Showing papers in "Inverse Problems in 2007"
TL;DR: It is shown that ‘1 minimization recovers x 0 exactly when the number of measurements exceeds m Const ·µ 2 (U) ·S · logn, where S is the numberof nonzero components in x 0, and µ is the largest entry in U properly normalized: µ(U) = p n · maxk,j |Uk,j|.
Abstract: We consider the problem of reconstructing a sparse signal x 0 2 R n from a limited number of linear measurements. Given m randomly selected samples of Ux 0 , where U is an orthonormal matrix, we show that ‘1 minimization recovers x 0 exactly when the number of measurements exceeds m Const ·µ 2 (U) ·S · logn, where S is the number of nonzero components in x 0 , and µ is the largest entry in U properly normalized: µ(U) = p n · maxk,j |Uk,j|. The smaller µ, the fewer samples needed. The result holds for “most” sparse signals x 0 supported on a fixed (but arbitrary) set T. Given T, if the sign of x 0 for each nonzero entry on T and the observed values of Ux 0 are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples.
2,187 citations
TL;DR: This paper describes two prior classes, analysis-based and synthesis-based, and shows that although when reducing to the complete and under-complete formulations the two become equivalent, in their more interesting overcomplete formulation the two types depart.
Abstract: The concept of prior probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors assign probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. The algebraic similarity between the two suggests that they could be strongly related; however, in the absence of a detailed study, contradicting approaches have emerged. While the computationally intensive synthesis approach is receiving ever-increasing attention and is notably preferred, other works hypothesize that the two might actually be much closer, going as far as to suggest that one can approximate the other. In this paper we describe the two prior classes in detail, focusing on the distinction between them. We show that although in the simpler complete and undercomplete formulations the two approaches are equivalent, in their overcomplete formulation they depart. Focusing on the l1 case, we present a novel approach for comparing the two types of priors based on high-dimensional polytopal geometry. We arrive at a series of theoretical and numerical results establishing the existence of an unbridgeable gap between the two.
777 citations
TL;DR: This work generalizes the result of showed that lp minimization with 0 < p < 1 recovers sparse signals from fewer linear measurements than does l1 minimization to an lp variant of the restricted isometry property, and determines how many random, Gaussian measurements are sufficient for the condition to hold with high probability.
Abstract: The recently emerged field known as compressive sensing has produced powerful results showing the ability to recover sparse signals from surprisingly few linear measurements, using l1 minimization. In previous work, numerical experiments showed that lp minimization with 0 < p < 1 recovers sparse signals from fewer linear measurements than does l1 minimization. It was also shown that a weaker restricted isometry property is sufficient to guarantee perfect recovery in the lp case. In this work, we generalize this result to an lp variant of the restricted isometry property, and then determine how many random, Gaussian measurements are sufficient for the condition to hold with high probability. The resulting sufficient condition is met by fewer measurements for smaller p. This adds to the theoretical justification for the methods already being applied to replacing high-dose CT scans with a small number of x-rays and reducing MRI scanning time. The potential benefits extend to any application of compressive sensing.
611 citations
TL;DR: In this article, the authors show that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively, and they take this observation and weaken the smoothing assumptions on the operator and prove a novel convergence rate result.
Abstract: There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear ill-posed operator equations. The first convergence rates results for such problems were developed by Engl, Kunisch and Neubauer in 1989. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and co-workers presented a modification of the convergence rates result of Burger and Osher which turns out to be a complete generalization of the rates result of Engl and co-workers. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence rate result. The most significant difference in this result from the previous ones is that the source condition is formulated as a variational inequality and not as an equation as previously. As examples, we present a phase retrieval problem and a specific inverse option pricing problem, both previously studied in the literature. For the inverse finance problem, the new approach allows us to bridge the gap to a singular case, where the operator smoothness degenerates just when the degree of ill-posedness is minimal.
384 citations
TL;DR: In this paper, a convex variational framework is proposed for solving inverse problems in Hilbert spaces with a priori information on the representation of the target solution in a frame, and the objective function to be minimized consists of a separable term penalizing each frame coefficient individually, and a smooth term modelling the data formation model as well as other constraints.
Abstract: A convex variational framework is proposed for solving inverse problems in Hilbert spaces with a priori information on the representation of the target solution in a frame. The objective function to be minimized consists of a separable term penalizing each frame coefficient individually, and a smooth term modelling the data formation model as well as other constraints. Sparsity-constrained and Bayesian formulations are examined as special cases. A splitting algorithm is presented to solve this problem and its convergence is established in infinite-dimensional spaces under mild conditions on the penalization functions, which need not be differentiable. Numerical simulations demonstrate applications to frame-based image restoration.
285 citations
TL;DR: In this article, the authors derived explicit formulae for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform.
Abstract: We derive explicit formulae for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulae are important for problems of thermo- and photo-acoustic tomography. A closed-form inversion formula of a filtrationbackprojection type is found for the case when the centres of the integration spheres lie on a sphere in R n surrounding the support of the unknown function.
281 citations
TL;DR: The boundary control method as mentioned in this paper is an approach to inverse problems based on their relations to control theory (Belishev 1986), which solves the problems on unknown manifolds: given inverse data of a dynamical system associated with a manifold, it recovers the manifold, the operator governing the system and the states of the system defined on the manifold.
Abstract: The review covers the period 1997–2007 of development of the boundary control method, which is an approach to inverse problems based on their relations to control theory (Belishev 1986). The method solves the problems on unknown manifolds: given inverse data of a dynamical system associated with a manifold it recovers the manifold, the operator governing the system and the states of the system defined on the manifold. The main subject of the review is the extension of the boundary control method to the inverse problems of electrodynamics, elasticity theory, impedance tomography, problems on graphs as well as some new relations of the method to functional analysis and topology.
190 citations
TL;DR: In this article, the Schrodinger equation with Dirichlet boundary data was studied and a stability inequality between |p − q| and | ∂y(q)/∂ν − ∂ y(p)/ ∂ν| with appropriate norms was shown.
Abstract: We study the Schrodinger equation iy' + Δy + qy = 0 in Ω × (0, T) with Dirichlet boundary data y|∂Ω×(0,T) and initial condition y|Ω×{0} and we consider the inverse problem of determining the potential q(x), x Ω when ∂y/∂ν|Γ0 ×(0,T) is given. Here Ω is an open-bounded domain of N, Γ0 is an open subset of ∂Ω satisfying a suitable geometrical condition and T > 0. More precisely, from a global Carleman estimate we prove a stability inequality between |p − q| and | ∂y(q)/∂ν − ∂y(p)/∂ν| with appropriate norms.
180 citations
TL;DR: In this article, a line detector is used to integrate the measured acoustic pressure over a straight line and can be realized by a thin line of a piezoelectric film or by a laser beam as part of an interferometer.
Abstract: Line detectors integrate the measured acoustic pressure over a straight line and can be realized by a thin line of a piezoelectric film or by a laser beam as part of an interferometer. Photoacoustic imaging with integrating line detectors is performed by rotating a sample or the detectors around an axis perpendicular to the line detectors. The subsequent reconstruction is a two-step procedure: first, two-dimensional (2D) projections parallel to the line detector are reconstructed, then the three-dimensional (3D) initial pressure distribution is obtained by applying the 2D inverse Radon transform. The first step involves an inverse problem for the 2D wave equation. Wave propagation in two dimensions is significantly different from 3D wave propagation and reconstruction algorithms from 3D photoacoustic imaging cannot be used directly. By integrating recently established 3D formulae in the direction parallel to the line detector we obtain novel back-projection formulae in two dimensions. Numerical simulations demonstrate the capability of the derived reconstruction algorithms, also for noisy measurement data, limited angle problems and 3D reconstruction with integrating line detectors.
167 citations
TL;DR: In this article, a line integrals are acquired with an optical line sensor based on a Mach-Zehnder interferometer for the reconstruction of a three-dimensional image.
Abstract: A method for photoacoustic tomography is presented that uses line integrals over the acoustic wave field from a photoacoustic source for the reconstruction of a three-dimensional image. The line integrals are acquired with an optical line sensor based on a Mach–Zehnder interferometer. Image reconstruction is a two-step process. In the first step data from a scan of the line outside the object are used to reconstruct a linear projection of the source distribution. In the second step the inverse linear Radon transform is applied to multiple projections taken at different directions. This study focuses on the first step comparing two different open scan curves of the line detector around the object and corresponding two-dimensional reconstruction algorithms. An open curve, such as a 180° arc or an 'L' formed by two lines, establishes a limited view problem for which limitations exist concerning the stability of the reconstruction. Using experimental data from phantoms, time domain direct and iterative reconstruction algorithms as well as frequency domain (FD) algorithms are compared. The results indicate that although the image quality for all algorithms is similar, the FD algorithms offer much faster reconstruction speed, whereas the time domain algorithms allow for a refinement of image quality by including special correction procedures.
148 citations
TL;DR: In this paper, an exact and efficient, FFT-based reconstruction algorithm, that converts acoustic measurements recorded over a plane to a PAT image, is known, but to capture sufficient data for an exact PAT reconstruction with a planar geometry requires an infinitely wide array.
Abstract: Biomedical photoacoustic tomography (PAT) is a soft-tissue imaging modality which combines the high spatial resolution of ultrasound (US) with the contrast and spectroscopic opportunities afforded by imaging optical absorption. Planar US arrays composed of piezoelectric or optical detector elements with small element sizes and fast acquisition times are readily available, making them an attractive option for imaging applications. An exact and efficient, FFT-based PAT reconstruction algorithm, that converts acoustic measurements recorded over a plane to a PAT image, is known. However, to capture sufficient data for an exact PAT reconstruction with a planar geometry requires an infinitely wide array. In practice it will be finite, resulting in a loss of resolution and introducing artefacts into the image. To overcome this limitation it is proposed that acoustic image sources, provided by enclosing the target in a reverberant cavity, are used to generate a periodically repeating sound field. Measurements of this periodic sound field can be used to reconstruct a PAT image exactly from measurements of reverberation made over a finite aperture. The existing FFT-based PAT reconstruction algorithm with only minor additional modifications can be used to generate the image in this case.
TL;DR: A mathematical relationship between the photoacoustic pressure wavefield data on an aperture that encloses the object and the three-dimensional Fourier transform of the optical absorption distribution evaluated on a collection of concentric spheres is investigated in this article.
Abstract: Photoacoustic tomography (PAT), also known as optoacoustic or thermoacoustic tomography, is a hybrid imaging technique that possesses great potential for a wide range of biomedical imaging applications. Image reconstruction in PAT is tantamount to solving an inverse source problem, where the source represents the optical energy absorption distribution in the object that is induced by an interrogating pulsed optical waveform. In this work, we re-examine the PAT image reconstruction problem from a Fourier domain perspective by use of established time-harmonic inverse source concepts. A mathematical relationship between the photoacoustic pressure wavefield data on an aperture that encloses the object and the three-dimensional Fourier transform of the optical absorption distribution evaluated on a collection of concentric spheres is investigated. In addition to providing a framework for deriving both exact and approximate analytic reconstruction formulae, we demonstrate that this mapping provides an intuitive means of understanding certain spatial resolution characteristics of PAT.
TL;DR: In this article, an explicit series solution for the inversion of the spherical mean Radon transform is proposed for any closed measuring surface surrounding a region for which the eigenfunctions of the Dirichlet Laplacian are explicitly known.
Abstract: An explicit series solution is proposed for the inversion of the spherical mean Radon transform. Such an inversion is required in problems of thermo- and photo-acoustic tomography. Closed-form inversion formulae are currently known only for the case when the centres of the integration spheres lie on a sphere surrounding the support of the unknown function, or on certain unbounded surfaces. Our approach results in an explicit series solution for any closed measuring surface surrounding a region for which the eigenfunctions of the Dirichlet Laplacian are explicitly known—such as, for example, cube, finite cylinder, half-sphere etc. In addition, we present a fast reconstruction algorithm applicable in the case when the detectors (the centres of the integration spheres) lie on a surface of a cube. This algorithm reconstructs 3D images thousands times faster than backprojection-type methods.
TL;DR: In this paper, an analytic reconstruction formula for photoacoustic and thermoacoustic tomography is presented, for any geometry of point detectors placement along a closed surface and for variable sound speed satisfying a non-trapping condition.
Abstract: The paper contains an analytic reconstruction formula in thermoacoustic and photoacoustic tomography. It works for any geometry of point detectors placement along a closed surface and for variable sound speed satisfying a non-trapping condition. It is shown how this formula leads in particular to eigenfunction expansion reconstructions, including those recently obtained for the case of a uniform background. A uniqueness of reconstruction result is also obtained.
TL;DR: In this paper, the problem of imaging the conductivity from knowledge of one current and corresponding voltage on a part of the boundary of an inhomogeneous isotropic object and of the magnitude |J(x)| of the current density inside was reduced to a boundary value problem with partial data.
Abstract: We consider the problem of imaging the conductivity from knowledge of one current and corresponding voltage on a part of the boundary of an inhomogeneous isotropic object and of the magnitude |J(x)| of the current density inside. The internal data are obtained from magnetic resonance measurements. The problem is reduced to a boundary value problem with partial data for the equation ∇ |J(x)||∇u|−1∇u = 0. We show that equipotential surfaces are minimal surfaces in the conformal metric |J|2/(n−1)I. In two dimensions, we solve the Cauchy problem with partial data and show that the conductivity is uniquely determined in the region spanned by the characteristics originating from the part of the boundary where measurements are available. We formulate sufficient conditions on the Dirichlet data to guarantee the unique recovery of the conductivity throughout the domain. The proof of uniqueness is constructive and yields an efficient algorithm for conductivity imaging. The computational feasibility of this algorithm is demonstrated in numerical experiments.
TL;DR: In this paper, the Steklov-Poincare operator is characterized as a kernel operator and the severe ill-posedness degree of the Cauchy problem is established for smooth domains with corners.
Abstract: An answer to the ill-posedness degree issue of the Cauchy problem may be found in the theory of kernel operators. The foundation of the proof is the Steklov–Poincare approach introduced in Ben Belgacem and El Fekih (2005 Inverse Problems 21 1915–36), which consists of reformulating the Cauchy problem as a variational equation, in an appropriate Sobolev scale, and is set on the part of the boundary where data are missing. The linear (Steklov–Poincare) operator involved in that reduced problem turns out to be compact with a non-closed range; hence the ill-posedness. Conducting an accurate spectral analysis of this operator requires characterization of it as a kernel operator, which is obtained through Green's functions of the (Laplace) differential equation. The severe ill-posedness is then settled for smooth domains after showing a fast decaying towards zero of the eigenvalues of that Steklov–Poincare operator. This is achieved by applying the Weyl–Courant min–max principle and some polynomial approximation results. Addressing more general smooth domains with corners, we discuss the regularity of Green's function and we explain why there is a room to extend our analysis to this case and why we are optimistic that it will definitely establish the severe ill-posedness of the Cauchy problem.
TL;DR: In this paper, a Fourier transform based reconstruction algorithm for solving the inverse problem in optoacoustic imaging is presented, which improves reconstruction efficiency and image quality, but without the need of using time-consuming zero-padding.
Abstract: A novel Fourier transform based reconstruction algorithm for solving the inverse problem in optoacoustic imaging is presented, which improves reconstruction efficiency and image quality. Fourier algorithms make use of an interpolation law when signal Fourier components are mapped to source Fourier components. To overcome inadequacies affiliated with interpolation methods such as nearest neighbour, linear, cubic or spline interpolation, together with signal data zero padding, we present a regularized interpolation method based on a forward model explicitly formulated for the compactly supported signal data. Simulations performed on a digital tissue phantom reveal the potential of this novel reconstruction method, which results in images of enhanced quality but without the need of using time-consuming zero-padding.
TL;DR: In vivo, it is demonstrated that the difference in sO_2 between a typical artery and a typical vein is conserved before and after spectral compensation, and holds regardless of the animal's systemic physiological state.
Abstract: Quantitative measurements of the oxygen saturation of hemoglobin (sO_2) in a blood vessel in vivo presents a challenge in photoacoustic imaging. As a result of wavelength-dependent optical attenuation in the skin, the local fluence at a subcutaneous vessel varies with the optical wavelength in spectral measurement and hence needs to be compensated for so that the intrinsic absorption coefficient can be recovered. Here, by employing a simplified double-layer skin model, we demonstrate that although the absolute value of sO_2 in a vessel is seriously affected by the volume fraction of blood and the spatially averaged sO_2 in the dermis, the difference of sO_2 between neighboring vessels is minimally affected. Experimentally, we acquire compensational factors for the wavelength-dependent optical attenuation by measuring the PA spectrum of a subcutaneously inserted 25 µm thick black film using our PA microscope. We demonstrate in vivo that the difference in sO_2 between a typical artery and a typical vein is conserved before and after spectral compensation. This conservation holds regardless of the animal's systemic physiological state.
TL;DR: In this article, a generalized version of Krasnoselski-Mann's iteration for solving a broader class of problems than the original KM algorithm is presented. And convergence results for the proposed method are proved.
Abstract: This paper deals with a method for approximating a solution of the following fixed-point problem: find , where is a Hilbert space, P and T are two nonexpansive mappings on a closed convex subset D and projFix(T) denotes the metric projection on the set of fixed points of T. This amounts to saying that is the fixed point of T which satisfies a variational inequality depending on a given criterion P, namely: find , where NFix(T) denotes the normal cone to the set of fixed points of T. Convergence results for the proposed method are proved. It should be noted that the proposed method can be regarded as a generalized version of Krasnoselski–Mann's iteration for solving a broader class of problems than the original KM algorithm, namely hierarchical fixed-point problems. This class is very interesting because it covers monotone variational inequality on fixed-point sets, minimization problems over equilibrium constraints, hierarchical minimization problems,.... The special aspect of the algorithm together with convergence results makes it an original and theoretically interesting scheme. On the other hand, the framework is general enough and permits us to treat in a unified way several iterative schemes, recovering, developing and improving some known related convergence results in this field.
TL;DR: In this paper, the authors considered the scattering of time harmonic electromagnetic plane waves by a bounded inhomogeneous medium and showed that under certain assumptions a lower bound on the index of refraction can be obtained from a knowledge of the smallest transmission eigenvalue corresponding to the medium.
Abstract: We consider the scattering of time harmonic electromagnetic plane waves by a bounded inhomogeneous medium and show that under certain assumptions a lower bound on the index of refraction can be obtained from a knowledge of the smallest transmission eigenvalue corresponding to the medium. It is then shown by numerical examples that this eigenvalue can be determined from a knowledge of the far field pattern of the scattered wave, thus providing a practical method for estimating the index of refraction from far field data.
TL;DR: In this article, the inverse scattering transform for an integrable discretization of the defocusing nonlinear Schrodinger equation with nonvanishing boundary values at infinity is constructed, and a suitable transformation of the scattering problem is introduced to address the open issue of analyticity of eigenfunctions and scattering data.
Abstract: The inverse scattering transform for an integrable discretization of the defocusing nonlinear Schrodinger equation with nonvanishing boundary values at infinity is constructed. This problem had been previously studied, and many key results had been established. Here, a suitable transformation of the scattering problem is introduced in order to address the open issue of analyticity of eigenfunctions and scattering data. Moreover, the inverse problem is formulated as a Riemann–Hilbert problem on the unit circle, and a modification of the standard procedure is required in order to deal with the dependence of asymptotics of the eigenfunctions on the potentials. The discrete analog of Gel’fand–Levitan–Marchenko equations is also derived. Finally, soliton solutions and solutions in the small-amplitude limit are obtained and the continuum limit is discussed. (Some figures in this article are in colour only in the electronic version)
TL;DR: In this article, a Wronskian formulation of the Boussinesq equation is presented, which involves a broad set of sufficient conditions consisting of linear partial differential equations, and the representative systems of the differential equations in the sufficient conditions are explicitly solved.
Abstract: A Wronskian formulation is presented for the Boussinesq equation, which involves a broad set of sufficient conditions consisting of linear partial differential equations. The representative systems of the differential equations in the sufficient conditions are explicitly solved. The obtained solution formulae provide us with a comprehensive approach to construct exact and explicit solutions to the Boussinesq equation, by which solitons, negatons, positons and complexitons are computed for the Boussinesq equation.
TL;DR: In this paper, the results concerning the injectivity of, the characterization of the range of, and the inversion of the wave equation were discussed, and a close connection between mean values over spheres and solutions of initial value problems was made.
Abstract: Let B represent the ball of radius ρ in Rn and S its boundary; consider the map , where represents the mean value of f on a sphere of radius r centered at p. We summarize and discuss the results concerning the injectivity of , the characterization of the range of , and the inversion of . There is a close connection between mean values over spheres and solutions of initial value problems for the wave equation. We also summarize the results for the corresponding wave equation problem.
TL;DR: In this paper, a method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line, which can alternatively be written explicitly as algebraic combinations of exponential, trigonometric and polynomial functions of the spatial and temporal coordinates.
Abstract: A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric and polynomial functions of the spatial and temporal coordinates.
TL;DR: In this article, a reflection principle for the time-harmonic Maxwell equations is established, which is then used to derive a uniqueness result in inverse electromagnetic obstacle scattering with polyhedral-type scatterers of perfect conductors.
Abstract: A novel reflection principle for the time-harmonic Maxwell equations is established. It is then used to derive a uniqueness result in inverse electromagnetic obstacle scattering with polyhedral-type scatterers of perfect conductors. The underlying scatterers can be very general; e.g., they may consist of finitely many solid polyhedra along with finitely many subsets of two-dimensional planes.
TL;DR: This paper develops a finite volume adaptive grid refinement method for the solution of distributed parameter estimation problems with almost discontinuous coefficients and shows that local refinement can significantly reduce the computational effort of solving the problem, and that the resulting reconstructions can significantly improve resolution.
Abstract: In this paper we develop a finite volume adaptive grid refinement method for the solution of distributed parameter estimation problems with almost discontinuous coefficients. We discuss discretization on locally refined grids, as well as optimization and refinement criteria. An OcTree data structure is utilized. We show that local refinement can significantly reduce the computational effort of solving the problem, and that the resulting reconstructions can significantly improve resolution, even for noisy data and diffusive forward problems.
TL;DR: In this article, the stability issue for the inverse problem of determining a Robin coefficient on the inaccessible portion of the boundary by the electrostatic measurements performed on the accessible one was considered and a Lipschitz stability estimate under the further a priori assumption of a piecewise constant Robin coefficient was provided.
Abstract: We are concerned with a problem arising in corrosion detection. We consider the stability issue for the inverse problem of determining a Robin coefficient on the inaccessible portion of the boundary by the electrostatic measurements performed on the accessible one. We provide a Lipschitz stability estimate under the further a priori assumption of a piecewise constant Robin coefficient. Furthermore, we prove that the Lipschitz constant of the above-mentioned estimate behaves exponentially with respect to the number of the portions considered.
TL;DR: In this paper, the authors consider a size-structured model for cell division and address the question of determining the division rate from the measured stable size distribution of the population, and formulate such a question as an inverse problem for an integro-differential equation posed on the half line.
Abstract: We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We formulate such a question as an inverse problem for an integro-differential equation posed on the half line. We develop firstly a regular dependence theory for the solution in terms of the coefficients and, secondly, a novel regularization technique for tackling this inverse problem which takes into account the specific nature of the equation. Our results also rely on generalized relative entropy estimates and related Poincare inequalities.
TL;DR: In this paper, it was shown that the factorization method provides a characterization of an open inclusion (modulo its boundary) if each point inside the inhomogeneity has an open neighbourhood where the perturbation of the conductivity is strictly positive (or negative) definite.
Abstract: In electrical impedance tomography, one tries to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many practically important situations, the investigated object has known background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool for locating such inclusions. Earlier, it was shown that under suitable regularity conditions positive (or negative) inhomogeneities can be characterized by the factorization technique if the conductivity or one of its higher normal derivatives jumps on the boundaries of the inclusions. In this work, we use a monotonicity argument to generalize these results: we show that the factorization method provides a characterization of an open inclusion (modulo its boundary) if each point inside the inhomogeneity has an open neighbourhood where the perturbation of the conductivity is strictly positive (or negative) definite. In particular, we do not assume any regularity of the inclusion boundary or set any conditions on the behaviour of the perturbed conductivity at the inclusion boundary. Our theoretical findings are verified by two-dimensional numerical experiments.
TL;DR: In this article, a generalized conditional gradient method was proposed for the minimization of Tikhonov-type functionals, which occur in the regularization of nonlinear inverse problems with sparsity constraints.
Abstract: The intention of this paper is to show the applicability of a generalized conditional gradient method for the minimization of Tikhonov-type functionals, which occur in the regularization of nonlinear inverse problems with sparsity constraints. We consider functionals of Tikhonov type where the usual quadratic penalty term is replaced by the pth power of a weighted lp-norm. First of all, we analyze the convergence properties of a generalized gradient algorithm. Further, we show that the considered minimization method coincides under certain conditions with a surrogate method. Numerical results are then illustrated by means of the nonlinear single photon emission computed tomography problem.