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1930-8337
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Inverse Problems & Imaging
November 2013 , Volume 7 , Issue 4
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2013, 7(4): 1115-1122
doi: 10.3934/ipi.2013.7.1115
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Abstract:
We consider the AKNS (Ablowitz-Kaup-Newell-Segur) operator on the unit interval with potentials belonging to Sobolev spaces in the framework of inverse spectral theory. Precise sets of eigenvalues are given in order that they, together with the knowledge of the potentials on the side $(a,1)$ and partial informations on the potential on $(a-\varepsilon,a)$ for some arbitrary small $\varepsilon>0$, determine the potentials entirely on $(0,1)$. Naturally, the smaller is $a$ and the more partial informations are known, the less is the number of the needed eigenvalues.
We consider the AKNS (Ablowitz-Kaup-Newell-Segur) operator on the unit interval with potentials belonging to Sobolev spaces in the framework of inverse spectral theory. Precise sets of eigenvalues are given in order that they, together with the knowledge of the potentials on the side $(a,1)$ and partial informations on the potential on $(a-\varepsilon,a)$ for some arbitrary small $\varepsilon>0$, determine the potentials entirely on $(0,1)$. Naturally, the smaller is $a$ and the more partial informations are known, the less is the number of the needed eigenvalues.
2013, 7(4): 1123-1138
doi: 10.3934/ipi.2013.7.1123
+[Abstract](1663)
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Abstract:
We use the Factorization method to retrieve the shape of cracks with impedance boundary conditions from farfields associated with incident plane waves at a fixed frequency. This work is an extension of the study initiated by Kirsch and Ritter [Inverse Problems, 16, pp. 89-105, 2000] where the case of sound soft cracks is considered. We address here the scalar problem and provide theoretical validation of the method when the impedance boundary conditions hold on both sides of the crack. We then deduce an inversion algorithm and present some validating numerical results in the case of simply and multiply connected cracks.
We use the Factorization method to retrieve the shape of cracks with impedance boundary conditions from farfields associated with incident plane waves at a fixed frequency. This work is an extension of the study initiated by Kirsch and Ritter [Inverse Problems, 16, pp. 89-105, 2000] where the case of sound soft cracks is considered. We address here the scalar problem and provide theoretical validation of the method when the impedance boundary conditions hold on both sides of the crack. We then deduce an inversion algorithm and present some validating numerical results in the case of simply and multiply connected cracks.
2013, 7(4): 1139-1155
doi: 10.3934/ipi.2013.7.1139
+[Abstract](2053)
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Abstract:
Continuing our previous work [6, Inverse Problems, 2012, 28, 055002] and [5, Inverse Problems, 2012, 28, 055001], we address the ill-posedness of the inverse scattering problem of electromagnetic waves due to an inhomogeneous medium by studying the Hessian of the data misfit. We derive and analyze the Hessian in both Hölder and Sobolev spaces. Using an integral equation approach based on Newton potential theory and compact embeddings in Hölder and Sobolev spaces, we show that the Hessian can be decomposed into three components, all of which are shown to be compact operators. The implication of the compactness of the Hessian is that for small data noise and model error, the discrete Hessian can be approximated by a low-rank matrix. This in turn enables fast solution of an appropriately regularized inverse problem, as well as Gaussian-based quantification of uncertainty in the estimated inhomogeneity.
Continuing our previous work [6, Inverse Problems, 2012, 28, 055002] and [5, Inverse Problems, 2012, 28, 055001], we address the ill-posedness of the inverse scattering problem of electromagnetic waves due to an inhomogeneous medium by studying the Hessian of the data misfit. We derive and analyze the Hessian in both Hölder and Sobolev spaces. Using an integral equation approach based on Newton potential theory and compact embeddings in Hölder and Sobolev spaces, we show that the Hessian can be decomposed into three components, all of which are shown to be compact operators. The implication of the compactness of the Hessian is that for small data noise and model error, the discrete Hessian can be approximated by a low-rank matrix. This in turn enables fast solution of an appropriately regularized inverse problem, as well as Gaussian-based quantification of uncertainty in the estimated inhomogeneity.
2013, 7(4): 1157-1182
doi: 10.3934/ipi.2013.7.1157
+[Abstract](1776)
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Abstract:
The aim of this paper is to formulate a class of inverse problems of particular relevance in crowded motion, namely the simultaneous identification of entropies and mobilities. We study a model case of this class, which is the identification from flux-based measurements in a stationary setup. This leads to an inverse problem for a nonlinear transport-diffusion model, where boundary values and possibly an external potential can be varied. In specific settings we provide a detailed theory for the forward map and an adjoint problem useful in the analysis and numerical solution. We further verify the simultaneous identifiability of the nonlinearities and present several numerical tests yielding further insight into the way variations in boundary values and external potential affect the quality of reconstructions.
The aim of this paper is to formulate a class of inverse problems of particular relevance in crowded motion, namely the simultaneous identification of entropies and mobilities. We study a model case of this class, which is the identification from flux-based measurements in a stationary setup. This leads to an inverse problem for a nonlinear transport-diffusion model, where boundary values and possibly an external potential can be varied. In specific settings we provide a detailed theory for the forward map and an adjoint problem useful in the analysis and numerical solution. We further verify the simultaneous identifiability of the nonlinearities and present several numerical tests yielding further insight into the way variations in boundary values and external potential affect the quality of reconstructions.
2013, 7(4): 1183-1214
doi: 10.3934/ipi.2013.7.1183
+[Abstract](3535)
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Abstract:
We propose a nonsmooth PDE-constrained optimization approach for the determination of the correct noise model in total variation (TV) image denoising. An optimization problem for the determination of the weights corresponding to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems.
We propose a nonsmooth PDE-constrained optimization approach for the determination of the correct noise model in total variation (TV) image denoising. An optimization problem for the determination of the weights corresponding to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems.
2013, 7(4): 1215-1233
doi: 10.3934/ipi.2013.7.1215
+[Abstract](1845)
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Abstract:
Rapid acquisition of magnetic resonance (MR) images via reconstruction from undersampled $k$-space data has the potential to greatly decrease MRI scan time on existing medical hardware. To this end, iterative image reconstruction based on the technique of compressed sensing has become the method choice for many researchers [1]. However, while conventional compressed sensing relies on random measurements from a discrete Fourier transform, actual MR scans often suffer from off-resonance effects and thus generate data by way of a non-Fourier operator [2]. Correcting for these effects requires that one employs more sophisticated image reconstruction methods and introduces computational bottlenecks that are not encountered in standard compressed sensing.
In this work, we demonstrate how one may accelerate the convergence of algorithms for solving the image reconstruction problem, \begin{equation}\label{eq:caneq} (1) \underset{\rho}{argmin} J(\rho) subject to A \rho = s \end{equation} by opting for a regularization of the form: \begin{equation}\label{eq:hybrid} (2) J(\rho) = | \nabla \rho| + \nu | F \rho | \end{equation} when $F$ is a tight frame and $A$ is only approximately a Fourier transform. In our experiments, reconstructing field-corrected MR images with the hybrid regularization of 2 provides a speedup of roughly one order of magnitude when compared with an approach based solely on total-variation and may produce higher quality images than an approach based solely on tight frames.
Rapid acquisition of magnetic resonance (MR) images via reconstruction from undersampled $k$-space data has the potential to greatly decrease MRI scan time on existing medical hardware. To this end, iterative image reconstruction based on the technique of compressed sensing has become the method choice for many researchers [1]. However, while conventional compressed sensing relies on random measurements from a discrete Fourier transform, actual MR scans often suffer from off-resonance effects and thus generate data by way of a non-Fourier operator [2]. Correcting for these effects requires that one employs more sophisticated image reconstruction methods and introduces computational bottlenecks that are not encountered in standard compressed sensing.
In this work, we demonstrate how one may accelerate the convergence of algorithms for solving the image reconstruction problem, \begin{equation}\label{eq:caneq} (1) \underset{\rho}{argmin} J(\rho) subject to A \rho = s \end{equation} by opting for a regularization of the form: \begin{equation}\label{eq:hybrid} (2) J(\rho) = | \nabla \rho| + \nu | F \rho | \end{equation} when $F$ is a tight frame and $A$ is only approximately a Fourier transform. In our experiments, reconstructing field-corrected MR images with the hybrid regularization of 2 provides a speedup of roughly one order of magnitude when compared with an approach based solely on total-variation and may produce higher quality images than an approach based solely on tight frames.
2013, 7(4): 1235-1250
doi: 10.3934/ipi.2013.7.1235
+[Abstract](1607)
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Abstract:
We consider a mathematical model of thermoacoustic tomography and other multi-wave imaging techniques with variable sound speed and attenuation. We find that a Neumann series reconstruction algorithm, previously studied under the assumption of zero attenuation, still converges if attenuation is sufficiently small. With complete boundary data, we show the inverse problem has a unique solution, and modified time reversal provides a stable reconstruction. We also consider partial boundary data, and in this case study those singularities that can be stably recovered.
We consider a mathematical model of thermoacoustic tomography and other multi-wave imaging techniques with variable sound speed and attenuation. We find that a Neumann series reconstruction algorithm, previously studied under the assumption of zero attenuation, still converges if attenuation is sufficiently small. With complete boundary data, we show the inverse problem has a unique solution, and modified time reversal provides a stable reconstruction. We also consider partial boundary data, and in this case study those singularities that can be stably recovered.
2013, 7(4): 1251-1270
doi: 10.3934/ipi.2013.7.1251
+[Abstract](1547)
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Abstract:
Source imaging maps back boundary measurements to underlying generators within the domain; e.g., retrieving the parameters of the generating dipoles from electrical potential measurements on the scalp such as in electroencephalography (EEG). Fitting such a parametric source model is non-linear in the positions of the sources and renewed interest in mathematical imaging has led to several promising approaches.
One important step in these methods is the application of a sensing principle that links the boundary measurements to volumetric information about the sources. This principle is based on the divergence theorem and a mathematical test function that needs to be an homogeneous solution of the governing equations (i.e., Poisson's equation). For a specific choice of the test function, we have devised an algebraic non-iterative source localization technique for which we have coined the term ``analytic sensing''.
Until now, this sensing principle has been applied to homogeneous-conductivity spherical models only. Here, we extend it for multi-layer spherical models that are commonly applied in EEG. We obtain a closed-form expression for the test function that can then be applied for subsequent localization. A simulation study show the feasibility of the proposed approach.
Source imaging maps back boundary measurements to underlying generators within the domain; e.g., retrieving the parameters of the generating dipoles from electrical potential measurements on the scalp such as in electroencephalography (EEG). Fitting such a parametric source model is non-linear in the positions of the sources and renewed interest in mathematical imaging has led to several promising approaches.
One important step in these methods is the application of a sensing principle that links the boundary measurements to volumetric information about the sources. This principle is based on the divergence theorem and a mathematical test function that needs to be an homogeneous solution of the governing equations (i.e., Poisson's equation). For a specific choice of the test function, we have devised an algebraic non-iterative source localization technique for which we have coined the term ``analytic sensing''.
Until now, this sensing principle has been applied to homogeneous-conductivity spherical models only. Here, we extend it for multi-layer spherical models that are commonly applied in EEG. We obtain a closed-form expression for the test function that can then be applied for subsequent localization. A simulation study show the feasibility of the proposed approach.
2013, 7(4): 1271-1293
doi: 10.3934/ipi.2013.7.1271
+[Abstract](2412)
+[PDF](763.8KB)
Abstract:
We propose an imaging technique for the detection of porous inclusions in a stationary flow based on the Factorization method. The stationary flow is described by the Stokes-Brinkmann equations, a standard model for a flow through a (partially) porous medium, involving the deformation tensor of the flow and the permeability tensor of the porous inclusion. On the boundary of the domain we prescribe Robin boundary conditions that provide the freedom to model, e.g., in- or outlets for the flow. The direct Stokes-Brinkmann problem to find a velocity field and a pressure for given boundary data is a mixed variational problem lacking coercivity due to the indefinite pressure part. It is well-known that indefinite problems are difficult to tackle theoretically using Factorization methods. Interestingly, the Factorization method can nevertheless be applied to this non-coercive problem, as long as one uses data consisting only of velocity measurements. We provide numerical experiments showing the feasibility of the proposed technique.
We propose an imaging technique for the detection of porous inclusions in a stationary flow based on the Factorization method. The stationary flow is described by the Stokes-Brinkmann equations, a standard model for a flow through a (partially) porous medium, involving the deformation tensor of the flow and the permeability tensor of the porous inclusion. On the boundary of the domain we prescribe Robin boundary conditions that provide the freedom to model, e.g., in- or outlets for the flow. The direct Stokes-Brinkmann problem to find a velocity field and a pressure for given boundary data is a mixed variational problem lacking coercivity due to the indefinite pressure part. It is well-known that indefinite problems are difficult to tackle theoretically using Factorization methods. Interestingly, the Factorization method can nevertheless be applied to this non-coercive problem, as long as one uses data consisting only of velocity measurements. We provide numerical experiments showing the feasibility of the proposed technique.
2013, 7(4): 1295-1305
doi: 10.3934/ipi.2013.7.1295
+[Abstract](1689)
+[PDF](654.2KB)
Abstract:
This paper presents a new Synthetic Aperture Radar (SAR) imaging system based on compressive sampling scheme and $l_1$ minimization. The compressive sampling scheme comprises of randomization and integration of radar echoes which slows down the analog-to-digital converters (ADC) rate significantly without an aliasing in image formation. Numerical experiments indicate that the resolution of SAR images retrieved by our method outperform that obtained by conventional methods. The results also reveal that the new SAR imaging system can still retrieve non-ambiguous images even when the data rate is $\frac{1}{10}$ of the original one. Finally, we applied the new method on raw data of RADARSAT-1 to testify its practicability.
This paper presents a new Synthetic Aperture Radar (SAR) imaging system based on compressive sampling scheme and $l_1$ minimization. The compressive sampling scheme comprises of randomization and integration of radar echoes which slows down the analog-to-digital converters (ADC) rate significantly without an aliasing in image formation. Numerical experiments indicate that the resolution of SAR images retrieved by our method outperform that obtained by conventional methods. The results also reveal that the new SAR imaging system can still retrieve non-ambiguous images even when the data rate is $\frac{1}{10}$ of the original one. Finally, we applied the new method on raw data of RADARSAT-1 to testify its practicability.
2013, 7(4): 1307-1329
doi: 10.3934/ipi.2013.7.1307
+[Abstract](1500)
+[PDF](446.9KB)
Abstract:
We study the stability for the direct acoustic scattering problem with sound-hard scatterers with minimal regularity assumptions on the scatterers. The main tool we use for this purpose is the convergence in the sense of Mosco.
We obtain uniform decay estimates for scattered fields and we investigate how a sound-hard screen may be approximated by thin sound-hard obstacles.
We study the stability for the direct acoustic scattering problem with sound-hard scatterers with minimal regularity assumptions on the scatterers. The main tool we use for this purpose is the convergence in the sense of Mosco.
We obtain uniform decay estimates for scattered fields and we investigate how a sound-hard screen may be approximated by thin sound-hard obstacles.
2013, 7(4): 1331-1366
doi: 10.3934/ipi.2013.7.1331
+[Abstract](1549)
+[PDF](1505.7KB)
Abstract:
The standard approach for image reconstruction is to stabilize the problem by including an edge-preserving roughness penalty in addition to faithfulness to the data. However, this methodology produces noisy object boundaries and creates a staircase effect. State-of-the-art methods to correct these undesirable effects either have weak convergence guarantees or are limited to specific situations; furthermore, most of them use a quadratic data-fidelity term. In this paper, we propose a simple alternative regularization model to improve contour regularity and to reduce the staircase effect-our model incorporates the smoothness of the edge field in an implicit way by adding a simple penalty term defined in the wavelet domain. We also derive an efficient half-quadratic algorithm to solve the resulting optimization problem, including the case when the data-fidelity term is not quadratic and the cost function is not convex. Our approach either extends or supplements existing methods and offers strong convergence guarantees. Numerical experiments show that it outperforms first-order total variation regularization as well as state-of-the-art second-order regularization techniques.
The standard approach for image reconstruction is to stabilize the problem by including an edge-preserving roughness penalty in addition to faithfulness to the data. However, this methodology produces noisy object boundaries and creates a staircase effect. State-of-the-art methods to correct these undesirable effects either have weak convergence guarantees or are limited to specific situations; furthermore, most of them use a quadratic data-fidelity term. In this paper, we propose a simple alternative regularization model to improve contour regularity and to reduce the staircase effect-our model incorporates the smoothness of the edge field in an implicit way by adding a simple penalty term defined in the wavelet domain. We also derive an efficient half-quadratic algorithm to solve the resulting optimization problem, including the case when the data-fidelity term is not quadratic and the cost function is not convex. Our approach either extends or supplements existing methods and offers strong convergence guarantees. Numerical experiments show that it outperforms first-order total variation regularization as well as state-of-the-art second-order regularization techniques.
2013, 7(4): 1367-1377
doi: 10.3934/ipi.2013.7.1367
+[Abstract](1708)
+[PDF](350.7KB)
Abstract:
In this paper we consider the linearized problem of recovering both the sound speed and the thermal absorption arising in thermoacoustic and photoacoustic tomography. We show that the problem is unstable in any scale of Sobolev spaces.
In this paper we consider the linearized problem of recovering both the sound speed and the thermal absorption arising in thermoacoustic and photoacoustic tomography. We show that the problem is unstable in any scale of Sobolev spaces.
2013, 7(4): 1379-1392
doi: 10.3934/ipi.2013.7.1379
+[Abstract](2761)
+[PDF](1801.1KB)
Abstract:
In seismic processing, one goal is to recover missing traces when the data is sparsely and incompletely sampled. We present a method which treats this reconstruction problem from a novel perspective. By utilizing its connection with the general matrix completion (MC) problem, we build an approximately low-rank matrix, which can be reconstructed through solving a proper nuclear norm minimization problem. Two efficient algorithms, accelerated proximal gradient method (APG) and low-rank matrix fitting (LMaFit) are discussed in this paper. The seismic data can then be recovered by the conversion of the completed matrix into the original signal space. Numerical experiments show the efficiency and high performance of data recovery for our model compared with other models.
In seismic processing, one goal is to recover missing traces when the data is sparsely and incompletely sampled. We present a method which treats this reconstruction problem from a novel perspective. By utilizing its connection with the general matrix completion (MC) problem, we build an approximately low-rank matrix, which can be reconstructed through solving a proper nuclear norm minimization problem. Two efficient algorithms, accelerated proximal gradient method (APG) and low-rank matrix fitting (LMaFit) are discussed in this paper. The seismic data can then be recovered by the conversion of the completed matrix into the original signal space. Numerical experiments show the efficiency and high performance of data recovery for our model compared with other models.
2013, 7(4): 1393-1407
doi: 10.3934/ipi.2013.7.1393
+[Abstract](1871)
+[PDF](1832.3KB)
Abstract:
This paper is concerned with the inverse problem of recovering a penetrable grating profile in the TM-polarization case from the scattered field measured only above the structure, corresponding to a countably infinite number of incident quasi-periodic waves. A sampling method is proposed to reconstruct the penetrable grating profile based on a near field linear operator equation in $l^2$. The mathematical justification of the sampling method is established and numerical results are presented to show the validity of the inversion algorithm.
This paper is concerned with the inverse problem of recovering a penetrable grating profile in the TM-polarization case from the scattered field measured only above the structure, corresponding to a countably infinite number of incident quasi-periodic waves. A sampling method is proposed to reconstruct the penetrable grating profile based on a near field linear operator equation in $l^2$. The mathematical justification of the sampling method is established and numerical results are presented to show the validity of the inversion algorithm.
2013, 7(4): 1409-1432
doi: 10.3934/ipi.2013.7.1409
+[Abstract](2088)
+[PDF](1112.5KB)
Abstract:
High order derivative information has been widely used in developing variational models in image processing to accomplish more advanced tasks. However, it is a nontrivial issue to construct efficient numerical algorithms to deal with the minimization of these variational models due to the associated high order Euler-Lagrange equations. In this paper, we propose an efficient numerical method for a mean curvature based image denoising model using the augmented Lagrangian method. A special technique is introduced to handle the mean curvature model for the augmented Lagrangian scheme. We detail the procedures of finding the related saddle-points of the functional. We present numerical experiments to illustrate the effectiveness and efficiency of the proposed numerical method, and show a few important features of the image denoising model such as keeping corners and image contrast. Moreover, a comparison with the gradient descent method further demonstrates the efficiency of the proposed augmented Lagrangian method.
High order derivative information has been widely used in developing variational models in image processing to accomplish more advanced tasks. However, it is a nontrivial issue to construct efficient numerical algorithms to deal with the minimization of these variational models due to the associated high order Euler-Lagrange equations. In this paper, we propose an efficient numerical method for a mean curvature based image denoising model using the augmented Lagrangian method. A special technique is introduced to handle the mean curvature model for the augmented Lagrangian scheme. We detail the procedures of finding the related saddle-points of the functional. We present numerical experiments to illustrate the effectiveness and efficiency of the proposed numerical method, and show a few important features of the image denoising model such as keeping corners and image contrast. Moreover, a comparison with the gradient descent method further demonstrates the efficiency of the proposed augmented Lagrangian method.
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