MCMC and variational approaches for Bayesian inversion in diffraction imaging

International audienceThe term “diffraction imaging” is meant, herein, in the sense of an “inverse scattering problem” where the goal is to build up an image of an unknown object from measurements of the scattered field that results from its interaction with a known probing wave. This type of problem occurs in many imaging and non-destructive testing applications. It corresponds to the situation where looking for a good trade-off between the image resolution and the penetration of the incident wave in the probed medium, leads to choosing the frequency of the latter in such a way that its wavelength lies in the “resonance” domain, in the sense that it is approximately of the same order of magnitude as the characteristic dimensions of the inhomogeneities of the inspected object. In this situation the wave-object interaction gives rise to important diffraction phenomena. This is the case for the two applications considered herein, where the interrogating waves are electromagnetic waves with wavelengths in the microwave and optical domains, whereas the characteristic dimensions of the sought object are 1 cm and 1 μm, respectively.The solution of an inverse problem obviously requires previous construction of a forward model that expresses the scattered field as a function of the parameters of the sought object. In this model, diffraction phenomena are taken into account by means of domain integral representations of the electric fields. The forward model is then described by two coupled integral equations, whose discrete versions are obtained using a method of moments and whose inversion leads to a non-linear problem.Concerning inversion, at the beginning of the 1980s, accounting for the diffraction phenomena has been the subject of much attention in the field of acoustic imaging for applications in geophysics, non-destructive testing or biomedical imaging. It led to techniques such as diffraction tomography, a term that denotes “applications that employs diffracting wavefields in the tomographic reconstruction process” , but which generally implies reconstruction processes based on the generalized projection-slice theorem, an extension to the diffraction case of the projection-slice theorem of the classical computed tomography whose forward model is given by a Radon transform . This theorem is based upon first- order linearizing assumptions such as the Born’s or Rytov’s approximations. So, the term diffraction tomography was paradoxically used to describe reconstruction techniques adapted to weakly scattering environments that do not provide quantitative information on highly contrasted dielectric objects such as those encountered in the applications considered herein, where multiple diffraction cannot be ignored.Furthermore, the resolution of these techniques is limited because evanescent waves are not taken into consideration. These limitations have led researchers to develop inversion algorithms able to deal with non-linear problems, at the beginning of the 1990s for microwave imaging and more recently for optical imaging. Many studies have focused on the development of deterministic methods, such as the Newton-Kantorovich algorithm, the modified gradient method (MGM) or the contrast-source inversion technique (CSI), where the solution is sought for by means of an iterative minimization by a gradient method of a cost functional that expresses the difference between the scattered field and the estimated model output. But, in addition to be non-linear, inverse scattering problems are also known to be ill-posed, which means that their resolution requires a regularization which generally consists in introducing prior information on the sought object. In the present case, for example, we look for man-made objects that are composed of homogeneous and compact regions made of a finite number of different materials, and with the aforementioned deterministic methods, it is not easy to take into account such prior information because it must be introduced into the cost functional to be minimized.On the contrary, the probabilistic framework of Bayesian estimation, basis of the model presented herein, is especially well suited for this situation. Prior information is appropriately introduced via a probabilistic Gauss-Markov-Potts model. The marginal contrast distribution is modeled as a mixture of Gaussians, where each Gaussian distribution represents a class of materials and the compactness of the regions is taken into account using a hidden Markov model. Estimation of the unknowns and parameters introduced into the prior model is performed via an unsupervised joint approach.Two iterative algorithms are proposed. The first one, denoted as the MCMC algorithm (Monte-Carlo Markov Chain), is rather classic ; it consists in expressing all the joint posterior or conditional distributions of all the unknowns and, then, using a Gibbs sampling algorithm for estimating the posterior mean of the unknowns. This algorithm yields good results, however, it is computationally intensive mainly because Gibbs sampling requires a significant number of samples.The second algorithm is based upon the variational Bayesian approximation (VBA). The latter was first introduced in the field of Bayesian inference for applications to neural networks, learning graphic models and model parameter estimation. Its appearance in the field of inverse problems is relatively recent, starting with source separation and image restoration. It consists in approximating the joint posterior distribution of all the unknowns by a free-form separable distribution that minimizes, with respect to the posterior law, the Kullback-Leibler divergence which has interesting properties for optimization and leads to an implicit parametric optimization scheme. Once the approximate distribution is built up, the estimator can be easily obtained.A solution to this functional optimization problem can be found in terms of exponential distributions whose shape parameters are estimated iteratively. It can be noted that, at each iteration, the updating expression for these parameters is similar to the one that could be obtained if a gradient method was used to solve the optimization problem. Moreover, the gradient and the step size have an interpretation in terms of statistical moments (means, variances, etc.).Both algorithms introduced herein are applied to two quite different configurations. The one related to microwave imaging is quasi-optimal: data are quasi-complete and frequency diverse. This means that the scattered fields are measured all around the object for several directions of illumination and several frequencies. The configuration used in optical imaging is less favorable since only aspect-limited data are available at a single frequency. This means that illuminations and measurements can only be performed in a limited angular sector. This limited aspect reinforces the ill-posedness of the inverse problem and makes essential the introduction of prior information. However, it will be shown that, in both cases, satisfactory results are obtained

Microwave tomography for breast cancer detection within a Variational Bayesian Approach

International audienceWe consider a nonlinear inverse scattering problem where the goal is to detect breast cancer from measurements of the scattered field that results from its interaction with a known wave in the microwave frequency range. The modeling of the wave-object interaction is tackled through a domain integral representation of the electric field in a 2D-TM configuration. The inverse problem is solved in a Bayesian framework where the prior information is introduced via a Gauss-Markov-Potts model. A Variational Bayesian Approximation (VBA) technique is adapted to complex valued contrast and applied to compute the posterior estimators and reconstruct maps of both the permittivity and conductivity. Results obtained by means of this approach from synthetic data are compared with those given by a deterministic contrast source inversion method

Variational Bayesian inversion for microwave breast imaging

International audienceMicrowave imaging is considered as a nonlinear inverse scattering problem and tackled in a Bayesian estimation framework. The object under test (a breast affected by a tumor) is assumed to be composed of compact regions made of a restricted number of different homogeneous materials. This a priori knowledge is defined by a Gauss-Markov-Potts distribution. First, we express the joint posterior of all the unknowns; then, we present in detail the variational Bayesian approximation used to compute the estimators and reconstruct both permittivity and conductivity maps. This approximation consists of the best separable probability law that approximates the true posterior distribution in the Kullback-Leibler sense. This leads to an implicit parametric optimization scheme which is solved iteratively. Some preliminary results, obtained by applying the proposed method to synthetic data, are presented and compared with those obtained by means of the classical contrast source inversion method

Variational Bayesian inversion for microwave breast imaging

International audienceMicrowave imaging is considered as a nonlinear inverse scattering problem and tackled in a Bayesian estimation framework. The object under test (a breast affected by a tumor) is assumed to be composed of compact regions made of a restricted number of different homogeneous materials. This a priori knowledge is defined by a Gauss-Markov-Potts distribution. First, we express the joint posterior of all the unknowns; then, we present in detail the variational Bayesian approximation used to compute the estimators and reconstruct both permittivity and conductivity maps. This approximation consists of the best separable probability law that approximates the true posterior distribution in the Kullback-Leibler sense. This leads to an implicit parametric optimization scheme which is solved iteratively. Some preliminary results, obtained by applying the proposed method to synthetic data, are presented and compared with those obtained by means of the classical contrast source inversion method

Binary Stars in the Orion Nebula Cluster

We report on a high-spatial-resolution survey for binary stars in the periphery of the Orion Nebula Cluster, at 5 - 15 arcmin (0.65 - 2 pc) from the cluster center. We observed 228 stars with adaptive optics systems, in order to find companions at separations of 0.13" - 1.12" (60 - 500 AU), and detected 13 new binaries. Combined with the results of Petr (1998), we have a sample of 275 objects, about half of which have masses from the literature and high probabilities to be cluster members. We used an improved method to derive the completeness limits of the observations, which takes into account the elongated point spread function of stars at relatively large distances from the adaptive optics guide star. The multiplicity of stars with masses >2 M_sun is found to be significantly larger than that of low-mass stars. The companion star frequency of low-mass stars is comparable to that of main-sequence M-dwarfs, less than half that of solar-type main-sequence stars, and 3.5 to 5 times lower than in the Taurus-Auriga and Scorpius-Centaurus star-forming regions. We find the binary frequency of low-mass stars in the periphery of the cluster to be the same or only slightly higher than for stars in the cluster core (<3 arcmin from theta1C Ori). This is in contrast to the prediction of the theory that the low binary frequency in the cluster is caused by the disruption of binaries due to dynamical interactions. There are two ways out of this dilemma: Either the initial binary frequency in the Orion Nebula Cluster was lower than in Taurus-Auriga, or the Orion Nebula Cluster was originally much denser and dynamically more active.Comment: 20 page

Wave operator bounds for 1-dimensional Schr\"odinger operators with singular potentials and applications

Boundedness of wave operators for Schr\"odinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive estimates and commutator bounds.Comment: 16 pages, 0 figure

Asymptotic models for the generation of internal waves by a moving ship, and the dead-water phenomenon

This paper deals with the dead-water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. More generally, we study the generation of internal waves by a disturbance moving at constant speed on top of two layers of fluids of different densities. Starting from the full Euler equations, we present several nonlinear asymptotic models, in the long wave regime. These models are rigorously justified by consistency or convergence results. A careful theoretical and numerical analysis is then provided, in order to predict the behavior of the flow and in which situations the dead-water effect appears.Comment: To appear in Nonlinearit

The EU and Asia within an evolving global order: what is Europe? Where is Asia?

The papers in this special edition are a very small selection from those presented at the EU-NESCA (Network of European Studies Centres in Asia) conference on "the EU and East Asia within an Evolving Global Order: Ideas, Actors and Processes" in November 2008 in Brussels. The conference was the culmination of three years of research activity involving workshops and conferences bringing together scholars from both regions primarily to discuss relations between Europe and Asia, perceptions of Europe in Asia, and the relationship between the European regional project and emerging regional forms in Asia. But although this was the last of the three major conferences organised by the consortium, it in many ways represented a starting point rather than the end; an opportunity to reflect on the conclusions of the first phase of collaboration and point towards new and continuing research agendas for the future

Gas and dust in the Beta Pictoris Moving Group as seen by the Herschel Space Observatory

Context. Debris discs are thought to be formed through the collisional grinding of planetesimals, and can be considered as the outcome of planet formation. Understanding the properties of gas and dust in debris discs can help us to comprehend the architecture of extrasolar planetary systems. Herschel Space Observatory far-infrared (IR) photometry and spectroscopy have provided a valuable dataset for the study of debris discs gas and dust composition. This paper is part of a series of papers devoted to the study of Herschel PACS observations of young stellar associations. Aims. This work aims at studying the properties of discs in the Beta Pictoris Moving Group (BPMG) through far-IR PACS observations of dust and gas. Methods. We obtained Herschel-PACS far-IR photometric observations at 70, 100 and 160 microns of 19 BPMG members, together with spectroscopic observations of four of them. Spectroscopic observations were centred at 63.18 microns and 157 microns, aiming to detect [OI] and [CII] emission. We incorporated the new far-IR observations in the SED of BPMG members and fitted modified blackbody models to better characterise the dust content. Results. We have detected far-IR excess emission toward nine BPMG members, including the first detection of an IR excess toward HD 29391.The star HD 172555, shows [OI] emission, while HD 181296, shows [CII] emission, expanding the short list of debris discs with a gas detection. No debris disc in BPMG is detected in both [OI] and [CII]. The discs show dust temperatures in the range 55 to 264 K, with low dust masses (6.6*10^{-5} MEarth to 0.2 MEarth) and radii from blackbody models in the range 3 to 82 AU. All the objects with a gas detection are early spectral type stars with a hot dust component.Comment: 12 pages, 7 figures, 6 table

Spatially resolved mid-infrared observations of the triple system T Tauri

To enhance our knowledge of the characteristics and distribution of the circumstellar dust associated with the individual components of the young hierarchical triple system T Tau, observations in the N-band with MIDI at the VLTI were performed. Our study is based on both the interferometric and the spectrophotometric measurements and is supplemented by new visual and infrared photometry. Also, the phases were investigated to determine the dominating mid-infrared source in the close southern binary. The data were fit with the help of a sophisticated physical disc model. This model utilises the radiative transfer code MC3D that is based on the Monte-Carlo method. Extended mid-infrared emission is found around all three components of the system. Simultaneous fits to the photometric and interferometric data confirm the picture of an almost face-on circumstellar disc around T Tau N. Towards this star, the silicate band is seen in emission. This emission feature is used to model the dust content of the circumstellar disc. Clear signs of dust processing are found. Towards T Tau S, the silicate band is seen in absorption. This absorption is strongly pronounced towards the infrared companion T Tau Sa as can be seen from the first individual N-band spectra for the two southern components. Our fits support the previous suggestion that an almost edge-on disc is present around T Tau Sa. This disc is thus misaligned with respect to the circumstellar disc around T Tau N. The interferometric data indicate that the disc around T Tau Sa is oriented in the north-south direction, which favours this source as launching site for the east-western jet. We further determine from the interferometric data the relative positions of the components of the southern binary.Comment: 24 pages, 19 figures, accepted for publication in A&