Small defects reconstruction in waveguides from multifrequency one-side scattering data

<p style='text-indent:20px;'>Localization and reconstruction of small defects in acoustic or electromagnetic waveguides is of crucial interest in nondestructive evaluation of structures. The aim of this work is to present a new multi-frequency inversion method to reconstruct small defects in a 2D waveguide. Given one-side multi-frequency wave field measurements of propagating modes, we use a Born approximation to provide a <inline-formula><tex-math id="M1">\begin{document}$ \text{L}^2 $\end{document}</tex-math></inline-formula>-stable reconstruction of three types of defects: a local perturbation inside the waveguide, a bending of the waveguide, and a localized defect in the geometry of the waveguide. This method is based on a mode-by-mode spacial Fourier inversion from the available partial data in the Fourier domain. Indeed, in the available data, some high and low spatial frequency information on the defect are missing. We overcome this issue using both a compact support hypothesis and a minimal smoothness hypothesis on the defects. We also provide a suitable numerical method for efficient reconstruction of such defects and we discuss its applications and limits.</p>

An inverse problem for a fractional diffusion equation with fractional power type nonlinearities

<p style='text-indent:20px;'>We study the well-posedness of a semi-linear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map. Our arguments are based on a first order linearization as well as the parabolic Runge approximation property.</p>

Weighted area constraints-based breast lesion segmentation in ultrasound image analysis

<p style='text-indent:20px;'>Breast ultrasound segmentation is a challenging task in practice due to speckle noise, low contrast and blurry boundaries. Although numerous methods have been developed to solve this problem, most of them can not produce a satisfying result due to uncertainty of the segmented region without specialized domain knowledge. In this paper, we propose a novel breast ultrasound image segmentation method that incorporates weighted area constraints using level set representations. Specifically, we first use speckle reducing anisotropic diffusion filter to suppress speckle noise, and apply the Grabcut on them to provide an initial segmentation result. In order to refine the resulting image mask, we propose a weighted area constraints-based level set formulation (WACLSF) to extract a more accurate tumor boundary. The major contribution of this paper is the introduction of a simple nonlinear constraint for the regularization of probability scores from a classifier, which can speed up the motion of zero level set to move to a desired boundary. Comparisons with other state-of-the-art methods, such as FCN-AlexNet and U-Net, show the advantages of our proposed WACLSF-based strategy in terms of visual view and accuracy.</p>

Uniqueness and numerical reconstruction for inverse problems dealing with interval size search

<p style='text-indent:20px;'>We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.</p>

On new surface-localized transmission eigenmodes

<p style='text-indent:20px;'>Consider the transmission eigenvalue problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (\Delta+k^2\mathbf{n}^2) w = 0, \ \ (\Delta+k^2)v = 0\ \ \mbox{in}\ \ \Omega;\quad w = v, \ \ \partial_\nu w = \partial_\nu v\ \ \mbox{on} \ \partial\Omega. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>It is shown in [<xref ref-type="bibr" rid="b16">16</xref>] that there exists a sequence of eigenfunctions <inline-formula><tex-math id="M1">\begin{document}$ (w_m, v_m)_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> associated with <inline-formula><tex-math id="M2">\begin{document}$ k_m\rightarrow \infty $\end{document}</tex-math></inline-formula> such that either <inline-formula><tex-math id="M3">\begin{document}$ \{w_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M4">\begin{document}$ \{v_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> are surface-localized, depending on <inline-formula><tex-math id="M5">\begin{document}$ \mathbf{n}&gt;1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M6">\begin{document}$ 0&lt;\mathbf{n}&lt;1 $\end{document}</tex-math></inline-formula>. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions <inline-formula><tex-math id="M7">\begin{document}$ (w_m, v_m)_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> associated with <inline-formula><tex-math id="M8">\begin{document}$ k_m\rightarrow \infty $\end{document}</tex-math></inline-formula> such that both <inline-formula><tex-math id="M9">\begin{document}$ \{w_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \{v_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> are surface-localized, no matter <inline-formula><tex-math id="M11">\begin{document}$ \mathbf{n}&gt;1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ 0&lt;\mathbf{n}&lt;1 $\end{document}</tex-math></inline-formula>. Though our study is confined within the radial geometry, the construction is subtle and technical.</p>

Euler equations and trace properties of minimizers of a functional for motion compensated inpainting

<p style='text-indent:20px;'>We compute the Euler equations of a functional useful for simultaneous video inpainting and motion estimation, which was obtained in [<xref ref-type="bibr" rid="b17">17</xref>] as the relaxation of a modified version of the functional proposed in [<xref ref-type="bibr" rid="b16">16</xref>]. The functional is defined on vectorial functions of bounded variations, therefore we also get the Euler equations holding on the singular sets of minimizers, highlighting in particular the conditions on the jump sets. Such conditions are expressed by means of traces of geometrically meaningful vector fields and characterized as pointwise limits of averages on cylinders with axes parallel to the unit normals to the jump sets.</p>

An inverse source problem for the stochastic wave equation

<p style='text-indent:20px;'>This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. For the direct problem, it is shown to be well-posed with a unique mild solution. For the inverse problem, the uniqueness is proved for a certain class of functions and the instability is characterized. Numerical experiments are presented to illustrate the reconstructions by using a truncation-based regularization method.</p>

A new approach to the inverse discrete transmission eigenvalue problem

<p style='text-indent:20px;'>A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove uniqueness of solution, global solvability, local solvability, and stability. Our approach is based on the reduction of the discrete transmission eigenvalue problem to a linear system with polynomials of the spectral parameter in the boundary condition.</p>

Counterexamples to inverse problems for the wave equation

<p style='text-indent:20px;'>We construct counterexamples to inverse problems for the wave operator on domains in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^{n+1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n \ge 2 $\end{document}</tex-math></inline-formula>, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^{n+1} $\end{document}</tex-math></inline-formula> the metrics are conformal to the Minkowski metric.</p>

Automated filtering in the nonlinear Fourier domain of systematic artifacts in 2D electrical impedance tomography

<p style='text-indent:20px;'>For patients undergoing mechanical ventilation due to respiratory failure, 2D electrical impedance tomography (EIT) is emerging as a means to provide functional monitoring of pulmonary processes. In EIT, electrical current is applied to the body, and the internal conductivity distribution is reconstructed based on subsequent voltage measurements. However, EIT images are known to often suffer from large systematic artifacts arising from various limitations and exacerbated by the ill-posedness of the inverse problem. The direct D-bar reconstruction method admits a nonlinear Fourier analysis of the EIT problem, providing the ability to process and filter reconstructions in the nonphysical frequency regime. In this work, a technique is introduced for automated Fourier-domain filtering of known systematic artifacts in 2D D-bar reconstructions. The new method is validated using three numerically simulated static thoracic datasets with induced artifacts, plus two experimental dynamic human ventilation datasets containing systematic artifacts. Application of the method is shown to significantly reduce the appearance of artifacts and improve the shape of the lung regions in all datasets.</p>