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Inverse Problems & Imaging

2016 , Volume 10 , Issue 3

Special issue on ALCOMA'15

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On the stability of some imaging functionals
Guillaume Bal, Olivier Pinaud and Lenya Ryzhik
2016, 10(3): 585-616 doi: 10.3934/ipi.2016013 +[Abstract](88) +[PDF](522.6KB)
This work is devoted to the stability/resolution analysis of several imaging functionals in complex environments. We consider both linear functionals in the wavefield as well as quadratic functionals based on wavefield correlations. Using simplified measurement settings and reduced functionals that retain the main features of functionals used in practice, we obtain optimal asymptotic estimates of the signal-to-noise ratios depending on the main physical parameters of the problem. We consider random media with possibly long-range dependence and with a correlation length that is less than or equal to the central wavelength of the source we aim to reconstruct. This corresponds to the wave propagation regimes of radiative transfer or homogenization.
Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators
Radu Ioan Boţ and Christopher Hendrich
2016, 10(3): 617-640 doi: 10.3934/ipi.2016014 +[Abstract](108) +[PDF](833.3KB)
The aim of this article is to present two different primal-dual methods for solving structured monotone inclusions involving parallel sums of compositions of maximally monotone operators with linear bounded operators. By employing some elaborated splitting techniques, all of the operators occurring in the problem formulation are processed individually via forward or backward steps. The treatment of parallel sums of linearly composed maximally monotone operators is motivated by applications in imaging which involve first- and second-order total variation functionals, to which a special attention is given.
The inverse problem for electroseismic conversion: Stable recovery of the conductivity and the electrokinetic mobility parameter
Jie Chen and Maarten de Hoop
2016, 10(3): 641-658 doi: 10.3934/ipi.2016015 +[Abstract](117) +[PDF](409.4KB)
Pride (1994, Phys. Rev. B 50 15678-96) derived the governing model of electroseismic conversion, in which Maxwell's equations are coupled with Biot's equations through an electrokinetic mobility parameter. The inverse problem of electroseismic conversion was first studied by Chen and Yang (2013, Inverse Problem 29 115006). By following the construction of Complex Geometrical Optics (CGO) solutions to a matrix Schrödinger equation introduced by Ola and Somersalo (1996, SIAM J. Appl. Math. 56 No. 4 1129-1145), we analyze the recovering of conductivity, permittivity and the electrokinetic mobility parameter in Maxwell's equations with internal measurements, while allowing the magnetic permeability $\mu$ to be a variable function. We show that knowledge of two internal data sets associated with well-chosen boundary electrical sources uniquely determines these parameters. Moreover, a Lipschitz-type stability is obtained based on the same set.
Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds
Thierry Daudé, Damien Gobin and François Nicoleau
2016, 10(3): 659-688 doi: 10.3934/ipi.2016016 +[Abstract](96) +[PDF](551.1KB)
In this paper, we adapt the well-known local uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schrödinger equation to prove local uniqueness results in the setting of inverse metric problems. More specifically, we consider a class of spherically symmetric manifolds having two asymptotically hyperbolic ends and study the scattering properties of massless Dirac waves evolving on such manifolds. Using the spherical symmetry of the model, the stationary scattering is encoded by a countable family of one-dimensional Dirac equations. This allows us to define the corresponding transmission coefficients $T(\lambda,n)$ and reflection coefficients $L(\lambda,n)$ and $R(\lambda,n)$ of a Dirac wave having a fixed energy $\lambda$ and angular momentum $n$. For instance, the reflection coefficients $L(\lambda,n)$ correspond to the scattering experiment in which a wave is sent from the left end in the remote past and measured in the same left end in the future. The main result of this paper is an inverse uniqueness result local in nature. Namely, we prove that for a fixed $\lambda \not=0$, the knowledge of the reflection coefficients $L(\lambda,n)$ (resp. $R(\lambda,n)$) - up to a precise error term of the form $O(e^{-2nB})$ with $B>0$ - determines the manifold in a neighbourhood of the left (resp. right) end, the size of this neighbourhood depending on the magnitude $B$ of the error term. The crucial ingredients in the proof of this result are the Complex Angular Momentum method as well as some useful uniqueness results for Laplace transforms.
An efficient projection method for nonlinear inverse problems with sparsity constraints
Deren Han, Zehui Jia, Yongzhong Song and David Z. W. Wang
2016, 10(3): 689-709 doi: 10.3934/ipi.2016017 +[Abstract](136) +[PDF](1280.5KB)
In this paper, we propose a modification of the accelerated projective steepest descent method for solving nonlinear inverse problems with an $\ell_1$ constraint on the variable, which was recently proposed by Teschke and Borries (2010 Inverse Problems 26 025007). In their method, there are some parameters need to be estimated, which is a difficult task for many applications. We overcome this difficulty by introducing a self-adaptive strategy in choosing the parameters. Theoretically, the convergence of their algorithm was guaranteed under the assumption that the underlying mapping $F$ is twice Fréchet differentiable together with some other conditions, while we can prove weak and strong convergence of the proposed algorithm under the condition that $F$ is Fréchet differentiable, which is a relatively weak condition. We also report some preliminary computational results and compare our algorithm with that of Teschke and Borries, which indicate that our method is efficient.
Reconstructing a function on the sphere from its means along vertical slices
Ralf Hielscher and Michael Quellmalz
2016, 10(3): 711-739 doi: 10.3934/ipi.2016018 +[Abstract](145) +[PDF](6790.1KB)
We present a novel algorithm for the inversion of the vertical slice transform, i.e. the transform that associates to a function on the two-dimensional unit sphere all integrals along circles that are parallel to one fixed direction. Our approach makes use of the singular value decomposition and resembles the mollifier approach by applying numerical integration with a reconstruction kernel via a quadrature rule. Considering the inversion problem as a statistical inverse problem, we find a family of asymptotically optimal mollifiers that minimize the maximum risk of the mean integrated error for functions within a Sobolev ball. By using fast spherical Fourier transforms and the fast Legendre transform, our algorithm can be implemented with almost linear complexity. In numerical experiments, we compare our algorithm with other approaches and illustrate our theoretical findings.
Lavrentiev's regularization method in Hilbert spaces revisited
Bernd Hofmann, Barbara Kaltenbacher and Elena Resmerita
2016, 10(3): 741-764 doi: 10.3934/ipi.2016019 +[Abstract](196) +[PDF](487.4KB)
In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the derivative. There are plenty of qualitative and quantitative convergence results in the literature, both in Hilbert and Banach spaces. Our aim here is mainly to contribute to convergence rates results in Hilbert spaces based on some types of error estimates derived under various source conditions and to interpret them in some settings. In particular, we propose and investigate new variational source conditions adapted to these Lavrentiev-type techniques. Another focus of this paper is to exploit the concept of approximate source conditions.
Reconstruction of penetrable obstacles in the anisotropic acoustic scattering
Yi-Hsuan Lin
2016, 10(3): 765-780 doi: 10.3934/ipi.2016020 +[Abstract](132) +[PDF](388.0KB)
We develop an enclosure-type reconstruction scheme to identify penetrable obstacles in acoustic waves with anisotropic medium in $\mathbb{R}^{3}$. The main difficulty of treating this problem lies in the fact that there are no complex geometrical optics solutions available for the acoustic equation with anisotropic medium in $\mathbb{R}^{3}$. Instead, we will use another type of special solutions called oscillating-decaying solutions. Even though that oscillating-decaying solutions are defined only on the half space, we are able to give necessary boundary inputs by the Runge approximation property. Moreover, since we are considering a Helmholtz-type equation, we turn to Meyers' $L^{p}$ estimate to compare the integrals coming from oscillating-decaying solutions and those from reflected solutions.
A gradient-based method for atmospheric tomography
Daniela Saxenhuber and Ronny Ramlau
2016, 10(3): 781-805 doi: 10.3934/ipi.2016021 +[Abstract](121) +[PDF](917.2KB)
In Adaptive Optics (AO) systems for ground-based telescopes, one aims at mechanically correcting for atmospheric aberrations by means of quickly moving deformable mirrors. In complex AO systems, which are using several light sources and aim at a good reconstruction in a large field of view, the derivation of optimal mirror commands from measured light typically includes the problem of atmospheric tomography. As the computational effort for such a limited-angle tomography problem is strongly increasing for growing telescope sizes, fast algorithms are needed. We present a novel algorithm for atmospheric tomography that takes real-life effects such as tip/tilt indetermination, cone effect and spot elongation into account. Furthermore, we discuss two models for the tip and tilt components of an incoming wavefront and incorporate them into the reconstruction. We find a fast step size choice for our Gradient-based iteration and compare it with different existing step size choices. Numerical results are demonstrated for two different AO systems on a 42 m telescope, using the European Southern Observatory's end-to-end simulation tool, OCTOPUS.
Image segmentation based on the hybrid total variation model and the K-means clustering strategy
Baoli Shi, Zhi-Feng Pang and Jing Xu
2016, 10(3): 807-828 doi: 10.3934/ipi.2016022 +[Abstract](163) +[PDF](975.0KB)
The performance of image segmentation highly relies on the original inputting image. When the image is contaminated by some noises or blurs, we can not obtain the efficient segmentation result by using direct segmentation methods. In order to efficiently segment the contaminated image, this paper proposes a two step method based on the hybrid total variation model with a box constraint and the K-means clustering method. In the first step, the hybrid model is based on the weighted convex combination between the total variation functional and the high-order total variation as the regularization term to obtain the original clustering data. In order to deal with non-smooth regularization term, we solve this model by employing the alternating split Bregman method. Then, in the second step, the segmentation can be obtained by thresholding this clustering data into different phases, where the thresholds can be given by using the K-means clustering method. Numerical comparisons show that our proposed model can provide more efficient segmentation results dealing with the noise image and blurring image.
Error bounds and stability in the $l_{0}$ regularized for CT reconstruction from small projections
Chengxiang Wang and Li Zeng
2016, 10(3): 829-853 doi: 10.3934/ipi.2016023 +[Abstract](299) +[PDF](875.3KB)
Due to the restriction of the scanning environment and the energy of X-ray, few projections of an object can be obtained in some practical applications of computed tomography (CT). In these situations, the projection data are incomplete and inconsistent, and the conventional analytic algorithm such as filtered backprojection (FBP) algorithm will not work. The streak artifacts can be significantly reduced in few-view reconstruction if the total variation minimization (TVM) based CT reconstruction algorithm is used. However, in the premise of preserving the resolution of image, it will not effectively suppress slope artifacts and metal artifacts when dealing with some few-view of the limited-angle reconstruction problems. To solve this problem, we focus on the image reconstruction algorithm base on $\ell_{0}$ regularized of wavelet coefficients. In this paper, the error bound between the reference or desire image and the reconstructed result, and the stability of solution were shown in theoretical and experimental, a reconstruction experiment on metal laths from few-view of the limited-angle projections was given. The experimental results indicate that this algorithm outperforms classical CT reconstruction algorithms in preserving the resolution of reconstructed image and suppressing the metal artifacts.
The reciprocity gap method for a cavity in an inhomogeneous medium
Fang Zeng, Xiaodong Liu, Jiguang Sun and Liwei Xu
2016, 10(3): 855-868 doi: 10.3934/ipi.2016024 +[Abstract](159) +[PDF](446.0KB)
We consider an interior inverse medium problem of reconstructing the shape of a cavity. Both the measurement locations and point sources are inside the cavity. Due to the lack of a priori knowledge of physical prosperities of the medium inside the cavity and to avoid the computation of background Green's functions, the reciprocity gap method is employed. We prove the related theory and present some numerical examples for validation.

2016  Impact Factor: 1.094




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