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Inverse Problems and Imaging

April 2017 , Volume 11 , Issue 2

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Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces
Guozhi Dong, Bert Jüttler, Otmar Scherzer and Thomas Takacs
2017, 11(2): 221-246 doi: 10.3934/ipi.2017011 +[Abstract](4202) +[HTML](64) +[PDF](769.8KB)

We study Tikhonov regularization for solving ill-posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account perturbations of the surfaces, in particular when the surfaces are approximated by spline surfaces. Another contribution is that we highlight the analysis of regularization for functions with range in vector bundles over surfaces. We also present some practical applications, such as an inverse problem of gravimetry and an imaging problem for denoising vector fields on surfaces, and show the numerical verification.

A two-step mixed inpainting method with curvature-based anisotropy and spatial adaptivity
Francisco J. Ibarrola and Ruben D. Spies
2017, 11(2): 247-262 doi: 10.3934/ipi.2017012 +[Abstract](3369) +[HTML](58) +[PDF](940.5KB)

The image inpainting problem consists of restoring an image from a (possibly noisy) observation, in which data from one or more regions are missing. Several inpainting models to perform this task have been developed, and although some of them perform reasonably well in certain types of images, quite a few issues are yet to be sorted out. For instance, if the image is expected to be smooth, the inpainting can be made with very good results by means of a Bayesian approach and a maximum a posteriori computation [2]. For non-smooth images, however, such an approach is far from being satisfactory. Even though the introduction of anisotropy by prior smooth gradient inpainting to the latter methodology is known to produce satisfactory results for slim missing regions [2], the quality of the restoration decays as the occluded regions widen. On the other hand, Total Variation (TV) inpainting models based on high order PDE diffusion equations can be used whenever edge restoration is a priority. More recently, the introduction of spatially variant conductivity coefficients on these models, such as in the case of Curvature-Driven Diffusion (CDD) [4], has allowed inpainted images with well defined edges and enhanced object connectivity. The CDD approach, nonetheless, is not quite suitable wherever the image is smooth, as it tends to produce piecewise constant restorations.

In this work we present a two-step inpainting process. The first step consists of using a CDD inpainting to build a pilot image from which to infer a-priori structural information on the image gradient. The second step is inpainting the image by minimizing a mixed spatially variant anisotropic functional, whose weight and penalization directions are based upon the aforementioned pilot image. Results are presented along with comparison measures in order to illustrate the performance of this inpainting method.

A phaseless inverse scattering problem for the 3-D Helmholtz equation
Michael V. Klibanov
2017, 11(2): 263-276 doi: 10.3934/ipi.2017013 +[Abstract](3863) +[HTML](54) +[PDF](448.5KB)

An inverse scattering problem for the 3-D Helmholtz equation is considered. Only the modulus of the complex valued scattered wave field is assumed to be measured and the phase is not measured. This problem naturally arises in the lensless quality control of fabricated nanostructures. Uniqueness theorem is proved.

Applications of CGO solutions to coupled-physics inverse problems
Ilker Kocyigit, Ru-Yu Lai, Lingyun Qiu, Yang Yang and Ting Zhou
2017, 11(2): 277-304 doi: 10.3934/ipi.2017014 +[Abstract](4383) +[HTML](55) +[PDF](577.6KB)

This paper surveys inverse problems arising in several coupled-physics imaging modalities for both medical and geophysical purposes. These include Photo-acoustic Tomography (PAT), Thermo-acoustic Tomography (TAT), Electro-Seismic Conversion, Transient Elastrography (TE) and Acousto-Electric Tomography (AET). These inverse problems typically consists of multiple inverse steps, each of which corresponds to one of the wave propagations involved. The review focuses on those steps known as the inverse problems with internal data, in which the complex geometrical optics (CGO) solutions to the underlying equations turn out to be useful in showing the uniqueness and stability in determining the desired information.

Optical flow on evolving sphere-like surfaces
Lukas F. Lang and Otmar Scherzer
2017, 11(2): 305-338 doi: 10.3934/ipi.2017015 +[Abstract](4569) +[HTML](49) +[PDF](577.6KB)

In this work we consider optical flow on evolving Riemannian 2-manifolds which can be parametrised from the 2-sphere. Our main motivation is to estimate cell motion in time-lapse volumetric microscopy images depicting fluorescently labelled cells of a live zebrafish embryo. We exploit the fact that the recorded cells float on the surface of the embryo and allow for the extraction of an image sequence together with a sphere-like surface. We solve the resulting variational problem by means of a Galerkin method based on vector spherical harmonics and present numerical results computed from the aforementioned microscopy data.

Stability in conductivity imaging from partial measurements of one interior current
Carlos Montalto and Alexandru Tamasan
2017, 11(2): 339-353 doi: 10.3934/ipi.2017016 +[Abstract](3143) +[HTML](51) +[PDF](413.8KB)

We prove a stability result in the hybrid inverse problem of recovering the electrical conductivity from partial knowledge of one current density field generated inside a body by an imposed boundary voltage. The region of stable reconstruction is well defined by a combination of the exact and perturbed data. This work explains the high resolution and accuracy reconstructions in some existing numerical experiments that use partial interior data.

Localization of the interior transmission eigenvalues for a ball
Vesselin Petkov and Georgi Vodev
2017, 11(2): 355-372 doi: 10.3934/ipi.2017017 +[Abstract](2755) +[HTML](53) +[PDF](437.7KB)

We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball \begin{document} $\{x ∈ \mathbb{R}^d:\: |x| ≤ 1\}, \: d≥ 2,$ \end{document} and the coefficients \begin{document} $c_j(x), \: j =1,2, $ \end{document} and the indices of refraction \begin{document} $n_j(x), \: j =1,2,$ \end{document} are constants near the boundary \begin{document} $|x| = 1$ \end{document}. We prove that in this case the eigenvalue-free region obtained in [17] for strictly concave domains can be significantly improved. In particular, if \begin{document} $c_j(x), n_j(x), j = 1,2$ \end{document} are constants for \begin{document} $|x| ≤ 1$ \end{document}, we show that all (ITEs) lie in a strip \begin{document} $|\operatorname{Im} λ| ≤ C$ \end{document}.

Multiwave tomography with reflectors: Landweber's iteration
Plamen Stefanov and Yang Yang
2017, 11(2): 373-401 doi: 10.3934/ipi.2017018 +[Abstract](3653) +[HTML](53) +[PDF](2236.4KB)

We use the Landweber method for numerical simulations for the multiwave tomography problem with a reflecting boundary and compare it with the averaged time reversal method. We also analyze the rate of convergence and the dependence on the step size for the Landweber iterations on a Hilbert space.

Some novel linear regularization methods for a deblurring problem
Xiangtuan Xiong, Jinmei Li and Jin Wen
2017, 11(2): 403-426 doi: 10.3934/ipi.2017019 +[Abstract](4266) +[HTML](60) +[PDF](1071.4KB)

In this article, we consider a fractional backward heat conduction problem (BHCP) in the two-dimensional space which is associated with a deblurring problem. It is well-known that the classical Tikhonov method is the most important regularization method for linear ill-posed problems. However, the classical Tikhonov method over-smooths the solution. As a remedy, we propose two quasi-boundary regularization methods and their variants. We prove that these two methods are better than Tikhonov method in the absence of noise in the data. Deblurring experiment is conducted by comparing with some classical linear filtering methods for BHCP and the total variation method with the proposed methods.

2021 Impact Factor: 1.483
5 Year Impact Factor: 1.462
2021 CiteScore: 2.6




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