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1930-8337
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Inverse Problems & Imaging
August 2014 , Volume 8 , Issue 3
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2014, 8(3): 611-644
doi: 10.3934/ipi.2014.8.611
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Abstract:
In this paper we consider the problem of determining an unknown pair $\lambda$, $\mu$ of piecewise constant Lamé parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous dependence of $\lambda$ and $\mu$ from the Dirichlet to Neumann map.
In this paper we consider the problem of determining an unknown pair $\lambda$, $\mu$ of piecewise constant Lamé parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous dependence of $\lambda$ and $\mu$ from the Dirichlet to Neumann map.
2014, 8(3): 645-683
doi: 10.3934/ipi.2014.8.645
+[Abstract](1039)
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Abstract:
It was shown in [Garnier et al., SIAM J. Imaging Sciences 2 (2009), 396] that it is possible to image reflectors by backpropagating cross correlations of signals generated by ambient noise sources and recorded at passive sensor arrays. The resolution of the image depends on the directional diversity of the noise signals relative to the locations of the sensor array and the reflector. When directional diversity is limited it is possible to enhance it by exploiting the scattering properties of the medium since scatterers will act as secondary noise sources. However, scattering increases the fluctuation level of the cross correlations and therefore tends to destabilize the image by reducing its signal-to-noise ratio. In this paper we study the trade-off in passive, correlation-based imaging between resolution enhancement and signal-to-noise ratio reduction that is due to scattering.
It was shown in [Garnier et al., SIAM J. Imaging Sciences 2 (2009), 396] that it is possible to image reflectors by backpropagating cross correlations of signals generated by ambient noise sources and recorded at passive sensor arrays. The resolution of the image depends on the directional diversity of the noise signals relative to the locations of the sensor array and the reflector. When directional diversity is limited it is possible to enhance it by exploiting the scattering properties of the medium since scatterers will act as secondary noise sources. However, scattering increases the fluctuation level of the cross correlations and therefore tends to destabilize the image by reducing its signal-to-noise ratio. In this paper we study the trade-off in passive, correlation-based imaging between resolution enhancement and signal-to-noise ratio reduction that is due to scattering.
2014, 8(3): 685-711
doi: 10.3934/ipi.2014.8.685
+[Abstract](1562)
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Abstract:
The first order optimality system of a total variation regularization based variational model with $L^2$-data-fitting in image denoising ($L^2$-TV problem) can be expressed as an elliptic variational inequality of the second kind. For a finite element discretization of the variational inequality problem, an a posteriori error residual based error estimator is derived and its reliability and (partial) efficiency are established. The results are applied to solve the $L^2$-TV problem by means of the adaptive finite element method. The adaptive mesh refinement relies on the newly derived a posteriori error estimator and on an additional heuristic providing a local variance estimator to cope with noisy data. The numerical solution of the discrete problem on each level of refinement is obtained by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques and which is stable with respect to noise in the data. Numerical results justifying the advantage of adaptive finite elements solutions are presented.
The first order optimality system of a total variation regularization based variational model with $L^2$-data-fitting in image denoising ($L^2$-TV problem) can be expressed as an elliptic variational inequality of the second kind. For a finite element discretization of the variational inequality problem, an a posteriori error residual based error estimator is derived and its reliability and (partial) efficiency are established. The results are applied to solve the $L^2$-TV problem by means of the adaptive finite element method. The adaptive mesh refinement relies on the newly derived a posteriori error estimator and on an additional heuristic providing a local variance estimator to cope with noisy data. The numerical solution of the discrete problem on each level of refinement is obtained by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques and which is stable with respect to noise in the data. Numerical results justifying the advantage of adaptive finite elements solutions are presented.
2014, 8(3): 713-732
doi: 10.3934/ipi.2014.8.713
+[Abstract](1163)
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Abstract:
We consider the stability of the inverse problem consisting of the determination of a coefficient of order zero $q$, appearing in the Dirichlet initial-boundary value problem for a wave equation $\partial_t^2u-\Delta u+q(x)u=0$ in $(0,T)\times\Omega$, with $\Omega=\omega\times\mathbb{R}$ an unbounded cylindrical waveguide and $\omega$ a bounded smooth domain of $\mathbb{R}^2$, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to the wave equation. Using suitable geometric optics solutions, we prove a Hölder stability estimate in the determination of $q$ from the Dirichlet to Neumann map. Moreover, provided that the coefficient $q$ is lying in a set of functions $\mathcal A$, where, for any $q_1,q_2\in\mathcal A$, $|q_1-q_2|$ attains its maximum in a fixed bounded subset of $\overline{\Omega}$, we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary $(0,T)\times\partial\Omega$.
We consider the stability of the inverse problem consisting of the determination of a coefficient of order zero $q$, appearing in the Dirichlet initial-boundary value problem for a wave equation $\partial_t^2u-\Delta u+q(x)u=0$ in $(0,T)\times\Omega$, with $\Omega=\omega\times\mathbb{R}$ an unbounded cylindrical waveguide and $\omega$ a bounded smooth domain of $\mathbb{R}^2$, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to the wave equation. Using suitable geometric optics solutions, we prove a Hölder stability estimate in the determination of $q$ from the Dirichlet to Neumann map. Moreover, provided that the coefficient $q$ is lying in a set of functions $\mathcal A$, where, for any $q_1,q_2\in\mathcal A$, $|q_1-q_2|$ attains its maximum in a fixed bounded subset of $\overline{\Omega}$, we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary $(0,T)\times\partial\Omega$.
2014, 8(3): 733-760
doi: 10.3934/ipi.2014.8.733
+[Abstract](1411)
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Abstract:
In this paper, we focus on the restoration of images acquired with a new active imaging concept. This new instrument generates a mosaic of active imaging acquisitions. We first describe a simplified forward model of this so-called ``mosaic active imaging''. We also assume a prior on the distribution of images, using the (TV), and deduce a restoration algorithm. This algorithm is a two-stage iterative process which alternates between: i) the estimation of the restored image; ii) the estimation of the acquisition parameters. We then provide the details useful to the implementation of these two steps. In particular, we show that the image estimation can be performed with graph cuts. This allows a fast resolution of this image estimation step. Finally, we detail numerical experiments showing that acquisitions made with a mosaic active imaging device can be restored even under severe noise levels, with few acquisitions.
In this paper, we focus on the restoration of images acquired with a new active imaging concept. This new instrument generates a mosaic of active imaging acquisitions. We first describe a simplified forward model of this so-called ``mosaic active imaging''. We also assume a prior on the distribution of images, using the (TV), and deduce a restoration algorithm. This algorithm is a two-stage iterative process which alternates between: i) the estimation of the restored image; ii) the estimation of the acquisition parameters. We then provide the details useful to the implementation of these two steps. In particular, we show that the image estimation can be performed with graph cuts. This allows a fast resolution of this image estimation step. Finally, we detail numerical experiments showing that acquisitions made with a mosaic active imaging device can be restored even under severe noise levels, with few acquisitions.
2014, 8(3): 761-777
doi: 10.3934/ipi.2014.8.761
+[Abstract](1577)
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Abstract:
Our aim of this article is to reconstruct a signal from undersampled data in the situation that the signal is sparse in terms of a tight frame. We present a condition, which is independent of the coherence of the tight frame, to guarantee accurate recovery of signals which are sparse in the tight frame, from undersampled data with minimal $l_1$-norm of transform coefficients. This improves the result in [4]. Also, the $l_q$-minimization $(0 < q < 1)$ approaches are introduced. We show that under a suitable condition, there exists a value $q_0\in(0,1]$ such that for any $q\in(0,q_0)$, each solution of the $l_q$-minimization is approximately well to the true signal. In particular, when the tight frame is an identity matrix or an orthonormal basis, all results obtained in this paper appeared in [18] and [17].
Our aim of this article is to reconstruct a signal from undersampled data in the situation that the signal is sparse in terms of a tight frame. We present a condition, which is independent of the coherence of the tight frame, to guarantee accurate recovery of signals which are sparse in the tight frame, from undersampled data with minimal $l_1$-norm of transform coefficients. This improves the result in [4]. Also, the $l_q$-minimization $(0 < q < 1)$ approaches are introduced. We show that under a suitable condition, there exists a value $q_0\in(0,1]$ such that for any $q\in(0,q_0)$, each solution of the $l_q$-minimization is approximately well to the true signal. In particular, when the tight frame is an identity matrix or an orthonormal basis, all results obtained in this paper appeared in [18] and [17].
2014, 8(3): 779-794
doi: 10.3934/ipi.2014.8.779
+[Abstract](1109)
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Abstract:
In this paper we develop and analyse a direct method, based on the so called reciprocity gap functional, for retrieving the location of an elastic inclusion. The method requires a pair of displacement (imposed) and traction (measured) data, on an accessible part of the boundary. We provide a criterion for the choice of displacement that, in one hand, provides more accurate results and on the other, does not require the Lamé parameters of background medium. We use this property to develop a two boundary measurements direct method for retrieving both Lamé parameters. Several numerical examples are presented in order to illustrate the accuracy and stability of the proposed methods.
In this paper we develop and analyse a direct method, based on the so called reciprocity gap functional, for retrieving the location of an elastic inclusion. The method requires a pair of displacement (imposed) and traction (measured) data, on an accessible part of the boundary. We provide a criterion for the choice of displacement that, in one hand, provides more accurate results and on the other, does not require the Lamé parameters of background medium. We use this property to develop a two boundary measurements direct method for retrieving both Lamé parameters. Several numerical examples are presented in order to illustrate the accuracy and stability of the proposed methods.
2014, 8(3): 795-810
doi: 10.3934/ipi.2014.8.795
+[Abstract](1292)
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Abstract:
We prove Weyl-type asymptotic formulas for the real and the complex internal transmission eigenvalues when the domain is a ball and the index of refraction is constant.
We prove Weyl-type asymptotic formulas for the real and the complex internal transmission eigenvalues when the domain is a ball and the index of refraction is constant.
2014, 8(3): 811-829
doi: 10.3934/ipi.2014.8.811
+[Abstract](1763)
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Abstract:
Quantitative photoacoustic tomography is a hybrid imaging method, combining near-infrared optical and ultrasonic imaging. One of the interests of the method is the reconstruction of the optical absorption coefficient within the target. The measurement depends also on the uninteresting but often unknown optical scattering coefficient. In this work, we apply the approximation error method for handling uncertainty related to the unknown scattering and reconstruct the absorption only. This way the number of unknown parameters can be reduced in the inverse problem in comparison to the case of estimating all the unknown parameters. The approximation error approach is evaluated with data simulated using the diffusion approximation and Monte Carlo method. Estimates are inspected in four two-dimensional cases with biologically relevant parameter values. Estimates obtained with the approximation error approach are compared to estimates where both the absorption and scattering coefficient are reconstructed, as well to estimates where the absorption is reconstructed, but the scattering is assumed (incorrect) fixed value. The approximation error approach is found to give better estimates for absorption in comparison to estimates with the conventional measurement error model using fixed scattering. When the true scattering contains stronger variations, improvement of the approximation error method over fixed scattering assumption is more significant.
Quantitative photoacoustic tomography is a hybrid imaging method, combining near-infrared optical and ultrasonic imaging. One of the interests of the method is the reconstruction of the optical absorption coefficient within the target. The measurement depends also on the uninteresting but often unknown optical scattering coefficient. In this work, we apply the approximation error method for handling uncertainty related to the unknown scattering and reconstruct the absorption only. This way the number of unknown parameters can be reduced in the inverse problem in comparison to the case of estimating all the unknown parameters. The approximation error approach is evaluated with data simulated using the diffusion approximation and Monte Carlo method. Estimates are inspected in four two-dimensional cases with biologically relevant parameter values. Estimates obtained with the approximation error approach are compared to estimates where both the absorption and scattering coefficient are reconstructed, as well to estimates where the absorption is reconstructed, but the scattering is assumed (incorrect) fixed value. The approximation error approach is found to give better estimates for absorption in comparison to estimates with the conventional measurement error model using fixed scattering. When the true scattering contains stronger variations, improvement of the approximation error method over fixed scattering assumption is more significant.
2014, 8(3): 831-844
doi: 10.3934/ipi.2014.8.831
+[Abstract](1578)
+[PDF](391.2KB)
Abstract:
We propose a novel framework for studying radar pulse compression with continuous waveforms. Our methodology is based on the recent developments of the mathematical theory of comparison of measurements. First we show that a radar measurement of a time-independent but spatially distributed radar target is rigorously more informative than another one if the modulus of the Fourier transform of the radar code is greater than or equal to the modulus of the Fourier transform of the second radar code. We then motivate the study by spreading a Gaussian pulse into a longer pulse with smaller peak power and re-compressing the spread pulse into its original form. We then review the basic concepts of the theory and pose the conditions for statistically equivalent radar experiments. We show that such experiments can be constructed by spreading the radar pulses via multiplication of their Fourier transforms by unimodular functions. Finally, we show by analytical and numerical methods some examples of the spreading and re-compression of certain simple pulses.
We propose a novel framework for studying radar pulse compression with continuous waveforms. Our methodology is based on the recent developments of the mathematical theory of comparison of measurements. First we show that a radar measurement of a time-independent but spatially distributed radar target is rigorously more informative than another one if the modulus of the Fourier transform of the radar code is greater than or equal to the modulus of the Fourier transform of the second radar code. We then motivate the study by spreading a Gaussian pulse into a longer pulse with smaller peak power and re-compressing the spread pulse into its original form. We then review the basic concepts of the theory and pose the conditions for statistically equivalent radar experiments. We show that such experiments can be constructed by spreading the radar pulses via multiplication of their Fourier transforms by unimodular functions. Finally, we show by analytical and numerical methods some examples of the spreading and re-compression of certain simple pulses.
2014, 8(3): 845-863
doi: 10.3934/ipi.2014.8.845
+[Abstract](1300)
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Abstract:
The level set method [33] is a commonly used framework for image segmentation algorithms. For edge detection and segmentation models, the standard level set method provides a flexible curve representation and implementation. However, one drawback has been in the types of curves that can be represented in this standard method. In the classical level set method, the curve must enclose an open set (i.e. loops or contours without endpoints). Thus the classical framework is limited to locating edge sets without endpoints. Using the curve representation from [37,36], we construct a segmentation and edge detection method which can locate arcs (i.e. curves with free endpoints) as well as standard contours. Within this new framework, the variational segmentation model presented here is able to detect general edge structures and linear objects. This energy is composed of two terms, an edge set regularizer and an edge attractor. Our variational model is related to the Mumford and Shah model [29] for joint segmentation and restoration in terms of an asymptotic limit, and in addition, is both general and flexible in terms of its uses and its applications. Numerical results are given on images with a variety of edge structures.
The level set method [33] is a commonly used framework for image segmentation algorithms. For edge detection and segmentation models, the standard level set method provides a flexible curve representation and implementation. However, one drawback has been in the types of curves that can be represented in this standard method. In the classical level set method, the curve must enclose an open set (i.e. loops or contours without endpoints). Thus the classical framework is limited to locating edge sets without endpoints. Using the curve representation from [37,36], we construct a segmentation and edge detection method which can locate arcs (i.e. curves with free endpoints) as well as standard contours. Within this new framework, the variational segmentation model presented here is able to detect general edge structures and linear objects. This energy is composed of two terms, an edge set regularizer and an edge attractor. Our variational model is related to the Mumford and Shah model [29] for joint segmentation and restoration in terms of an asymptotic limit, and in addition, is both general and flexible in terms of its uses and its applications. Numerical results are given on images with a variety of edge structures.
2014, 8(3): 865-883
doi: 10.3934/ipi.2014.8.865
+[Abstract](1311)
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Abstract:
We give several generalizations of Rellich's classical uniqueness theorem to unbounded domains. We give a natural half-space generalization for super-exponentially decaying inhomogeneities using real variable techniques. We also prove under super-exponential decay a discrete generalization where the inhomogeneity only needs to vanish in a suitable cone.
The more traditional complex variable techniques are used to prove the half-space result again, but with less exponential decay, and a variant with polynomial decay, but with supports exponentially thin at infinity. As an application, we prove the discreteness of non-scattering energies for non-compactly supported potentials with suitable asymptotic behaviours and supports.
We give several generalizations of Rellich's classical uniqueness theorem to unbounded domains. We give a natural half-space generalization for super-exponentially decaying inhomogeneities using real variable techniques. We also prove under super-exponential decay a discrete generalization where the inhomogeneity only needs to vanish in a suitable cone.
The more traditional complex variable techniques are used to prove the half-space result again, but with less exponential decay, and a variant with polynomial decay, but with supports exponentially thin at infinity. As an application, we prove the discreteness of non-scattering energies for non-compactly supported potentials with suitable asymptotic behaviours and supports.
2014, 8(3): 885-900
doi: 10.3934/ipi.2014.8.885
+[Abstract](1896)
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Abstract:
We discuss shape reconstruction methods for data presented in various image spaces. We demonstrate the usefulness of the Fourier transform in transferring image data and shape model projections to a domain more suitable for shape inversion. Using boundary contours in images to represent minimal information, we present uniqueness results for shapes recoverable from interferometric and range-Doppler data. We present applications of our methods to adaptive optics, interferometry, and range-Doppler images.
We discuss shape reconstruction methods for data presented in various image spaces. We demonstrate the usefulness of the Fourier transform in transferring image data and shape model projections to a domain more suitable for shape inversion. Using boundary contours in images to represent minimal information, we present uniqueness results for shapes recoverable from interferometric and range-Doppler data. We present applications of our methods to adaptive optics, interferometry, and range-Doppler images.
2014, 8(3): 901-923
doi: 10.3934/ipi.2014.8.901
+[Abstract](1477)
+[PDF](832.8KB)
Abstract:
In signal acquisition, Toeplitz and circulant matrices are widely used as sensing operators. They correspond to discrete convolutions and are easily or even naturally realized in various applications. For compressive sensing, recent work has used random Toeplitz and circulant sensing matrices and proved their efficiency in theory, by computer simulations, as well as through physical optical experiments. Motivated by recent work [8], we propose models to learn a circulant sensing matrix/operator for one and higher dimensional signals. Given the dictionary of the signal(s) to be sensed, the learned circulant sensing matrix/operator is more effective than a randomly generated circulant sensing matrix/operator, and even slightly so than a (non-circulant) Gaussian random sensing matrix. In addition, by exploiting the circulant structure, we improve the learning from the patch scale in [8] to the much large image scale. Furthermore, we test learning the circulant sensing matrix/operator and the nonparametric dictionary altogether and obtain even better performance. We demonstrate these results using both synthetic sparse signals and real images.
In signal acquisition, Toeplitz and circulant matrices are widely used as sensing operators. They correspond to discrete convolutions and are easily or even naturally realized in various applications. For compressive sensing, recent work has used random Toeplitz and circulant sensing matrices and proved their efficiency in theory, by computer simulations, as well as through physical optical experiments. Motivated by recent work [8], we propose models to learn a circulant sensing matrix/operator for one and higher dimensional signals. Given the dictionary of the signal(s) to be sensed, the learned circulant sensing matrix/operator is more effective than a randomly generated circulant sensing matrix/operator, and even slightly so than a (non-circulant) Gaussian random sensing matrix. In addition, by exploiting the circulant structure, we improve the learning from the patch scale in [8] to the much large image scale. Furthermore, we test learning the circulant sensing matrix/operator and the nonparametric dictionary altogether and obtain even better performance. We demonstrate these results using both synthetic sparse signals and real images.
2014, 8(3): 925-937
doi: 10.3934/ipi.2014.8.925
+[Abstract](1704)
+[PDF](2802.5KB)
Abstract:
The problem of compressive-sensing (CS) L2-L1-TV reconstruction of magnetic resonance (MR) scans from undersampled $k$-space data has been addressed in numerous studies. However, the regularization parameters in models of CS L2-L1-TV reconstruction are rarely studied. Once the regularization parameters are given, the solution for an MR reconstruction model is fixed and is less effective in the case of strong noise. To overcome this shortcoming, we present a new alternating formulation to replace the standard L2-L1-TV reconstruction model. A weighted-average alternating minimization method is proposed based on this new formulation and a convergence analysis of the method is carried out. The advantages of and the motivation for the proposed alternating formulation are explained. Experimental results demonstrate that the proposed formulation yields better reconstruction results in the case of strong noise and can improve image reconstruction via flexible parameter selection.
The problem of compressive-sensing (CS) L2-L1-TV reconstruction of magnetic resonance (MR) scans from undersampled $k$-space data has been addressed in numerous studies. However, the regularization parameters in models of CS L2-L1-TV reconstruction are rarely studied. Once the regularization parameters are given, the solution for an MR reconstruction model is fixed and is less effective in the case of strong noise. To overcome this shortcoming, we present a new alternating formulation to replace the standard L2-L1-TV reconstruction model. A weighted-average alternating minimization method is proposed based on this new formulation and a convergence analysis of the method is carried out. The advantages of and the motivation for the proposed alternating formulation are explained. Experimental results demonstrate that the proposed formulation yields better reconstruction results in the case of strong noise and can improve image reconstruction via flexible parameter selection.
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