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Volume 1, 2007

Inverse Problems & Imaging

2007 , Volume 1 , Issue 1

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Editorial
Lassi Päivärinta , Matti Lassas and  Jackie (Jianhong) Shen
2007, 1(1): i-iii doi: 10.3934/ipi.2007.1.1i +[Abstract](28) +[PDF](36.5KB)
Abstract:
The fields of inverse problems and imaging are new and flourishing branches of both pure and applied mathematics. In particular, these areas are concerned with recovering information about an object from indirect, incomplete or noisy observations and have become one of the most important and topical fields of modern applied mathematics.
    The modern study of inverse problems and imaging applies a wide range of geometric and analytic methods which in turn creates new connections to various fields of mathematics, ranging from geometry, microlocal analysis and control theory to mathematical physics, stochastics and numerical analysis. Research in inverse problems has shown that many results of pure mathematics are in fact crucial components of practical algorithms. For example,a theoretical understanding of the structures that ideal measurements should reveal, or of the non-uniqueness of solutions,can lead to a dramatic increase in the quality of imaging applications. On the other hand,inverse problems have also raised many new mathematical problems. For example, the invention of the inverse spectral method to solve the Korteweg-de Vries equation gave rise to the field of integrable systems and the mathematical theory of solitons.

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Microlocal sequential regularization in imaging
Daniela Calvetti and  Erkki Somersalo
2007, 1(1): 1-11 doi: 10.3934/ipi.2007.1.1 +[Abstract](74) +[PDF](232.6KB)
Abstract:
In this article, we consider imaging problems in which the data consist of noisy observations of the true image through a linear filter such as blurring, sparse sampling or tomographic projections. The image restoration problem is ill-posed and in order to obtain a meaningful result, the problem needs to be regularized or augmented by additional information. In this article, we consider Tikhonov regularization by a class of non-linear smoothness filters that are capable of detecting and restoring edges in the image. The regularization function is microlocal in the sense that it is sensitive to the location and the direction of the non-smoothness of the image. The implementation of the algorithm leads to a sequence of simple linear least squares problems, the penalty term being calculated as a direction-sensitive weighted finite difference approximation of the Laplacian. The algorithm is applied to two classical imaging problems, image zooming and limited angle tomography.
The interior transmission problem
David Colton , Lassi Päivärinta and  John Sylvester
2007, 1(1): 13-28 doi: 10.3934/ipi.2007.1.13 +[Abstract](90) +[PDF](204.8KB)
Abstract:
The interior transmission problem is a boundary value problem that plays a basic role in inverse scattering theory but unfortunately does not seem to be included in any existing theory in partial differential equations.This paper presents old and new results for the interior transmission problem ,in particular its relation to inverse scattering theory and new results on the spectral theory associated with this class of boundary value problems.
Iteratively solving linear inverse problems under general convex constraints
Ingrid Daubechies , Gerd Teschke and  Luminita Vese
2007, 1(1): 29-46 doi: 10.3934/ipi.2007.1.29 +[Abstract](65) +[PDF](270.3KB)
Abstract:
We consider linear inverse problems where the solution is assumed to fulfill some general homogeneous convex constraint. We develop an algorithm that amounts to a projected Landweber iteration and that provides and iterative approach to the solution of this inverse problem. For relatively moderate assumptions on the constraint we can always prove weak convergence of the iterative scheme. In certain cases, i.e. for special families of convex constraints, weak convergence implies norm convergence. The presented approach covers a wide range of problems, e.g. Besov-- or BV--restoration for which we present also numerical experiments in the context of image processing.
Integrodifferential equations for continuous multiscale wavelet shrinkage
Stephan Didas and  Joachim Weickert
2007, 1(1): 47-62 doi: 10.3934/ipi.2007.1.47 +[Abstract](108) +[PDF](210.9KB)
Abstract:
The relations between wavelet shrinkage and nonlinear diffusion for discontinuity-preserving signal denoising are fairly well-understood for single-scale wavelet shrinkage, but not for the practically relevant multiscale case. In this paper we show that 1-D multiscale continuous wavelet shrinkage can be linked to novel integrodifferential equations. They differ from nonlinear diffusion filtering and corresponding regularisation methods by the fact that they involve smoothed derivative operators and perform a weighted averaging over all scales. Moreover, by expressing the convolution-based smoothed derivative operators by power series of differential operators, we show that multiscale wavelet shrinkage can also be regarded as averaging over pseudodifferential equations.
Reconstruction of obstacles immersed in an incompressible fluid
Horst Heck , Gunther Uhlmann and  Jenn-Nan Wang
2007, 1(1): 63-76 doi: 10.3934/ipi.2007.1.63 +[Abstract](38) +[PDF](192.9KB)
Abstract:
We consider the reconstruction of obstacles inside a bounded domain filled with an incompressible fluid. Our method relies on special complex geometrical optics solutions for the stationary Stokes equation with a variable viscosity.
Approximation errors in nonstationary inverse problems
Janne M.J. Huttunen and  J. P. Kaipio
2007, 1(1): 77-93 doi: 10.3934/ipi.2007.1.77 +[Abstract](135) +[PDF](583.5KB)
Abstract:
Inverse problems are known to be very intolerant to both data errors and errors in the forward model. With several inverse problems the adequately accurate forward model can turn out to be computationally excessively complex. The Bayesian framework for inverse problems has recently been shown to enable the adoption of highly approximate forward models. This approach is based on the modelling of the associated approximation errors that are incorporated in the construction of the computational model. In this paper we investigate the extension of the approximation error theory to nonstationary inverse problems. We develop the basic framework for linear nonstationary inverse problems that allows one to use both highly reduced states and extended time steps. As an example we study the one dimensional heat equation.
On uniqueness in the inverse conductivity problem with local data
Victor Isakov
2007, 1(1): 95-105 doi: 10.3934/ipi.2007.1.95 +[Abstract](202) +[PDF](156.4KB)
Abstract:
We show that the Dirichlet-to-Neumann map given on an arbitrary part of the boundary of a three-dimensional domain with zero Dirichlet (or Neumann) data on the remaining (spherical or plane) part of the boundary uniquely determines conductivity or potential coefficients. This is the first uniqueness result for the Calderon problem with zero data on unaccessible part of the boundary. Proofs use some modification of the method of complex geometrical solutions due to Calderon-Sylvester-Uhlmann.
Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography
Hiroshi Isozaki
2007, 1(1): 107-134 doi: 10.3934/ipi.2007.1.107 +[Abstract](53) +[PDF](292.8KB)
Abstract:
We consider a boundary value problem for the Schrödinger operator $- \Delta + q(x)$ in a ball $\Omega : (x_1 + R)^2 + x_2^2 + (x_3 - r)^2 < r^2$, whose boundary we regard as a horosphere in the hyperbolic space $ H^3$ realized in the upper half space $ R^3_+$. Let $S = \{|x| = R, x_3 > 0\}$ be a hemisphere, which is generated by a family of geodesics in $ H^3$. By imposing a suitable boundary condition on $\partial\Omega$ in terms of a pseudo-differential operator, we compute the integral mean of $q(x)$ over $S\cap\Omega$ from the local knowledge of the associated (generalized) Robin-to-Dirichlet map for $- \Delta + q(x)$ around $S\cap\partial\Omega$. The potential $q(x)$ is then reconstructed by virtue of the inverse Radon transform on hyperbolic space. If the support of $q(x)$ has a positive distance from $\partial\Omega$, one can construct this generalized Robin-to-Dirichlet map from the usual Dirichlet-to-Neumann map. These results explain the mathematical background of the well-known Barber-Brown algorithm in electrical impedance tomography.
Stability of boundary distance representation and reconstruction of Riemannian manifolds
Atsushi Katsuda , Yaroslav Kurylev and  Matti Lassas
2007, 1(1): 135-157 doi: 10.3934/ipi.2007.1.135 +[Abstract](36) +[PDF](317.9KB)
Abstract:
A boundary distance representation of a Riemannian manifold with boundary $(M,g,$∂$\M)$ is the set of functions $\{r_x\in C $ (∂$\M$) $:\ x\in M\}$, where $r_x$ are the distance functions to the boundary, $r_x(z)=d(x, z)$, $z\in$∂$M$. In this paper we study the question whether this representation determines the Riemannian manifold in a stable way if this manifold satisfies some a priori geometric bounds. The answer is affermative, moreover, given a discrete set of approximate boundary distance functions, we construct a finite metric space that approximates the manifold $(M,g)$ in the Gromov-Hausdorff topology.
    In applications, the boundary distance representation appears in many inverse problems, where measurements are made on the boundary of the object under investigation. As an example, for the heat equation with an unknown heat conductivity the boundary measurements determine the boundary distance representation of the Riemannian metric which corresponds to this conductivity.
An integral equation approach and the interior transmission problem for Maxwell's equations
Andreas Kirsch
2007, 1(1): 159-179 doi: 10.3934/ipi.2007.1.159 +[Abstract](42) +[PDF](269.6KB)
Abstract:
In the first part of this paper we recall the direct scattering problem for time harmonic electromagnetic fields where arbitrary incident fields are scattered by a medium described by a space dependent permittivity, permeability, and conductivity. We present an integral equation approach and recall its basic features. In the second part we investigate the corresponding interior transmission eigenvalue problem and prove that the spectrum is discrete. Finally, we study the inhomogeneous interior transmission problem and show that it is uniquely solvable provided $k^2$ is not an interior eigenvalue.
Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation
John C. Schotland and  Vadim A. Markel
2007, 1(1): 181-188 doi: 10.3934/ipi.2007.1.181 +[Abstract](56) +[PDF](308.5KB)
Abstract:
We consider the inverse scattering problem for the radiative transport equation. We show that the linearized form of this problem can be formulated in terms of the inversion of a suitably defined Fourier-Laplace transform. This generalizes a previous result obtained within the diffusion approximation to the radiative transport equation.
Zeros of OPUC and long time asymptotics of Schur and related flows
Barry Simon
2007, 1(1): 189-215 doi: 10.3934/ipi.2007.1.189 +[Abstract](29) +[PDF](330.6KB)
Abstract:
We provide a complete analysis of the asymptotics for the semi-infinite Schur flow: $\alpha_j(t)=(1-|\alpha_j(t)|^2) (\alpha_{j+1}(t)-\alpha_{j-1}(t))$ for $\alpha_{-1}(t)= 1$ boundary conditions and $n=0,1,2,...$, with initial condition $\alpha_j(0)\in (-1,1)$. We also provide examples with $\alpha_j(0)\in\bbD$ for which $\alpha_0(t)$ does not have a limit. The proofs depend on the solution via a direct/inverse spectral transform.
Inverse scattering at a fixed energy for long-range potentials
Ricardo Weder and  Dimitri Yafaev
2007, 1(1): 217-224 doi: 10.3934/ipi.2007.1.217 +[Abstract](30) +[PDF](149.6KB)
Abstract:
In this paper we consider the inverse scattering problem at a fixed energy for the Schrödinger equation with a long-range potential in $R^d, d\geq 3$. We prove that the long-range part can be uniquely reconstructed from the leading forward singularity of the scattering amplitude at some positive energy.
A remark on inverse problems for resonances
Maciej Zworski
2007, 1(1): 225-227 doi: 10.3934/ipi.2007.1.225 +[Abstract](40) +[PDF](82.5KB)
Abstract:
Trace formulæ have been a powerful tool of inverse spectral theory on compact manifolds. We explain how the information from singularities away from zero immediately translates to the setting of resonances producing similar inverse results.

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