorithms (MCMC), at the expense of a higher computational cost.

Speckle based imaging consists of forming a super-resolved reconstruction of an unknown sample from low-resolution images obtained under random inhomogeneous illuminations (speckles). In a blind context where the illuminations are unknown, we study the intrinsic capacity of speckle-based imagers to recover spatial frequencies outside the frequency support of the data, with minimal assumptions about the sample. We demonstrate that, under physically realistic conditions, the covariance of the data has a super-resolution power corresponding to the squared magnitude of the imager point spread function. This theoretical result is important for many practical imaging systems such as acoustic and electromagnetic tomographs, fluorescence and photoacoustic microscopes, or synthetic aperture radar imaging. A numerical validation is presented in the case of fluorescence microscopy.

The blind structured illumination microscopy (SIM) strategy proposed in (Mudry et al., 1992) is fully re-founded in this paper, unveiling the central role of the sparsity of the illumination patterns in the mechanism that drives super-resolution in the method. A numerical analysis shows that the resolving power of the method can be further enhanced with optimized one-photon or two-photon speckle illuminations. A much improved numerical implementation is provided for the reconstruction problem under the image positivity constraint. This algorithm rests on a new preconditioned proximal iteration faster than existing solutions, paving the way to 3D and real-time 2D reconstruction

The Hapke model has been widely used to describe the photometrical behavior of planetary surface through the Bi-directional Reflectance Distribution Function (BRDF), but the uncertainties about retrieved parameters has been difficult to handle so far. A recent study proposed to estimate the uncertainties using a Bayesian approach (Schmidt et al., Icarus 2015). In the present article, we first propose an improved numerical implementation to speed up the uncertainties estimation. Then, we conduct two synthetic studies about photometric measurements in order to analyze the influence of observation geometry: First, we introduce the concept of \"efficiency\" of a set of geometries to sample the photometric behavior. A set of angular sampling elements (noted as geometry) is efficient if the retrieved Hapke parameters are close to the expected ones. We compared different geometries and found that the principal plane with high incidence is the most efficient geometry among the tested ones. In particular, such geometries are better than poorly sampled full BRDF. Second, we test the analysis scheme of a collection of photometric data acquired from various locations in order to answer the question: are these locations photometrically homogeneous or not? For instance, this question arises when combining data from an entire planetary body, where each spatial position is sampled at a single geometry. We tested the ability of the Bayesian method to decipher two situations, in the presence of noise: (i) a photometrically homogeneous surface (all observations with the same photometric behavior), or (ii) an heterogeneous surface with two distinct photometrical properties (half observations with photometric behavior 1, other half with photometric behavior 2). We show that the naive interpretation of the results provided by Bayesian method is not able to solve this problem.