Yves Goussard

Regularized reconstruction in electrical impedance tomography using a variance uniformization constraint

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 conductivities. 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.

On global and local convergence of half-quadratic algorithms

This paper provides original results on the global and local convergence properties of half-quadratic (HQ) algorithms resulting from the Geman and Yang (GY) and Geman and Reynolds (GR) primal-dual constructions. First, we show that the convergence domain of the GY algorithm can be extended with the benefit of an improved convergence rate. Second, we provide a precise comparison of the convergence rates for both algorithms. This analysis shows that the GR form does not benefit from a better convergence rate in general. Moreover, the GY iterates often take advantage of a low cost implementation. In this case, the GY form is usually faster than the GR form from the CPU time viewpoint.

Multichannel seismic deconvolution

Deals with Bayesian estimation of 2D stratified structures from echosounding signals. This problem is of interest in seismic exploration, but also for nondestructive testing or medical imaging. The proposed approach consists of a multichannel Bayesian deconvolution method of the 2D reflectivity based upon a theoretically sound prior stochastic model. The Markov-Bernoulli random field representation introduced by Idier et al. (1993) is used to model the geometric properties of the reflectivity, and 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. >

A new algorithm for iterative deconvolution of sparse spike trains

An iterative algorithm for deconvolution of Bernoulli-Gaussian processes is presented. This detection-estimation problem is formulated as that of a change of initial conditions in linear least-squares estimation. An algorithm with a very simple structure is obtained. It allows the evaluation of either marginal or joint likelihood criteria without any approximation; the resulting method is easy to implement and computationally inexpensive and remains nearly optimal.<<ETX>>

Three-dimensional edge-preserving image enhancement for computed tomography

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.

GCV and ML Methods of Determining Parameters in Image Restoration by Regularization: Fast Computation in the Spatial Domain and Experimental Comparison

Abstract Many linear image restoration methods minimize a compound criterion which balances some fidelity to the observed data via a least-squares measure, and some fidelity to prior information on the unknown object via a smoothing function. In the case of quadratic criteria, this regularization scheme can be interpreted as a Bayesian estimation of an object which is modeled as a Gaussian random field. The global degree of regularization is controlled by a scalar smoothing parameter and by the prior covariance matrix of the object. These quantities are usually unknown and should also be determined from the observed image. Two ways of choosing these parameters are discussed and compared on both synthetic and real images: maximum likelihood (ML) and generalized cross-validation (GCV). Particular attention is paid to implementation problems. Both criteria are evaluated in the spatial domain using fast pixel-recursive techniques. Results show that GCV is more robust than ML with respect to modeling assumptions, and should therefore be preferred in real-world applications.

Time-recursive solution to the inverse problem of electrocardiography: a model-based approach

This paper presents a new approach to the estimation of epicardial potentials from measured body surface potentials. This problem is ill-posed, and regularization techniques, through incorporation of a priori information on the solution, provide an efficient way of improving the quality of the estimates. During a cardiac cycle, the time-evolution of the epicardial potentials presents a very structured character which is the consequence of underlying propagation phenomena. In order to account for this a priori information, we introduce a linear prediction model that explicitly relates the epicardial potentials at two consecutive time instants. The linear prediction model, along with the usual relationship between epicardial and body surface potentials, makes up a state-space representation of the system. Epicardial potentials can then be estimated using efficient time-recursive techniques such as Kalman filtering. Simulation results obtained with real epicardial data indicate the validity of the linear prediction model; comparison of the reconstructed epicardial potentials with those produced by existing methods confirm the interest of the ap-

Reduction of Beam-Hardening Artifacts in X-Ray CT

Our goal is to decrease the importance of beam-hardening artifacts in X-ray computed tomography by addressing the polyenergetic nature of the X-ray source. We use the same physical model as De Man and al (1998). We next adopt an estimation framework for the reconstruction: the attenuation coefficients are determined by a likelihood-based estimator. This approach leads to minimization of an objective function which exhibits a complex structure. Nonetheless, we develop a numerical procedure with satisfactory numerical efficiency: we use a nonlinear conjugate gradient method. The three major contributions of this communication are: the positivity of the solution ensured by a change of variables, the convergence properties of the algorithm, and a satisfying computation time

Mapping of Equivalent Currents on High-Speed Digital Printed Circuit Boards Based on Near-Field Measurements

In this paper, a method to build equivalent models of radiating printed circuit boards based on complex E-field measurements taken in the close vicinity of the device under test is explored. It is shown that the inverse problem to be solved to retrieve the currents from the field data is ill-posed. An innovative regularization approach implementing a penalty on abrupt spatial variations of the currents is proposed to alleviate this difficulty. Various schemes to mesh the equivalent current distribution are also explored. Combining these with measurements, it is shown that accurate estimation of equivalent current models can be achieved, thereby allowing the identification of the emission sources. The method is tested for different circuit configurations with both synthetic and real data. Obtained results demonstrate the efficiency of the method.

Time-space regularization of the inverse problem of electrocardiography

Our goal is to reconstruct the epicardial potentials (EPs) from measured body surface potentials. This non-invasive technique is useful for the diagnostics of cardio-vascular diseases. In our approach, the time-correlation between EPs is used as the regularizing a priori information. This information is introduced via the state-space representation previously proposed. Our contribution is the derivation of maximum likelihood estimators for identification of the parameters of the state-space model, and in a second stage, for determination of the EPs. Therefore, all unknown quantities are determined from the only available data: the body surface potentials.