Inverse Problems and Imaging最新文献

In this paper, we show that it is possible to overcome one of the fundamental limitations of super-resolution microscopy: the necessity to be in an optically homogeneous environment. Using recent modal approximation results from [10, 7], we show, as a proof of concept, that it is possible to recover the position of a single point-like emitter in a known resonant environment from far-field measurements, with a precision two orders of magnitude below the classical Rayleigh limit. The procedure does not involve solving any partial differential equation, is computationally light (optimisation in begin{document}$ mathbb{R}^d $end{document} with begin{document}$ d $end{document} of the order of begin{document}$ 10 $end{document}) and is therefore suited for the recovery of a very large number of single emitters.

We construct counterexamples for the partial data inverse problem for the fractional conductivity equation in all dimensions on general bounded open sets. In particular, we show that for any bounded domain begin{document}$ Omega subset {mathbb R}^n $end{document} and any disjoint open sets begin{document}$ W_1, W_2 Subset {mathbb R}^n setminus overline{Omega} $end{document} there always exist two positive, bounded, smooth, conductivities begin{document}$ gamma_1, gamma_2 $end{document}, begin{document}$ gamma_1 neq gamma_2 $end{document}, with equal partial exterior Dirichlet-to-Neumann maps begin{document}$ Lambda_{gamma_1}f|_{W_2} = Lambda_{gamma_2}f|_{W_2} $end{document} for all begin{document}$ f in C_c^{infty}(W_1) $end{document}. The proof uses the characterization of equal exterior data from another work of the authors in combination with the maximum principle of fractional Laplacians. The main technical difficulty arises from the requirement that the conductivities should be strictly positive and have a special regularity property begin{document}$ gamma_i^{1/2}-1 in H^{2s, frac{n}{2s}}( {mathbb R}^n) $end{document} for begin{document}$ i = 1, 2 $end{document}. We also provide counterexamples on domains that are bounded in one direction when begin{document}$ n geq 4 $end{document} or begin{document}$ s in (0, n/4] $end{document} when begin{document}$ n = 2, 3 $end{document} using a modification of the argument on bounded domains.

In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in begin{document}$ {mathbb R}^m $end{document}, begin{document}$ m = 2, 3 $end{document} from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open begin{document}$ m-1 $end{document} dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [20]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach

We study the Dirichlet-to-Neumann map for the stationary linear equation of elasticity in a bounded domain in begin{document}$ mathbb{R}^d $end{document}, begin{document}$ dge 2 $end{document}, with smooth boundary. We show that it can be approximated by a pseudodifferential operator on the boundary with a matrix-valued symbol and we compute the principal symbol modulo conjugation by unitary matrices.

We propose a new variational framework to remove a mixture of Gaussian and impulse noise from images. This framework is based on a non-convex PDE-constrained with a fractional-order operator. The non-convex norm is applied to the impulse component controlled by a weighted parameter begin{document}$ gamma $end{document}, which depends on the level of the impulse noise and image feature. Furthermore, the fractional operator is used to preserve image texture and edges. In a first part, we study the theoretical properties of the proposed PDE-constrained, and we show some well-posdnees results. In a second part, after having demonstrated how to numerically find a minimizer, a proximal linearized algorithm combined with a Primal-Dual approach is introduced. Moreover, a bi-level optimization framework with a projected gradient algorithm is proposed in order to automatically select the parameter begin{document}$ gamma $end{document}. Denoising tests confirm that the non-convex term and learned parameter begin{document}$ gamma $end{document} lead in general to an improved reconstruction when compared to results of convex norm and other competitive denoising methods. Finally, we show extensive denoising experiments on various images and noise intensities and we report conventional numerical results which confirm the validity of the non-convex PDE-constrained, its analysis and also the proposed bi-level optimization with learning data.