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Inverse Problems & Imaging

June 2020 , Volume 14 , Issue 3

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$ \chi^2 $ test for total variation regularization parameter selection
J. Mead
2020, 14(3): 401-421 doi: 10.3934/ipi.2020019 +[Abstract](76) +[HTML](33) +[PDF](8427.6KB)

Total Variation (TV) is an effective method of removing noise in digital image processing while preserving edges. The scaling or regularization parameter in the TV process defines the amount of denoising, with a value of zero giving a result equivalent to the input signal. The discrepancy principle is a classical method for regularization parameter selection whereby data is fit to a specified tolerance. The tolerance is often identified based on the fact that the least squares data fit is known to follow a \begin{document}$ \chi^2 $\end{document} distribution. However, this approach fails when the number of parameters is greater than or equal to the number of data. Typically, heuristics are employed to identify the tolerance in the discrepancy principle and this leads to oversmoothing. In this work we identify a \begin{document}$ \chi^2 $\end{document} test for TV regularization parameter selection assuming the blurring matrix is full rank. In particular, we prove that the degrees of freedom in the TV regularized residual is the number of data and this is used to identify the appropriate tolerance. The importance of this work lies in the fact that the \begin{document}$ \chi^2 $\end{document} test introduced here for TV automates the choice of regularization parameter selection and can straightforwardly be incorporated into any TV algorithm. Results are given for three test images and compared to results using the discrepancy principle and MAP estimates.

On range condition of the tensor x-ray transform in $ \mathbb R^n $
Aleksander Denisiuk
2020, 14(3): 423-435 doi: 10.3934/ipi.2020020 +[Abstract](68) +[HTML](42) +[PDF](373.06KB)

Consider the problem of the range description of the tensor x-ray transform in \begin{document}$ \mathbb R^n $\end{document}, \begin{document}$ n\ge3 $\end{document}. In this paper we use the relation between the x-ray transform and the Radon transform to obtain a geometrical interpretation of the range condition and related John differential operator. As a corollary, it is proved that the range of the \begin{document}$ m $\end{document}-tensor x-ray transform in \begin{document}$ \mathbb R^n $\end{document} can be described by \begin{document}$ \binom{n+m-2}{m+1} $\end{document} linear differential equations of order \begin{document}$ 2(m+1) $\end{document}.

Regularization of inverse problems via box constrained minimization
Philipp Hungerländer, Barbara Kaltenbacher and Franz Rendl
2020, 14(3): 437-461 doi: 10.3934/ipi.2020021 +[Abstract](77) +[HTML](37) +[PDF](4927.89KB)

In the present paper we consider minimization based formulations of inverse problems \begin{document}$ (x, \Phi)\in \mbox{argmin}\left\{{ \mathcal{J}(x, \Phi;y)}:{(x, \Phi)\in M_{ad}(y)}\right\} $\end{document} for the specific but highly relevant case that the admissible set \begin{document}$ M_{ad}^\delta(y^\delta) $\end{document} is defined by pointwise bounds, which is the case, e.g., if \begin{document}$ L^\infty $\end{document} constraints on the parameter are imposed in the sense of Ivanov regularization, and the \begin{document}$ L^\infty $\end{document} noise level in the observations is prescribed in the sense of Morozov regularization. As application examples for this setting we consider three coefficient identification problems in elliptic boundary value problems.

Discretization of \begin{document}$ (x, \Phi) $\end{document} with piecewise constant and piecewise linear finite elements, respectively, leads to finite dimensional nonlinear box constrained minimization problems that can numerically be solved via Gauss-Newton type SQP methods. In our computational experiments we revisit the suggested application examples. In order to speed up the computations and obtain exact numerical solutions we use recently developed active set methods for solving strictly convex quadratic programs with box constraints as subroutines within our Gauss-Newton type SQP approach.

Uniqueness and stability for the recovery of a time-dependent source in elastodynamics
Guanghui Hu and Yavar Kian
2020, 14(3): 463-487 doi: 10.3934/ipi.2020022 +[Abstract](73) +[HTML](31) +[PDF](901.64KB)

This paper is concerned with inverse source problems for the time-dependent Lamé system in an unbounded domain corresponding to either the exterior of a bounded cavity or the full space \begin{document}$ {\mathbb{R}}^3 $\end{document}. If the time and spatial variables of the source term can be separated with compact support, we prove that the vector valued spatial source term can be uniquely determined by boundary Dirichlet data in the exterior of a given cavity. Uniqueness and stability for recovering some class of time-dependent source terms are also obtained by using, respectively, partial and full boundary data.

Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency
Xiaoxu Xu, Bo Zhang and Haiwen Zhang
2020, 14(3): 489-510 doi: 10.3934/ipi.2020023 +[Abstract](67) +[HTML](44) +[PDF](552.29KB)

This paper is concerned with uniqueness results in inverse acoustic and electromagnetic scattering problems with phaseless total-field data at a fixed frequency. We use superpositions of two point sources as the incident fields at a fixed frequency and measure the modulus of the acoustic total-field (called phaseless acoustic near-field data) on two spheres containing the scatterers generated by such incident fields on the two spheres. Based on this idea, we prove that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined from the phaseless acoustic near-field data at a fixed frequency. Moreover, the idea is also applied to the electromagnetic case, and it is proved that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined by the phaseless electric near-field data at a fixed frequency, that is, the modulus of the tangential component with the orientations \begin{document}$ \boldsymbol{e}_\phi $\end{document} and \begin{document}$ \boldsymbol{e}_\theta $\end{document}, respectively, of the electric total-field measured on a sphere enclosing the scatters and generated by superpositions of two electric dipoles at a fixed frequency located on the measurement sphere and another bigger sphere with the polarization vectors \begin{document}$ \boldsymbol{e}_\phi $\end{document} and \begin{document}$ \boldsymbol{e}_\theta $\end{document}, respectively. As far as we know, this is the first uniqueness result for three-dimensional inverse electromagnetic scattering with phaseless near-field data.

Statistical characterization of scattering delay in synthetic aperture radar imaging
Mikhail Gilman and Semyon Tsynkov
2020, 14(3): 511-533 doi: 10.3934/ipi.2020024 +[Abstract](57) +[HTML](34) +[PDF](3574.45KB)

Distinguishing between the instantaneous and delayed scatterers in synthetic aperture radar (SAR) images is important for target identification and characterization. To perform this task, one can use the autocorrelation analysis of coordinate-delay images. However, due to the range-delay ambiguity the difference in the correlation properties between the instantaneous and delayed targets may be small. Moreover, the reliability of discrimination is affected by speckle, which is ubiquitous in SAR images, and requires statistical treatment.

Previously, we have developed a maximum likelihood based approach for discriminating between the instantaneous and delayed targets in SAR images. To test it, we employed simple statistical models. They allowed us to simulate ensembles of images that depend on various parameters, including aperture width and target contrast.

In the current paper, we enhance our previously developed methodology by establishing confidence levels for the discrimination between the instantaneous and delayed scatterers. Our procedure takes into account the difference in thresholds for different target contrasts without making any assumptions about the statistics of those contrasts.

Robust reconstruction of fluorescence molecular tomography with an optimized illumination pattern
Yan Liu, Wuwei Ren and Habib Ammari
2020, 14(3): 535-568 doi: 10.3934/ipi.2020025 +[Abstract](101) +[HTML](49) +[PDF](4523.44KB)

Fluorescence molecular tomography (FMT) is an emerging tool for biomedical research. There are two factors that influence FMT reconstruction most effectively. The first one is regularization techniques. Traditional methods such as Tikhonov regularization suffer from low resolution and poor signal to noise ratio. Therefore, sparse regularization techniques have been introduced to improve the reconstruction quality. The second factor is the illumination pattern. A better illumination pattern ensures the quantity and quality of the information content of the data set, thus leading to better reconstructions. In this work, we take advantage of the discrete formulation of the forward problem to give a rigorous definition of an illumination pattern as well as the admissible set of patterns. We add restrictions in the admissible set as different types of regularizers to a discrepancy functional, generating another inverse problem with the illumination pattern as unknown. Both inverse problems of reconstructing the fluorescence distribution and finding the optimal illumination pattern are solved by efficient iterative algorithms. Numerical experiments have shown that with a suitable choice of regularization parameters, the two-step approach converges to an optimal illumination pattern quickly and the reconstruction result is improved significantly regardless of the initial illumination setting.

Unique determinations in inverse scattering problems with phaseless near-field measurements
Deyue Zhang, Yukun Guo, Fenglin Sun and Hongyu Liu
2020, 14(3): 569-582 doi: 10.3934/ipi.2020026 +[Abstract](73) +[HTML](64) +[PDF](391.26KB)

In this paper, we establish the unique determination results for several inverse acoustic scattering problems using the modulus of the near-field data. By utilizing the superpositions of point sources as the incident waves, we rigorously prove that the phaseless near-fields collected on an admissible surface can uniquely determine the location and shape of the obstacle as well as its boundary condition and the refractive index of a medium inclusion, respectively. We also establish the uniqueness in determining a locally rough surface from the phaseless near-field data due to superpositions of point sources. These are novel uniqueness results in inverse scattering with phaseless near-field data.

2018  Impact Factor: 1.469




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