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Original results on the global and local convergence properties of half-quadratic algorithms resulting from the Geman and Yang (GY) and Geman and Reynolds (GR) primal-dual constructions are provided. The analysis shows that convergence domain of the GY algorithm can be extended with the benefit of an improved convergence rate.

The equivalent magnetic currents method can be used to characterize radiating structures and to obtain information about the current distribution on the radiating device under test (DUT). In order to do so, the system is assumed to be linear and inversion from near-field measurements can be carried out using numerical techniques such as the singular value decomposition. However this approach is very sensitive to modeling and measurement errors, which is a consequence of the ill-conditioned nature of the inversion process. Here, we propose to account for the ill-conditioning by using a Tikhonov regularization technique to perform the inversion. This technique requires a regularization parameter to be set to an appropriate value, and we show that a generalized cross-validation approach yields adequate determination of the regularization parameter directly from the measured data. The proposed method can be tailored to account for specific characteristics of the DUT. This point is illustrated by the estimation of the current distribution of a bidimensional radiating structure for which prior information about the orientation of radiating edges is incorporated into the inversion method. Index Terms--Antenna measurements, equivalent magnetic currents (EqMC), generalized cross validation, ill-posed problems, inverse problems, near field (NF), Tikhonov regularization technique (TRT).

Computed tomography (CT) images exhibit a variable amount of noise and blur, depending on the physical characteristics of the apparatus and the selected reconstruction method. Standard algorithms tend to favor reconstruction speed over resolution, thereby jeopardizing applications where accuracy is critical. In this paper, we propose to enhance CT images by applying half-quadratic edge-preserving image restoration (or deconvolution) to them. This approach may be used with virtually any CT scanner, provided the overall point-spread function can be roughly estimated. In image restoration, Markov random fields (MRFs) have proven to be very flexible a priori models and to yield impressive results with edge-preserving penalization, but their implementation in clinical routine is limited because they are often viewed as complex and time consuming. For these practical reasons, we focused on numerical efficiency and developed a fast implementation based on a simple three-dimensional MRF model with convex edge-preserving potentials. The resulting restoration method provides good recovery of sharp discontinuities while using convex duality principles yields fairly simple implementation of the optimization. Further reduction of the computational load can be achieved if the point-spread function is assumed to be separable. Synthetic and real data experiments indicate that the method provides significant improvements over standard reconstruction techniques and compares well with convex-potential Markov-based reconstruction, while being more flexible and numerically efficient. Index Terms--Edge-preserving convex potentials, half-quadratic criteria, Markov random fields, three-dimensional image reconstruction, three-dimensional image restoration, x-ray tomography.

This paper describes a new approach to reconstruction of the conductivity field in electrical impedance tomography. Our goal is to improve the tradeoff between the quality of the images and the numerical complexity of the reconstruction method. In order to reduce the computational load, we adopt a linearized approximation to the forward problem that describes the relationship between the unknown conductivity and the measurements. In this framework, we focus on finding a proper way to cope with the ill-posed nature of the problem, mainly caused by strong attenuation phenomena; this is done by devising regularization techniques well suited to this particular problem. First, we propose a solution which is based on Tikhonov regularization of the problem. Second, we introduce an original regularized reconstruction method in which the regularization matrix is determined by space-uniformization of the variance of the reconstructed condictivities. Both methods are nonsupervised, i.e., all tuning parameters are automatically determined from the measured data. Tests performed on simulated and real data indicate that Tikhonov regularization provides results similar to those obtained with iterative methods, but with a much smaller amount of computations. Regularization using a variance uniformization constraint yields further improvements, particularly in the central region of the unknown object where attenuation is most severe. We anticipate that the variance uniformization approach could be adapted to iterative methods that preserve the nonlinearity of the forward problem. More generally, it appears as a useful tool for solving other severely ill-posed reconstruction problems such as eddy current tomography. Index Terms - Electrical impedance tomography, ill-posed inverse problem, reconstruction, regularization.

A stochastic version of the expectation-maximization algorithm (SEM) allows efficient determination of hyperparameters in an unsupervised restoration of an efficient sparse spike trains. A core algorithm used for supervised deconvolution allows effective implementation of the SEM method. Comparison of the statistical behavior of SEM with the classical maximum generalized likelihood (MGL) approach for unsupervised deconvolution shows the superiority of SEM over MGL.

This paper deals with Bayesian estimation of two-dimensional (2-D) stratified structures from echosounding signals. This problem is of interest in seismic exploration, but also for non-destructive testing or medical imaging. The proposed approach consists of a multichannel Bayesian deconvolution method of the 2-D reflectivity based upon a theoretically sound prior stochastic model. The Markov-Bernoulli random field representation introduced in (1) is used to model the geometric properties of the reflectivity, and the emphasis is placed on representation of the amplitudes and on deconvolution algorithms. It is shown that the algorithmic structure and computational complexity of the proposed multichannel methods are similar to those of single-channel B-G deconvolution procedures, but that explicit modeling of the stratified structure results in significantly better performances. Simulation results and examples of real-data processing illustrate the performances and the practicality of the multichannel approach.

Bayesian estimation of two-dimensional stratified structures is described. The major point addressed here is the derivation of a statistical prior model that adequately describes such layered media. This problem is of interest in application domains such as seismic exploration, nondestructive testing, and medical imaging, where the data are generally processed in one dimension only. In order to take local interactions into account, a Markovian description is used. The model is derived so as to fulfill a set of constraints which summarize physical and geometrical characteristics of the problem as well as practical requirements. The approach adopted is reminiscent of the work of Pickard. The resulting class of Markov random fields presents a unilateral structure on a nonrectangular lattice and a hierarchical organization which involves a line process. In addition, it is shown to be an extension of one-dimensional models already used in the application domains previously mentioned. After an investigation of the properties of the model, its practical interest is demonstrated by an application to seismic deconvolution. Simulation results show significant improvements with respect to the usual one-dimensional methods.

A proposed method for improving the numerical stability of the downdating operation of a Cholesky factor is an extended version of the algorithm to derive stabilized square-root forms of standard linear least square estimation. Results show that a linear prediction problem confirms the gain in numerical stability which involves a relatively small additional computational cost.

Magnetic induction tomography aims to reconstruct the passive electric properties of an object by measuring its scattered magnetic field. Current state-ofthe- art numerical techniques are based on differential formulations such as the finite element method. A formulation based on volume integral equations has not yet been applied to its biomedical field and could improve the reconstruction speed by reducing the number of unknowns. This paper investigates salient characteristics of the approach and offers a solution based on inhomogeneous Green's functions. [ABSTRACT FROM AUTHOR]