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Inverse Problems & Imaging

June 2018 , Volume 12 , Issue 3

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Recovering a large number of diffusion constants in a parabolic equation from energy measurements
Gianluca Mola
2018, 12(3): 527-543 doi: 10.3934/ipi.2018023 +[Abstract](562) +[HTML](179) +[PDF](426.47KB)
Abstract:

Let \begin{document} $\left(H, \left\langle { \cdot , \cdot } \right\rangle \right)$ \end{document} be a separable Hilbert space and \begin{document} $A_{i}:D(A_i) \to H$ \end{document} (\begin{document} $i = 1,···,n$ \end{document}) be a family of nonnegative and self-adjoint operators mutually commuting. We study the inverse problem consisting in the identification of a function \begin{document} $u:[0,T] \to H$ \end{document} and \begin{document} $n$ \end{document} constants \begin{document} $α_{1},···,α_{n} > 0$ \end{document}(diffusion coefficients) that fulfill the initial-value problem

and the additional conditions

where \begin{document} $\varphi_{i}$ \end{document} are given positive constants. Under suitable assumptions on the operators \begin{document} $A_{i}$ \end{document} and on the initial data \begin{document} $x ∈ H$ \end{document}, we shall prove that the solution of such a problem is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion constants in a heat equation and of the Lamé parameters in a elasticity problem on a plate.

Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction
Chengxiang Wang, Li Zeng, Wei Yu and Liwei Xu
2018, 12(3): 545-572 doi: 10.3934/ipi.2018024 +[Abstract](777) +[HTML](263) +[PDF](955.55KB)
Abstract:

In some practical applications of computed tomography (CT) imaging, the projections of an object are obtained within a limited-angle range due to the restriction of the scanning environment. In this situation, conventional analytic algorithms, such as filtered backprojection (FBP), will not work because the projections are incomplete. An image reconstruction algorithm based on total variation minimization (TVM) can significantly reduce streak artifacts in sparse-view reconstruction, but it will not effectively suppress slope artifacts when dealing with limited-angle reconstruction problems. To solve this problem, we consider a family of image reconstruction model based on \begin{document} $\ell_{0}$ \end{document} and \begin{document} $\ell_{2}$ \end{document} regularizations for limited-angle CT and prove the existence of a solution for two CT reconstruction models. The Alternating Direction Method of Multipliers (ADMM)-like method is utilized to solve our model. Furthermore, we prove the convergence of our algorithm under certain conditions. Some numerical experiments are used to evaluate the performance of our algorithm and the results indicate that our algorithm has advantage in suppressing slope artifacts.

Mathematical imaging using electric or magnetic nanoparticles as contrast agents
Durga Prasad Challa, Anupam Pal Choudhury and Mourad Sini
2018, 12(3): 573-605 doi: 10.3934/ipi.2018025 +[Abstract](548) +[HTML](174) +[PDF](513.15KB)
Abstract:

We analyse mathematically the imaging modality using electromagnetic nanoparticles as contrast agent. This method uses the electromagnetic fields, collected before and after injecting electromagnetic nanoparticles, to reconstruct the electrical permittivity. The particularity here is that these nanoparticles have high contrast electric or magnetic properties compared to the background media. First, we introduce the concept of electric (or magnetic) nanoparticles to describe the particles, of relative diameter \begin{document} $δ$ \end{document}(relative to the size of the imaging domain), having relative electric permittivity (or relative magnetic permeability) of order \begin{document} $δ^{-α}$ \end{document} with a certain \begin{document} $α>0$ \end{document}, as \begin{document} $0<δ<<1$ \end{document}. Examples of such material, used in the imaging community, are discussed. Second, we derive the asymptotic expansion of the electromagnetic fields due to such singular contrasts. We consider here the scalar electromagnetic model. Using these expansions, we extract the values of the total fields inside the domain of imaging from the scattered fields measured before and after injecting the nanoparticles. From these total fields, we derive the values of the electric permittivity at the expense of numerical differentiations.

Morozov principle for Kullback-Leibler residual term and Poisson noise
Bruno Sixou, Tom Hohweiller and Nicolas Ducros
2018, 12(3): 607-634 doi: 10.3934/ipi.2018026 +[Abstract](601) +[HTML](218) +[PDF](2729.24KB)
Abstract:

We study the properties of a regularization method for inverse problems corrupted by Poisson noise with Kullback-Leibler divergence as data term. The regularization parameter is chosen according to a Morozov type principle. We show that this method of choice of the parameter is well-defined. This a posteriori choice leads to a convergent regularization method. Convergences rates are obtained for this a posteriori choice of the regularization parameter when some source condition is satisfied.

A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers
Jun Lai, Ming Li, Peijun Li and Wei Li
2018, 12(3): 635-665 doi: 10.3934/ipi.2018027 +[Abstract](618) +[HTML](166) +[PDF](1358.85KB)
Abstract:

Consider the scattering of a time-harmonic plane wave by heterogeneous media consisting of linear or nonlinear point scatterers and extended obstacles. A generalized Foldy–Lax formulation is developed to take fully into account of the multiple scattering by the complex media. A new imaging function is proposed and an FFT-based direct imaging method is developed for the inverse obstacle scattering problem, which is to reconstruct the shape of the extended obstacles. The novel idea is to utilize the nonlinear point scatterers to excite high harmonic generation so that enhanced imaging resolution can be achieved. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

EIT in a layered anisotropic medium
Giovanni Alessandrini, Maarten V. de Hoop, Romina Gaburro and Eva Sincich
2018, 12(3): 667-676 doi: 10.3934/ipi.2018028 +[Abstract](656) +[HTML](165) +[PDF](360.85KB)
Abstract:

We consider the inverse problem in geophysics of imaging the subsurface of the Earth in cases where a region below the surface is known to be formed by strata of different materials and the depths and thicknesses of the strata and the (possibly anisotropic) conductivity of each of them need to be identified simultaneously. This problem is treated as a special case of the inverse problem of determining a family of nested inclusions in a medium \begin{document}$Ω\subset\mathbb{R}^n$\end{document}, \begin{document}$n ≥ 3$\end{document}.

On a gesture-computing technique using electromagnetic waves
Jingzhi Li, Hongyu Liu and Hongpeng Sun
2018, 12(3): 677-696 doi: 10.3934/ipi.2018029 +[Abstract](455) +[HTML](156) +[PDF](1141.82KB)
Abstract:

This paper is concerned with a conceptual gesture-based instruction/input technique using electromagnetic wave detection. The gestures are modeled as the shapes of some impenetrable or penetrable scatterers from a certain admissible class, called a dictionary. The gesture-computing device generates time-harmonic electromagnetic point signals for the gesture recognition and detection. It then collects the scattered wave in a relatively small backscattering aperture on a bounded surface containing the point sources. The recognition algorithm consists of two stages and requires only two incident waves of different wavenumbers. The location of the scatterer is first determined approximately by using the measured data at a small wavenumber and the shape of the scatterer is then identified using the computed location of the scatterer and the measured data at a regular wavenumber. We provide the corresponding mathematical principle with rigorous analysis. Numerical experiments show that the proposed device works effectively and efficiently.

SAR correlation imaging and anisotropic scattering
Kaitlyn (Voccola) Muller
2018, 12(3): 697-731 doi: 10.3934/ipi.2018030 +[Abstract](633) +[HTML](224) +[PDF](2287.45KB)
Abstract:

In this paper we investigate the ability of correlation synthetic-aperture radar (SAR) imaging to reconstruct isotropic and anisotropic scatterers. SAR correlation imaging was suggested by the author previously in [34]. Correlation imaging algorithms produce an image of a second-order quantity describing an object an interest, for example, the reflectivity function squared. In the previous work [34] it was argued that the effects of volume scattering clutter on the image can be minimized by choosing which pairs of collected data to correlate prior to applying a backprojection-type reconstruction algorithm. This choice of pairs for the correlation process is determined by what is known as the memory effect of scattering of waves by random scatterers [42,43,40,41,7,14]. It is the goal of this current work to determine the different imaging outcomes for an isotropic or point scatterer versus an anisotropic or dipole scatterer. In addition we aim to determine if removing contributions to the image due to the memory effect is necessary for diminishing the contributions of anisotropic or clutter scatterers to the scene of interest. Finally we extend the analysis of [34] to the polarimetric SAR case to determine whether the additional data provided by this modality contributes to decreasing the effects of clutter on the SAR image.

Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement
Xiaoping Fang and Youjun Deng
2018, 12(3): 733-743 doi: 10.3934/ipi.2018031 +[Abstract](384) +[HTML](162) +[PDF](324.16KB)
Abstract:

We consider the recovery of piecewise constant conductivity and an unknown inner core in inverse conductivity problem. We first show the unique recovery of the conductivity in a one layer structure without inner core by one measurement on any surface enclosing the unknown medium. Then we recover the unknown inner core in a one layer structure. We then show that in a two layer structure, the conductivity can be uniquely recovered by using one measurement.

Determination of singular time-dependent coefficients for wave equations from full and partial data
Guanghui Hu and Yavar Kian
2018, 12(3): 745-772 doi: 10.3934/ipi.2018032 +[Abstract](419) +[HTML](166) +[PDF](598.67KB)
Abstract:

We study the problem of determining uniquely a time-dependent singular potential \begin{document}$q$\end{document}, appearing in the wave equation \begin{document}$\partial_t^2u-Δ_x u+q(t,x)u = 0$\end{document} in \begin{document}$Q = (0,T)×Ω$\end{document} with \begin{document}$T>0$\end{document} and \begin{document}$Ω$\end{document} a \begin{document}$ \mathcal C^2$\end{document} bounded domain of \begin{document}$\mathbb{R}^n$\end{document}, \begin{document}$n≥2$\end{document}. We start by considering the unique determination of some general singular time-dependent coefficients. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first claiming unique determination of unbounded time-dependent coefficients, which is motivated by the problem of determining general nonlinear terms appearing in nonlinear wave equations.

Backward problem for a time-space fractional diffusion equation
Junxiong Jia, Jigen Peng, Jinghuai Gao and Yujiao Li
2018, 12(3): 773-799 doi: 10.3934/ipi.2018033 +[Abstract](1010) +[HTML](272) +[PDF](659.55KB)
Abstract:

In this paper, a backward problem for a time-space fractional diffusion process has been considered. For this problem, we propose to construct the initial function by minimizing data residual error in Fourier space domain with variable total variation (TV) regularizing term which can protect the edges as TV regularizing term and reduce staircasing effect. The well-posedness of this optimization problem is obtained under a very general setting. Actually, we rewrite the time-space fractional diffusion equation as an abstract fractional differential equation and deduce our results by using fractional operator semigroup theory, hence, our theoretical results can be applied to other backward problems for the differential equations with more general fractional operator. Then a modified Bregman iterative algorithm has been proposed to approximate the minimizer. The new features of this algorithm is that the regularizing term altered in each step and we need not to solve the complex Euler-Lagrange equation of variable TV regularizing term (just need to solve a simple Euler-Lagrange equation). The convergence of this algorithm and the strategy of choosing parameters are also obtained. Numerical implementations are provided to support our theoretical analysis to show the flexibility of our minimization model.

An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data
Sombuddha Bhattacharyya
2018, 12(3): 801-830 doi: 10.3934/ipi.2018034 +[Abstract](446) +[HTML](160) +[PDF](480.66KB)
Abstract:

We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schrödinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the boundary. The uniqueness proof relies on proving a suitable Carleman estimate for functions which vanish only on a part of boundary and constructing complex geometric optics solutions which vanish on a part of the boundary.

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