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Inverse Problems and Imaging

May 2011 , Volume 5 , Issue 2

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Identifying a space dependent coefficient in a reaction-diffusion equation
Elena Beretta and Cecilia Cavaterra
2011, 5(2): 285-296 doi: 10.3934/ipi.2011.5.285 +[Abstract](3190) +[PDF](331.2KB)
We consider a reaction-diffusion equation for the front motion $u$ in which the reaction term is given by $c(x)g(u)$. We formulate a suitable inverse problem for the unknowns $u$ and $c$, where $u$ satisfies homogeneous Neumann boundary conditions and the additional condition is of integral type on the time interval $[0,T]$. Uniqueness of the solution is proved in the case of a linear $g$. Assuming $g$ non linear, we show uniqueness for large $T$.
On an inverse problem in electromagnetism with local data: stability and uniqueness
Pedro Caro
2011, 5(2): 297-322 doi: 10.3934/ipi.2011.5.297 +[Abstract](3871) +[PDF](480.2KB)
In this paper we prove a stable determination of the coefficients of the time-harmonic Maxwell equations from local boundary data. The argument --due to Isakov-- requires some restrictions on the domain.
A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration
Ke Chen, Yiqiu Dong and Michael Hintermüller
2011, 5(2): 323-339 doi: 10.3934/ipi.2011.5.323 +[Abstract](4124) +[PDF](643.5KB)
Based on the Fenchel pre-dual of the total variation model, a nonlinear multigrid algorithm for image denoising is proposed. Due to the structure of the differential operator involved in the Euler-Lagrange equations of the dual models, line Gauss-Seidel-semismooth-Newton step is utilized as the smoother, which provides rather good smoothing rates. The paper ends with a report on numerical results and a comparison with a very recent nonlinear multigrid solver based on Chambolle's iteration [6].
3D coded aperture imaging, ill-posedness and link with incomplete data radon transform
Jean-François Crouzet
2011, 5(2): 341-353 doi: 10.3934/ipi.2011.5.341 +[Abstract](2853) +[PDF](374.0KB)
Coded Aperture Imaging is a cheap imaging process encountered in many fields of research like optics, medical imaging, astronomy, and that has led to several good results for two dimensional reconstruction methods. However, the three dimensional reconstruction problem remains nowadays severely ill-posed, and has not yet furnished satisfactory outcomes.
    In the present study, we propose an illustration of the poorness of the data in order to operate a good inversion in the 3D case. In the context of a far-field imaging, an inversion formula is derived when the detector screen can be widely translated. This reformulates the 3D inversion problem of coded aperture imaging in terms of classical Radon transform. In the sequel, we examine more accurately this reconstruction formula, and claim that it is equivalent to solve the limited angle Radon transform problem with very restricted data.
    We thus deduce that the performances of any numerical reconstruction will remain shrank, essentially because of the physical nature of the coding process, excepted when a very strong a priori knowledge is given for the 3D source.
Electrical impedance tomography using a point electrode inverse scheme for complete electrode data
Fabrice Delbary and Rainer Kress
2011, 5(2): 355-369 doi: 10.3934/ipi.2011.5.355 +[Abstract](3468) +[PDF](454.3KB)
For the two dimensional inverse electrical impedance problem in the case of piecewise constant conductivities with the currents injected at adjacent point electrodes and the resulting voltages measured between the remaining electrodes, in [3] the authors proposed a nonlinear integral equation approach that extends a method that has been suggested by Kress and Rundell [10] for the case of perfectly conducting inclusions. As the main motivation for using a point electrode method we emphasized on numerical difficulties arising in a corresponding approach by Eckel and Kress [4, 5] for the complete electrode model. Therefore, the purpose of the current paper is to illustrate that the inverse scheme based on point electrodes can be successfully employed when synthetic data from the complete electrode model are used.
Filtered Kirchhoff migration of cross correlations of ambient noise signals
Josselin Garnier and Knut Solna
2011, 5(2): 371-390 doi: 10.3934/ipi.2011.5.371 +[Abstract](2707) +[PDF](457.4KB)
In this paper we study passive sensor imaging with ambient noise sources by suitably migrating cross correlations of the recorded signals. We propose and study different imaging functionals. A new functional is introduced that is an inverse Radon transform applied to a special function of the cross correlation matrix. We analyze the properties of the new imaging functional in the high-frequency regime which shows that it produces sharper images than the usual Kirchhoff migration functional. Numerical simulations confirm the theoretical predictions.
A refinement and coarsening indicator algorithm for finding sparse solutions of inverse problems
Barbara Kaltenbacher and Jonas Offtermatt
2011, 5(2): 391-406 doi: 10.3934/ipi.2011.5.391 +[Abstract](3699) +[PDF](439.0KB)
In this paper we extend the idea of adaptive discretization by using refinement and coarsening indicators from papers by Chavent, Bissell, Benameur and Jaffré (cf., e.g., [5], [9]) to a general setting. This allows to make use of the relation between adaptive discretization and sparse paramerization in order to construct an algorithm for finding sparse solutions of inverse problems. We provide some first steps in the analysis of the proposed method and apply it to an inverse problem in systems biology, namely the reconstruction of gene networks in an ordinary differential equation (ODE) model. Here due to the fact that not all genes interact with each other, reconstruction of a sparse connectivity matrix is a key issue.
A regularized k-means and multiphase scale segmentation
Sung Ha Kang, Berta Sandberg and Andy M. Yip
2011, 5(2): 407-429 doi: 10.3934/ipi.2011.5.407 +[Abstract](4853) +[PDF](1666.6KB)
We propose a data clustering model reduced from variational approach. This new clustering model, a regularized k-means, is an extension from the classical k-means model. It uses the sum-of-squares error for assessing fidelity, and the number of data in each cluster is used as a regularizer. The model automatically gives a reasonable number of clusters by a choice of a parameter. We explore various properties of this classification model and present different numerical results. This model is motivated by an application to scale segmentation. A typical Mumford-Shah-based image segmentation is driven by the intensity of objects in a given image, and we consider image segmentation using additional scale information in this paper. Using the scale of objects, one can further classify objects in a given image from using only the intensity value. The scale of an object is not a local value, therefore the procedure for scale segmentation needs to be separated into two steps: multiphase segmentation and scale clustering. The first step requires a reliable multiphase segmentation where we applied unsupervised model, and apply a regularized k-means for a fast automatic data clustering for the second step. Various numerical results are presented to validate the model.
Recovering two Lamé kernels in a viscoelastic system
Alfredo Lorenzi and Vladimir G. Romanov
2011, 5(2): 431-464 doi: 10.3934/ipi.2011.5.431 +[Abstract](2821) +[PDF](352.2KB)
Let $\mathcal B$ be a viscoelastic body with a (smooth) bounded open reference set $\Omega$ in $\mathbb R^3$, with the equation of motion being described by the Lamé coefficients $\lambda_0$ and $\mu_0$ and the related viscoelastic coefficients $\lambda_1$ and $\mu_1$. The latter are assumed to be factorized with the same temporal part, i.e. $\lambda_1(t,x)=k(t)p(x)$ and $\mu_1(t,x)=k(t)q(x)$. Furthermore, it is assumed that the spatial parts $p$ and $q$ of $\lambda_1$ and $\mu_1$ are unknown and the three additional measurements $\sum_{j=1}^3\sigma_{i,j}^0(t,x)$n$_j(x) = g_i(t,x)$, $i=1,2,3$, are available on $(0,T)\times \partial \Omega$ for some (sufficiently large) subset $\Gamma\subset \partial \Omega$.
    The fundamental task of this paper is to show the uniqueness of the pair $(p,q)$ as well as its continuous dependence on the boundary conditions, the initial data being kept fixed and the initial velocity being suitably related to the initial displacement.
Near field sampling type methods for the inverse fluid--solid interaction problem
Peter Monk and Virginia Selgas
2011, 5(2): 465-483 doi: 10.3934/ipi.2011.5.465 +[Abstract](4135) +[PDF](944.1KB)
The inverse fluid--solid interaction problem considered here is to determine the shape of an elastic body from pressure measurements made in the near field. In particular we assume that the elastic body is probed by pressure waves due to point sources, and the resulting scattered field and the normal derivative of the scattered field is available for every source and receiver combination on the source and measurement curves. We provide an analysis of the Reciprocity Gap (RG) method in this case, as well as the Linear Sampling Method (LSM). A novelty of our analysis is that we exhibit a connection between the RG method and a non--standard LSM using sources and receivers on different curves. We provide numerical tests of the algorithms using both synthetic and real data.
Recovering conductivity at the boundary in three-dimensional electrical impedance tomography
Gen Nakamura, Päivi Ronkanen, Samuli Siltanen and Kazumi Tanuma
2011, 5(2): 485-510 doi: 10.3934/ipi.2011.5.485 +[Abstract](3534) +[PDF](935.6KB)
The aim of electrical impedance tomography (EIT) is to reconstruct the conductivity values inside a conductive object from electric measurements performed at the boundary of the object. EIT has applications in medical imaging, nondestructive testing, geological remote sensing and subsurface monitoring. Recovering the conductivity and its normal derivative at the boundary is a preliminary step in many EIT algorithms; Nakamura and Tanuma introduced formulae for recovering them approximately from localized voltage-to-current measurements in [Recent Development in Theories & Numerics, International Conference on Inverse Problems 2003]. The present study extends that work both theoretically and computationally. As a theoretical contribution, reconstruction formulas are proved in a more general setting. On the computational side, numerical implementation of the reconstruction formulae is presented in three-dimensional cylindrical geometry. These experiments, based on simulated noisy EIT data, suggest that the conductivity at the boundary can be recovered with reasonable accuracy using practically realizable measurements. Further, the normal derivative of the conductivity can also be recovered in a similar fashion if measurements from a homogeneous conductor (dummy load) are available for use in a calibration step.
Non-local regularization of inverse problems
Gabriel Peyré, Sébastien Bougleux and Laurent Cohen
2011, 5(2): 511-530 doi: 10.3934/ipi.2011.5.511 +[Abstract](5635) +[PDF](1841.8KB)
This article proposes a new framework to regularize imaging linear inverse problems using an adaptive non-local energy. A non-local graph is optimized to match the structures of the image to recover. This allows a better reconstruction of geometric edges and textures present in natural images. A fast algorithm computes iteratively both the solution of the regularization process and the non-local graph adapted to this solution. The graph adaptation is efficient to solve inverse problems with randomized measurements such as inpainting random pixels or compressive sensing recovery. Our non-local regularization gives state-of-the-art results for this class of inverse problems. On more challenging problems such as image super-resolution, our method gives results comparable to sparse regularization in a translation invariant wavelet frame.

2021 Impact Factor: 1.483
5 Year Impact Factor: 1.462
2021 CiteScore: 2.6




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