Image Reconstruction in Optical Interferometry

This tutorial paper describes the problem of image reconstruction from interferometric data with a particular focus on the specific problems encountered at optical (visible/IR) wavelengths. The challenging issues in image reconstruction from interferometric data are introduced in the general framework of inverse problem approach. This framework is then used to describe existing image reconstruction algorithms in radio interferometry and the new methods specifically developed for optical interferometry.Comment: accepted for publication in IEEE Signal Processing Magazin

Optimization Methods for Inverse Problems

Optimization plays an important role in solving many inverse problems. Indeed, the task of inversion often either involves or is fully cast as a solution of an optimization problem. In this light, the mere non-linear, non-convex, and large-scale nature of many of these inversions gives rise to some very challenging optimization problems. The inverse problem community has long been developing various techniques for solving such optimization tasks. However, other, seemingly disjoint communities, such as that of machine learning, have developed, almost in parallel, interesting alternative methods which might have stayed under the radar of the inverse problem community. In this survey, we aim to change that. In doing so, we first discuss current state-of-the-art optimization methods widely used in inverse problems. We then survey recent related advances in addressing similar challenges in problems faced by the machine learning community, and discuss their potential advantages for solving inverse problems. By highlighting the similarities among the optimization challenges faced by the inverse problem and the machine learning communities, we hope that this survey can serve as a bridge in bringing together these two communities and encourage cross fertilization of ideas.Comment: 13 page

On Algorithms Based on Joint Estimation of Currents and Contrast in Microwave Tomography

This paper deals with improvements to the contrast source inversion method which is widely used in microwave tomography. First, the method is reviewed and weaknesses of both the criterion form and the optimization strategy are underlined. Then, two new algorithms are proposed. Both of them are based on the same criterion, similar but more robust than the one used in contrast source inversion. The first technique keeps the main characteristics of the contrast source inversion optimization scheme but is based on a better exploitation of the conjugate gradient algorithm. The second technique is based on a preconditioned conjugate gradient algorithm and performs simultaneous updates of sets of unknowns that are normally processed sequentially. Both techniques are shown to be more efficient than original contrast source inversion.Comment: 12 pages, 12 figures, 5 table

Inverse polynomial optimization

We consider the inverse optimization problem associated with the polynomial program f^*=\min \{f(x): x\in K\}$and a given current feasible solution$y\in K$. We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial$\tilde{f}$(which may be of same degree as$f$if desired) with the following properties: (a)$y$is a global minimizer of$\tilde{f}$on$K$with a Putinar's certificate with an a priori degree bound$d$fixed, and (b),$\tilde{f}$minimizes$\Vert f-\tilde{f}\Vert$(which can be the$\ell_1$,$\ell_2$or$\ell_\infty$-norm of the coefficients) over all polynomials with such properties. Computing$\tilde{f}_d$reduces to solving a semidefinite program whose optimal value also provides a bound on how far is$f(\y)$from the unknown optimal value$f^*$. The size of the semidefinite program can be adapted to the computational capabilities available. Moreover, if one uses the$\ell_1$-norm, then$\tilde{f}$ takes a simple and explicit canonical form. Some variations are also discussed.Comment: 25 pages; to appear in Math. Oper. Res; Rapport LAAS no. 1114