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1930-8337
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1930-8345
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Inverse Problems & Imaging
February 2015 , Volume 9 , Issue 1
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2015, 9(1): 1-25
doi: 10.3934/ipi.2015.9.1
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Abstract:
We consider the inverse problem of the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions of the Maxwell's system in 3D with limited boundary observations of the electric field. The theoretical stability for the problem is provided by the Carleman estimates. For the numerical computations the problem is formulated as an optimization problem and hybrid finite element/difference method is used to solve the parameter identification problem.
We consider the inverse problem of the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions of the Maxwell's system in 3D with limited boundary observations of the electric field. The theoretical stability for the problem is provided by the Carleman estimates. For the numerical computations the problem is formulated as an optimization problem and hybrid finite element/difference method is used to solve the parameter identification problem.
2015, 9(1): 27-53
doi: 10.3934/ipi.2015.9.27
+[Abstract](1503)
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Abstract:
We present a scalable solver for approximating the maximum a posteriori (MAP) point of Bayesian inverse problems with Besov priors based on wavelet expansions with random coefficients. It is a subspace trust region interior reflective Newton conjugate gradient method for bound constrained optimization problems. The method combines the rapid locally-quadratic convergence rate properties of Newton's method, the effectiveness of trust region globalization for treating ill-conditioned problems, and the Eisenstat--Walker idea of preventing oversolving. We demonstrate the scalability of the proposed method on two inverse problems: a deconvolution problem and a coefficient inverse problem governed by elliptic partial differential equations. The numerical results show that the number of Newton iterations is independent of the number of wavelet coefficients $n$ and the computation time scales linearly in $n$. It will be numerically shown, under our implementations, that the proposed solver is two times faster than the split Bregman approach, and it is an order of magnitude less expensive than the interior path following primal-dual method. Our results also confirm the fact that the Besov $\mathbb{B}_{11}^1$ prior is sparsity promoting, discretization-invariant, and edge-preserving for both imaging and inverse problems governed by partial differential equations.
We present a scalable solver for approximating the maximum a posteriori (MAP) point of Bayesian inverse problems with Besov priors based on wavelet expansions with random coefficients. It is a subspace trust region interior reflective Newton conjugate gradient method for bound constrained optimization problems. The method combines the rapid locally-quadratic convergence rate properties of Newton's method, the effectiveness of trust region globalization for treating ill-conditioned problems, and the Eisenstat--Walker idea of preventing oversolving. We demonstrate the scalability of the proposed method on two inverse problems: a deconvolution problem and a coefficient inverse problem governed by elliptic partial differential equations. The numerical results show that the number of Newton iterations is independent of the number of wavelet coefficients $n$ and the computation time scales linearly in $n$. It will be numerically shown, under our implementations, that the proposed solver is two times faster than the split Bregman approach, and it is an order of magnitude less expensive than the interior path following primal-dual method. Our results also confirm the fact that the Besov $\mathbb{B}_{11}^1$ prior is sparsity promoting, discretization-invariant, and edge-preserving for both imaging and inverse problems governed by partial differential equations.
2015, 9(1): 55-77
doi: 10.3934/ipi.2015.9.55
+[Abstract](1618)
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Abstract:
In this paper, we consider tomographic reconstruction for axially symmetric objects from a single radiograph formed by fan-beam X-rays. All contemporary methods are based on the assumption that the density is piecewise constant or linear. From a practical viewpoint, this is quite a restrictive approximation. The method we propose is based on high-order total variation regularization. Its main advantage is to reduce the staircase effect while keeping sharp edges and enable the recovery of smoothly varying regions. The optimization problem is solved using the augmented Lagrangian method which has been recently applied in image processing. Furthermore, we use a one-dimensional (1D) technique for fan-beam X-rays to approximate 2D tomographic reconstruction for cone-beam X-rays. For the 2D problem, we treat the cone beam as fan beam located at parallel planes perpendicular to the symmetric axis. Then the density of the whole object is recovered layer by layer. Numerical results in 1D show that the proposed method has improved the preservation of edge location and the accuracy of the density level when compared with several other contemporary methods. The 2D numerical tests show that cylindrical symmetric objects can be recovered rather accurately by our high-order regularization model.
In this paper, we consider tomographic reconstruction for axially symmetric objects from a single radiograph formed by fan-beam X-rays. All contemporary methods are based on the assumption that the density is piecewise constant or linear. From a practical viewpoint, this is quite a restrictive approximation. The method we propose is based on high-order total variation regularization. Its main advantage is to reduce the staircase effect while keeping sharp edges and enable the recovery of smoothly varying regions. The optimization problem is solved using the augmented Lagrangian method which has been recently applied in image processing. Furthermore, we use a one-dimensional (1D) technique for fan-beam X-rays to approximate 2D tomographic reconstruction for cone-beam X-rays. For the 2D problem, we treat the cone beam as fan beam located at parallel planes perpendicular to the symmetric axis. Then the density of the whole object is recovered layer by layer. Numerical results in 1D show that the proposed method has improved the preservation of edge location and the accuracy of the density level when compared with several other contemporary methods. The 2D numerical tests show that cylindrical symmetric objects can be recovered rather accurately by our high-order regularization model.
2015, 9(1): 79-103
doi: 10.3934/ipi.2015.9.79
+[Abstract](1600)
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Abstract:
A novel variational model for deformable multi-modal image registration is presented in this work. As an alternative to the models based on maximizing mutual information, the Rényi's statistical dependence measure of two random variables is proposed as a measure of the goodness of matching in our objective functional. The proposed model does not require an estimation of the continuous joint probability density function. Instead, it only needs observed independent instances. Moreover, the theory of reproducing kernel Hilbert space is used to simplify the computation. Experimental results and comparisons with several existing methods are provided to show the effectiveness of the model.
A novel variational model for deformable multi-modal image registration is presented in this work. As an alternative to the models based on maximizing mutual information, the Rényi's statistical dependence measure of two random variables is proposed as a measure of the goodness of matching in our objective functional. The proposed model does not require an estimation of the continuous joint probability density function. Instead, it only needs observed independent instances. Moreover, the theory of reproducing kernel Hilbert space is used to simplify the computation. Experimental results and comparisons with several existing methods are provided to show the effectiveness of the model.
2015, 9(1): 105-125
doi: 10.3934/ipi.2015.9.105
+[Abstract](1237)
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Abstract:
In this article, we are interested in the study of the asymptotic behavior, in terms of finite-dimensional attractors, of a generalization of the Cahn--Hilliard equation with a fidelity term (integrated over $\Omega\backslash D$ instead of the entire domain $\Omega$, $D \subset \subset \Omega$). Such a model has, in particular, applications in image inpainting. The difficulty here is that we no longer have the conservation of mass, i.e. of the spatial average of the order parameter $u$, as in the Cahn--Hilliard equation. Instead, we prove that the spatial average of $u$ is dissipative. We finally give some numerical simulations which confirm previous ones on the efficiency of the model.
In this article, we are interested in the study of the asymptotic behavior, in terms of finite-dimensional attractors, of a generalization of the Cahn--Hilliard equation with a fidelity term (integrated over $\Omega\backslash D$ instead of the entire domain $\Omega$, $D \subset \subset \Omega$). Such a model has, in particular, applications in image inpainting. The difficulty here is that we no longer have the conservation of mass, i.e. of the spatial average of the order parameter $u$, as in the Cahn--Hilliard equation. Instead, we prove that the spatial average of $u$ is dissipative. We finally give some numerical simulations which confirm previous ones on the efficiency of the model.
2015, 9(1): 127-141
doi: 10.3934/ipi.2015.9.127
+[Abstract](1391)
+[PDF](456.4KB)
Abstract:
Consider the two-dimensional inverse elastic scattering problem of recovering a piecewise linear rigid rough or periodic surface of rectangular type for which the neighboring line segments are always perpendicular. We prove the global uniqueness with at most two incident elastic plane waves by using near-field data. If the Lamé constants satisfy a certain condition, then the data of a single plane wave is sufficient to imply the uniqueness. Our proof is based on a transcendental equation for the Navier equation, which is derived from the expansion of analytic solutions to the Helmholtz equation. The uniqueness results apply also to an inverse scattering problem for non-convex bounded rigid bodies of rectangular type.
Consider the two-dimensional inverse elastic scattering problem of recovering a piecewise linear rigid rough or periodic surface of rectangular type for which the neighboring line segments are always perpendicular. We prove the global uniqueness with at most two incident elastic plane waves by using near-field data. If the Lamé constants satisfy a certain condition, then the data of a single plane wave is sufficient to imply the uniqueness. Our proof is based on a transcendental equation for the Navier equation, which is derived from the expansion of analytic solutions to the Helmholtz equation. The uniqueness results apply also to an inverse scattering problem for non-convex bounded rigid bodies of rectangular type.
2015, 9(1): 143-161
doi: 10.3934/ipi.2015.9.143
+[Abstract](1320)
+[PDF](446.7KB)
Abstract:
We study the broken ray transform on $n$-dimensional Euclidean domains where the reflecting parts of the boundary are flat and establish injectivity and stability under certain conditions. Given a subset $E$ of the boundary $\partial \Omega$ such that $\partial \Omega \setminus E$ is itself flat (contained in a union of hyperplanes), we measure the attenuation of all broken rays starting and ending at $E$ with the standard optical reflection rule applied to $\partial \Omega \setminus E$. By localizing the measurement operator around broken rays which reflect off a fixed sequence of flat hyperplanes, we can apply the analytic microlocal approach of Frigyik, Stefanov, and Uhlmann ([7]) for the ordinary ray transform by means of a local path unfolding. This generalizes the author's previous result in [9], although we can no longer treat reflections from corner points. Similar to the result for the two dimensional square, we show that the normal operator is a classical pseudo differential operator of order $-1$ plus a smoothing term with $C_{0}^{\infty}$ Schwartz kernel.
We study the broken ray transform on $n$-dimensional Euclidean domains where the reflecting parts of the boundary are flat and establish injectivity and stability under certain conditions. Given a subset $E$ of the boundary $\partial \Omega$ such that $\partial \Omega \setminus E$ is itself flat (contained in a union of hyperplanes), we measure the attenuation of all broken rays starting and ending at $E$ with the standard optical reflection rule applied to $\partial \Omega \setminus E$. By localizing the measurement operator around broken rays which reflect off a fixed sequence of flat hyperplanes, we can apply the analytic microlocal approach of Frigyik, Stefanov, and Uhlmann ([7]) for the ordinary ray transform by means of a local path unfolding. This generalizes the author's previous result in [9], although we can no longer treat reflections from corner points. Similar to the result for the two dimensional square, we show that the normal operator is a classical pseudo differential operator of order $-1$ plus a smoothing term with $C_{0}^{\infty}$ Schwartz kernel.
2015, 9(1): 163-188
doi: 10.3934/ipi.2015.9.163
+[Abstract](1696)
+[PDF](1066.5KB)
Abstract:
We shall derive and propose several efficient overlapping domain decomposition methods for solving some typical linear inverse problems, including the identification of the flux, the source strength and the initial temperature in second order elliptic and parabolic systems. The methods are iterative, and computationally very efficient: only local forward and adjoint problems need to be solved in each subdomain, and the local minimizations have explicit solutions. Numerical experiments are provided to demonstrate the robustness and efficiency of the methods: the algorithms converge globally, even with rather poor initial guesses; and their convergences do not deteriorate or deteriorate only slightly when the meshes are refined.
We shall derive and propose several efficient overlapping domain decomposition methods for solving some typical linear inverse problems, including the identification of the flux, the source strength and the initial temperature in second order elliptic and parabolic systems. The methods are iterative, and computationally very efficient: only local forward and adjoint problems need to be solved in each subdomain, and the local minimizations have explicit solutions. Numerical experiments are provided to demonstrate the robustness and efficiency of the methods: the algorithms converge globally, even with rather poor initial guesses; and their convergences do not deteriorate or deteriorate only slightly when the meshes are refined.
2015, 9(1): 189-210
doi: 10.3934/ipi.2015.9.189
+[Abstract](1618)
+[PDF](560.4KB)
Abstract:
A novel method is developed for solving the inverse obstacle scattering problem in near-field imaging. The obstacle surface is assumed to be a small and smooth deformation of a circle. Using the transformed field expansion, the direct obstacle scattering problem is reduced to a successive sequence of two-point boundary value problems. Analytical solutions of these problems are derived by a Green's function method. The nonlinear inverse problem is linearized by dropping the higher order terms in the power series expansion. Based on the linear model and analytical solutions, an explicit reconstruction formula is obtained. In addition, a nonlinear correction scheme is devised to improve the results dramatically when the deformation is large. The method requires only a single incident wave at a fixed frequency. Numerical tests show that the method is stable and effective for near-field imaging of obstacles with subwavelength resolution.
A novel method is developed for solving the inverse obstacle scattering problem in near-field imaging. The obstacle surface is assumed to be a small and smooth deformation of a circle. Using the transformed field expansion, the direct obstacle scattering problem is reduced to a successive sequence of two-point boundary value problems. Analytical solutions of these problems are derived by a Green's function method. The nonlinear inverse problem is linearized by dropping the higher order terms in the power series expansion. Based on the linear model and analytical solutions, an explicit reconstruction formula is obtained. In addition, a nonlinear correction scheme is devised to improve the results dramatically when the deformation is large. The method requires only a single incident wave at a fixed frequency. Numerical tests show that the method is stable and effective for near-field imaging of obstacles with subwavelength resolution.
2015, 9(1): 211-229
doi: 10.3934/ipi.2015.9.211
+[Abstract](2107)
+[PDF](6005.7KB)
Abstract:
This paper proposes a novel approach to reconstruct changes in a target conductivity from electrical impedance tomography measurements. As in the conventional difference imaging, the reconstruction of the conductivity change is based on electrical potential measurements from the exterior boundary of the target before and after the change. In this paper, however, images of the conductivity before and after the change are reconstructed simultaneously based on the two data sets. The key feature of the approach is that the conductivity after the change is parameterized as a linear combination of the initial state and the change. This allows for modeling independently the spatial characteristics of the background conductivity and the change of the conductivity - by separate regularization functionals. The approach also allows in a straightforward way the restriction of the conductivity change to a localized region of interest inside the domain. While conventional difference imaging reconstruction is based on a global linearization of the observation model, the proposed approach amounts to solving a non-linear inverse problem. The feasibility of the proposed reconstruction method is tested experimentally and with a simulation which demonstrates a potential new medical application of electrical impedance tomography: imaging of vocal folds in voice loading studies.
This paper proposes a novel approach to reconstruct changes in a target conductivity from electrical impedance tomography measurements. As in the conventional difference imaging, the reconstruction of the conductivity change is based on electrical potential measurements from the exterior boundary of the target before and after the change. In this paper, however, images of the conductivity before and after the change are reconstructed simultaneously based on the two data sets. The key feature of the approach is that the conductivity after the change is parameterized as a linear combination of the initial state and the change. This allows for modeling independently the spatial characteristics of the background conductivity and the change of the conductivity - by separate regularization functionals. The approach also allows in a straightforward way the restriction of the conductivity change to a localized region of interest inside the domain. While conventional difference imaging reconstruction is based on a global linearization of the observation model, the proposed approach amounts to solving a non-linear inverse problem. The feasibility of the proposed reconstruction method is tested experimentally and with a simulation which demonstrates a potential new medical application of electrical impedance tomography: imaging of vocal folds in voice loading studies.
2015, 9(1): 231-238
doi: 10.3934/ipi.2015.9.231
+[Abstract](1331)
+[PDF](312.4KB)
Abstract:
Recently, many practical algorithms have been proposed to recover the sparse signal from fewer measurements. Orthogonal matching pursuit (OMP) is one of the most effective algorithm. In this paper, we use the restricted isometry property to analysis OMP. We show that, under certain conditions based on the restricted isometry property and the signals, OMP will recover the support of the sparse signal when measurements are corrupted by additive noise.
Recently, many practical algorithms have been proposed to recover the sparse signal from fewer measurements. Orthogonal matching pursuit (OMP) is one of the most effective algorithm. In this paper, we use the restricted isometry property to analysis OMP. We show that, under certain conditions based on the restricted isometry property and the signals, OMP will recover the support of the sparse signal when measurements are corrupted by additive noise.
2015, 9(1): 239-255
doi: 10.3934/ipi.2015.9.239
+[Abstract](1492)
+[PDF](375.4KB)
Abstract:
The inverse spectral-scattering problems for the radial Schrödinger equation on the half-line are considered with a real-valued integrable potential with a finite moment. It is shown that if the potential is sufficiently smooth in a neighborhood of the origin and its derivatives are known, then it is uniquely determined on the half-line in terms of the amplitude or scattering phase of the Jost function without bound state data, that is, the bound state data is missing.
The inverse spectral-scattering problems for the radial Schrödinger equation on the half-line are considered with a real-valued integrable potential with a finite moment. It is shown that if the potential is sufficiently smooth in a neighborhood of the origin and its derivatives are known, then it is uniquely determined on the half-line in terms of the amplitude or scattering phase of the Jost function without bound state data, that is, the bound state data is missing.
2015, 9(1): 257-272
doi: 10.3934/ipi.2015.9.257
+[Abstract](1705)
+[PDF](1854.5KB)
Abstract:
In this paper, we propose an iterative method for solving the $\ell_1$-regularized minimization problem $\min_{x\in\mathbb{R}^n} f(x)+\rho^T |x|$, which has great applications in the areas of compressive sensing. The construction of our method consists of two main steps: (1) to reformulate an $\ell_1$-problem into a nonsmooth equation $h^\tau(x)={\bf 0}$, where ${\bf 0}\in \mathbb{R}^n$ is the zero vector; and (2) to use $-h^\tau(x)$ as a search direction. The proposed method can be regarded as spectral residual method because we use the residual as a search direction at each iteration. Under mild conditions, we establish the global convergence of the method with a nonmonotone line search. Some numerical experiments are performed to compare the behavior of the proposed method with three other methods. The results positively support the proposed method.
In this paper, we propose an iterative method for solving the $\ell_1$-regularized minimization problem $\min_{x\in\mathbb{R}^n} f(x)+\rho^T |x|$, which has great applications in the areas of compressive sensing. The construction of our method consists of two main steps: (1) to reformulate an $\ell_1$-problem into a nonsmooth equation $h^\tau(x)={\bf 0}$, where ${\bf 0}\in \mathbb{R}^n$ is the zero vector; and (2) to use $-h^\tau(x)$ as a search direction. The proposed method can be regarded as spectral residual method because we use the residual as a search direction at each iteration. Under mild conditions, we establish the global convergence of the method with a nonmonotone line search. Some numerical experiments are performed to compare the behavior of the proposed method with three other methods. The results positively support the proposed method.
2015, 9(1): 273-287
doi: 10.3934/ipi.2015.9.273
+[Abstract](1400)
+[PDF](404.5KB)
Abstract:
The transmission eigenvalue problem on the interval $\left[ a,b\right] $ is considered. We show that the isolated transmission eigenvalues are continuous functions of the coefficients of the problem and that the transmission eigenfunctions can be normalized so that they depend continuously on all coefficients in the uniform norm. Throughout this work, our results are established without assumptions on the sign of the contrasts.
The transmission eigenvalue problem on the interval $\left[ a,b\right] $ is considered. We show that the isolated transmission eigenvalues are continuous functions of the coefficients of the problem and that the transmission eigenfunctions can be normalized so that they depend continuously on all coefficients in the uniform norm. Throughout this work, our results are established without assumptions on the sign of the contrasts.
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