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Inverse Problems & Imaging

2010 , Volume 4 , Issue 3

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On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations
Johann Baumeister, Barbara Kaltenbacher and Antonio Leitão
2010, 4(3): 335-350 doi: 10.3934/ipi.2010.4.335 +[Abstract](227) +[PDF](1416.7KB)
Abstract:
In this article a modified Levenberg-Marquardt method coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations is investigated. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a non-linear inverse doping problem based on a bipolar model.
A quasi-reversibility approach to solve the inverse obstacle problem
Laurent Bourgeois and Jérémi Dardé
2010, 4(3): 351-377 doi: 10.3934/ipi.2010.4.351 +[Abstract](319) +[PDF](684.7KB)
Abstract:
We introduce a new approach based on the coupling of the method of quasi-reversibility and a simple level set method in order to solve the inverse obstacle problem with Dirichlet boundary condition. We provide a theoretical justification of our approach and illustrate its feasibility with the help of numerical experiments in $2D$.
Simultaneous cartoon and texture inpainting
Jian-Feng Cai, Raymond H. Chan and Zuowei Shen
2010, 4(3): 379-395 doi: 10.3934/ipi.2010.4.379 +[Abstract](250) +[PDF](677.7KB)
Abstract:
Real images usually have two layers, namely, cartoons (the piecewise smooth part of the image) and textures (the oscillating pattern part of the image). Both these two layers have sparse approximations under some tight frame systems such as framelet, translation invariant wavelet, curvelet, and local DCTs. In this paper, we solve image inpainting problems by using two separate tight frame systems which can sparsely represent cartoons and textures respectively. Different from existing schemes in the literature which are either analysis-based or synthesis-based sparsity priors, our minimization formulation balances these two priors. We also derive iterative algorithms to find their solutions and prove their convergence. Numerical simulation examples are given to demonstrate the applicability and usefulness of our proposed algorithms in image inpainting.
Reconstructing the metric and magnetic field from the scattering relation
Nurlan Dairbekov and Gunther Uhlmann
2010, 4(3): 397-409 doi: 10.3934/ipi.2010.4.397 +[Abstract](171) +[PDF](195.3KB)
Abstract:
We develop a method for reconstructing the conformal factor of a Riemannian metric and the magnetic field on a surface from the scattering relation associated to the corresponding magnetic flow. The scattering relation maps a starting point and direction of a magnetic geodesic into its end point and direction. The key point in the reconstruction is the interplay between the magnetic ray transform, the fiberwise Hilbert transform on the circle bundle of the surface, and the Laplace-Beltrami operator of the underlying Riemannian metric.
On forward and inverse models in fluorescence diffuse optical tomography
Herbert Egger, Manuel Freiberger and Matthias Schlottbom
2010, 4(3): 411-427 doi: 10.3934/ipi.2010.4.411 +[Abstract](275) +[PDF](432.3KB)
Abstract:
This paper investigates forward and inverse problems in fluorescence optical tomography, with the aim to devise stable methods for the tomographic image reconstruction.
   We analyze solvability of a standard nonlinear forward model and two approximations by reduced models, which provide certain advantages for a theoretical as well as numerical treatment of the inverse problem. Important properties of the forward operators, that map the unknown fluorophore concentration on virtual measurements, are derived; in particular, the ill-posedness of the reconstruction problem is proved, and uniqueness issues are discussed.
   For the stable solution of the inverse problem, we consider Tikhonov-type regularization methods, and we prove that the forward operators have all the properties, that allow to apply standard regularization theory. We also investigate the applicability of nonlinear regularization methods, i.e., TV-regularization and a method of levelset-type, which are better suited for the reconstruction of localized or piecewise constant solutions.
   The theoretical results are supported by numerical tests, which demonstrate the viability of the reduced models for the treatment of the inverse problem, and the advantages of nonlinear regularization methods for reconstructing localized fluorophore distributions.
Convex source support in half-plane
Lauri Harhanen and Nuutti Hyvönen
2010, 4(3): 429-448 doi: 10.3934/ipi.2010.4.429 +[Abstract](197) +[PDF](325.8KB)
Abstract:
This work extends the concept of convex source support to the framework of inverse source problems for the Poisson equation in an insulated upper half-plane. The convex source support is, in essence, the smallest (nonempty) convex set that supports a source that produces the measured (nontrivial) data on the horizontal axis. In particular, it belongs to the convex hull of the support of any source that is compatible with the measurements. We modify a previously introduced method for reconstructing the convex source support in bounded domains to our unbounded setting. The performance of the resulting numerical algorithm is analyzed both for the inverse source problem and for electrical impedance tomography with single pair of boundary current and potential as the measurement data.
Inverse problems with partial data in a slab
Xiaosheng Li and Gunther Uhlmann
2010, 4(3): 449-462 doi: 10.3934/ipi.2010.4.449 +[Abstract](257) +[PDF](229.1KB)
Abstract:
In this paper we consider several inverse boundary value problems with partial data on an infinite slab. We prove the unique determination results of the coefficients for the Schrödinger equation and the conductivity equation when the corresponding Dirichlet and Neumann data are given either on the different boundary hyperplanes of the slab or on the same single hyperplane.
Nonstationary inversion of convection-diffusion problems - recovery from unknown nonstationary velocity fields
Antti Lipponen, Aku Seppänen, Jari Hämäläinen and Jari P. Kaipio
2010, 4(3): 463-483 doi: 10.3934/ipi.2010.4.463 +[Abstract](231) +[PDF](2606.7KB)
Abstract:
In this paper, we consider the reconstruction of time-varying concentration distributions under nonstationary flow conditions. Previous studies have shown that the state estimation approach that is based on stochastic process evolution models, facilitates reconstructions of rapidly time-varying targets. However, only cases with stationary velocity fields, or cases in which the velocity field can be completely specified by a velocity profile, have been studied. While simultaneous estimation of the time-varying concentration and low-dimensional representations of the flow field itself has been shown to be possible to some extent, this would be computationally too heavy for on-line process estimation and control. On the other hand, using an incorrect flow model in the evolution model may induce intolerable estimation errors. In this paper, we consider an approach in which the state evolution model is written to correspond to a stationary flow, while the actual flow is nonstationary. The associated modelling errors are handled by constructing the state noise process to accommodate to this discrepancy. We carry out a numerical feasibility study with different Reynolds numbers and show that the approach yields significant reduction of estimation errors and simultaneously facilitates using computationally efficient reduced order models.
Fortran linear inverse problem solver
Mikko Orispää and Markku Lehtinen
2010, 4(3): 485-503 doi: 10.3934/ipi.2010.4.485 +[Abstract](219) +[PDF](780.0KB)
Abstract:
FLIPS (Fortran Linear Inverse Problem Solver) is a Fortran 95 module for solving large-scale statistical linear systems. Instead of inverting large matrices, FLIPS transforms the system into a simpler one by using Givens rotations. This simplified system is then solved by FLIPS quickly and efficiently. FLIPS is also capable of calculating the full a posteriori covariance matrix. It is also possible to add or delete measurements and unknowns making it useful in time-dependent problems of the Kalman-filter type. The FLIPS implementation is explained and the advantages of using FLIPS, especially for overdetermined systems, are shown. Plans for future developments are discussed.
A variational setting for volume constrained image registration
Christiane Pöschl, Jan Modersitzki and Otmar Scherzer
2010, 4(3): 505-522 doi: 10.3934/ipi.2010.4.505 +[Abstract](275) +[PDF](469.5KB)
Abstract:
We consider image registration, which is the determination of a geometrical transformation between two data sets. In this paper we propose constrained variational methods which aim for controlling the change of area or volume under registration transformation. We prove an existence result, convergence of a finite element method, and present a simple numerical example for volume-preserving registration.
A two-level domain decomposition method for image restoration
Jing Xu, Xue-Cheng Tai and Li-Lian Wang
2010, 4(3): 523-545 doi: 10.3934/ipi.2010.4.523 +[Abstract](267) +[PDF](1348.2KB)
Abstract:
Image restoration has drawn much attention in recent years and a surge of research has been done on variational models and their numerical studies. However, there remains an urgent need to develop fast and robust methods for solving the minimization problems and the underlying nonlinear PDEs to process images of moderate to large size. This paper aims to propose a two-level domain decomposition method, which consists of an overlapping domain decomposition technique and a coarse mesh correction, for directly solving the total variational minimization problems. The iterative algorithm leads to a system of small size and better conditioning in each subspace, and is accelerated with a piecewise linear coarse mesh correction. Various numerical experiments and comparisons demonstrate that the proposed method is fast and robust particularly for images of large size.
Reconstructing electromagnetic obstacles by the enclosure method
Ting Zhou
2010, 4(3): 547-569 doi: 10.3934/ipi.2010.4.547 +[Abstract](163) +[PDF](358.3KB)
Abstract:
We show that one can determine Perfectly Magnetic Conductor obstacles, Perfectly Electric Conductor obstacles and obstacles satisfying impedance boundary condition, embedded in a known electromagnetic medium, by making electromagnetic measurements at the boundary of the medium. The boundary measurements are encoded in the impedance map that sends the tangential component of the electric field to the tangential component of the magnetic field. We do this by probing the medium with complex geometrical optics solutions to the corresponding Maxwell's equations and extend the enclosure method to this case.

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