# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

December 2019 , Volume 13 , Issue 6

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2019, 13(6): 1139-1159 doi: 10.3934/ipi.2019051 +[Abstract](477) +[HTML](139) +[PDF](2919.0KB)
Abstract:

It is known that EIT inversion is an ill-posed problem, meaning that the solution is unstable if noise exists in the measured data. Generally, a regularization scheme is needed to alleviate the ill-posedness. In this work, a multiplicative regularization scheme is applied to EIT inversion. In this regularization scheme, a cost functional is constructed in which the data misfit functional is multiplied by a regularization factor, and no regularization parameter is needed. The regularization factor is based on the weighted \begin{document}$L2$\end{document}-norm favoring 'blocky' profiles in the reconstructed images. Gauss–Newton method is used to minimize the cost functional iteratively. In the implementation of the multiplicative regularization scheme, the spatial gradient and divergence need to be computed on triangular meshes. For this purpose, the discrete exterior calculus (DEC) theory is applied to formulate the related discrete operators. Numerical and experimental results show good anti-noise performance of the multiplicative regularization scheme in EIT inverse problem.

2019, 13(6): 1161-1188 doi: 10.3934/ipi.2019052 +[Abstract](456) +[HTML](209) +[PDF](8363.67KB)
Abstract:

It is a long-standing problem to preserve fine scale features such as texture in the process of deblurring. In order to deal with this challenging but imperative issue, we establish a framework of nonlinear fractional diffusion equations, which performs well in deblurring images with textures. In the new model, a fractional gradient is used for regularization of the diffusion process to preserve texture features and a source term with blurring kernel is used for deblurring. This source term ensures that the model can handle various blurring kernels. The relation between the regularization parameter and the deblurring performance is investigated theoretically, which ensures a satisfactory recovery when the blur type is known. Moreover, we derive a digital fractional diffusion filter that lives on images. Experimental results and comparisons show the effectiveness of the proposed model for texture-preserving deblurring.

2019, 13(6): 1189-1212 doi: 10.3934/ipi.2019053 +[Abstract](369) +[HTML](142) +[PDF](2935.54KB)
Abstract:

Reliable estimation of parameters of chaotic dynamical systems is a long standing problem important in numerous applications. We present a robust method for parameter estimation and uncertainty quantification that requires neither the knowledge of initial values for the system nor good guesses for the unknown model parameters. The method uses a new distance concept recently introduced to characterize the variability of chaotic dynamical systems. We apply it to cases where more traditional methods, such as those based on state space filtering, are no more applicable. Indeed, the approach combines concepts from chaos theory, optimization and statistics in a way that enables solving problems considered as 'intractable and unsolved' in prior literature. We illustrate the results with a large number of chaotic test cases, and extend the method in ways that increase the accuracy of the estimation results.

2019, 13(6): 1213-1258 doi: 10.3934/ipi.2019054 +[Abstract](374) +[HTML](130) +[PDF](602.07KB)
Abstract:

In this paper, we establish conditional stability estimates for two inverse problems of determining metrics in two dimensional Laplace-Beltrami operators. As data, in the first inverse problem we adopt spectral data on an arbitrarily fixed subboundary, while in the second, we choose the Dirichlet-to-Neumann map limited on an arbitrarily fixed subboundary. The conditional stability estimates for the two inverse problems are stated as follows. If the difference between spectral data or Dirichlet-to-Neumann maps related to two metrics \begin{document}${\bf{g}}_1$\end{document} and \begin{document}${\bf{g}}_2$\end{document} is small, then \begin{document}${\bf{g}}_1$\end{document} and \begin{document}${\bf{g}}_2$\end{document} are close in \begin{document}$L^2(\Omega)$\end{document} modulo a suitable diffeomorphism within a priori bounds of \begin{document}${\bf{g}}_1$\end{document} and \begin{document}${\bf{g}}_2$\end{document}. Both stability estimates are of the same double logarithmic rate.

2019, 13(6): 1259-1282 doi: 10.3934/ipi.2019055 +[Abstract](323) +[HTML](173) +[PDF](12540.77KB)
Abstract:

Total variation denoising (TVD) is an effective technique for image denoising, in particular, for recovering blocky, discontinuous images from noisy background. The problem is formulated as an optimization problem in the space of bounded variation functions, and the solution is obtained by solving the associated Euler–Lagrange equation defined on the domain occupied by the entire image. The method offers high quality results, but is computationally expensive for large images, especially for three-dimensional problems. In this paper, we introduce a highly parallel version of the algorithm which formulates the problem as multiple overlapping, but independent, optimization problems, and each is defined on a portion of the image domain. This approach is similar to the overlapping Schwarz type domain decomposition method, but is non-iterative, for solving partial differential equations, and is highly scalable, without using any coarse grids, for parallel computers with a large number of processors. We show by a theory and also by some two- and three-dimensional numerical experiments that the new approach has similar numerical accuracy as the classical TVD approach, but is much more efficient on parallel computers.

2019, 13(6): 1283-1307 doi: 10.3934/ipi.2019056 +[Abstract](291) +[HTML](102) +[PDF](554.37KB)
Abstract:

We study the existence and suppression of artifacts for a Doppler-based Synthetic Aperture Radar (DSAR) system. The idealized air- or space-borne system transmits a continuous wave at a fixed frequency and a co-located receiver measures the resulting scattered waves; a windowed Fourier transform then converts the raw data into a function of two variables: slow time and frequency. Under simplifying assumptions, we analyze the linearized forward scattering map and the feasibility of inverting it via filtered backprojection, using techniques of microlocal analysis which robustly describe how sharp features in the target appear in the data. For DSAR with a straight flight path, there is, as with conventional SAR, a left-right ambiguity artifact in the DSAR image, which can be avoided via beam forming to the left or right. For a circular flight path, the artifact has a more complicated structure, but filtering out echoes coming from straight ahead or behind the transceiver, as well as those outside a critical range, produces an artifact-free image. We show that these results are qualitatively robust; although initially derived under an approximation widely used for range-based SAR, they are either structurally stable or robust with respect to a more accurate model.

2019, 13(6): 1309-1325 doi: 10.3934/ipi.2019057 +[Abstract](296) +[HTML](81) +[PDF](357.04KB)
Abstract:

We consider a linearized inverse boundary value problem for the elasticity system. From the linearized Dirichlet-to-Neumann map at zero frequency, we show that a transversely isotropic perturbation of a homogeneous isotropic elastic tensor can be uniquely determined. From the linearized Dirichlet-to-Neumann map at two distinct positive frequencies, we show that a transversely isotropic perturbation of a homogeneous isotropic density can be identified at the same time.

2019, 13(6): 1327-1348 doi: 10.3934/ipi.2019058 +[Abstract](264) +[HTML](88) +[PDF](368.97KB)
Abstract:

In this work we develop a new reproducing kernel Hilbert space (RKHS) framework for inverse scattering problems using the Born approximation. We assume we have backscattered data of a field that is dependent on an unknown scattering potential. Our goal is to reconstruct or image this scattering potential. Assuming the scattering potential lies in a RKHS, we find that the imaging equation can be rewritten as the inner product of the desired unknown function with the adjoint of the forward operator applied to the kernel of the imaging operator. We therefore may choose the kernel of the imaging operator such that the adjoint applied to this kernel is precisely the reproducing kernel of the Hilbert space the reflectivity function lies in. In this way we are able to obtain an alternative definition of an ideal image. We will demonstrate this theory using synthetic aperture radar imaging as an example, though there are other applicable imaging modalities i.e. inverse diffraction and diffraction tomography [1,6]. We choose SAR as it was the motivating application for this work. We will compare the RKHS ideal imaging technique to the standard microlocal analytic ideal image from backprojection theory. Note this method requires a variation of the standard SAR data model with the assumption of a full two dimensional data collection surface as opposed to a one dimensional flight path, however we are able to perform imaging with a single frequency and avoid the approximations made in the backprojection imaging operator derivation.

2019, 13(6): 1349-1365 doi: 10.3934/ipi.2019059 +[Abstract](238) +[HTML](85) +[PDF](1794.05KB)
Abstract:

In this work, the direct and inverse scattering problems of wave impenetrable scatterers in the three-layered ocean waveguide are under investigation. We have established the well-posedness of forward problem and proposed a novel direct sampling method for the inverse problem. The direct recovery approach only applies the matrix-vector operations to approximate the wave impenetrable obstacle from the received partial data. The method is capable of reconstructing the objects of different shapes and locations, computationally quite cheap and easy to carry out. The theoretical analysis and the novel direct recovery algorithm are expected to have wide applications in the direct and inverse scattering problems of submerged acoustics.

2019, 13(6): 1367-1393 doi: 10.3934/ipi.2019060 +[Abstract](223) +[HTML](94) +[PDF](541.67KB)
Abstract:

For the first time, a globally convergent numerical method is developed and Lipschitz stability estimate is obtained for the challenging problem of travel time tomography in 3D for formally determined incomplete data. The semidiscrete case is considered meaning that finite differences are involved with respect to two out of three variables. First, Lipschitz stability estimate is derived, which implies uniqueness. Next, a weighted globally strictly convex Tikhonov-like functional is constructed using a Carleman-like weight function for a Volterra integral operator. The gradient projection method is constructed to minimize this functional. It is proven that this method converges globally to the exact solution if the noise in the data tends to zero.

2018  Impact Factor: 1.469