Regularized D-bar method for the inverse conductivity problem
Kim Knudsen Matti Lassas Jennifer L. Mueller Samuli Siltanen
Inverse Problems & Imaging 2009, 3(4): 599-624 doi: 10.3934/ipi.2009.3.599
A strategy for regularizing the inversion procedure for the two-dimensional D-bar reconstruction algorithm based on the global uniqueness proof of Nachman [Ann. Math. 143 (1996)] for the ill-posed inverse conductivity problem is presented. The strategy utilizes truncation of the boundary integral equation and the scattering transform. It is shown that this leads to a bound on the error in the scattering transform and a stable reconstruction of the conductivity; an explicit rate of convergence in appropriate Banach spaces is derived as well. Numerical results are also included, demonstrating the convergence of the reconstructed conductivity to the true conductivity as the noise level tends to zero. The results provide a link between two traditions of inverse problems research: theory of regularization and inversion methods based on complex geometrical optics. Also, the procedure is a novel regularized imaging method for electrical impedance tomography.
keywords: ill-posed problem electrical impedance tomography inverse problem regularization. inverse conductivity problem
Recovering boundary shape and conductivity in electrical impedance tomography
Ville Kolehmainen Matti Lassas Petri Ola Samuli Siltanen
Inverse Problems & Imaging 2013, 7(1): 217-242 doi: 10.3934/ipi.2013.7.217
Electrical impedance tomography (EIT) aims to reconstruct the electric conductivity inside a physical body from current-to-voltage measurements at the boundary of the body. In practical EIT one often lacks exact knowledge of the domain boundary, and inaccurate modeling of the boundary causes artifacts in the reconstructions. A novel method is presented for recovering the boundary shape and an isotropic conductivity from EIT data. The first step is to determine the minimally anisotropic conductivity in a model domain reproducing the measured EIT data. Second, a Beltrami equation is solved, providing shape-deforming reconstruction. The algorithm is applied to simulated noisy data from a realistic electrode model, demonstrating that approximate recovery of the boundary shape and conductivity is feasible.
keywords: conformal deformation. electrical impedance tomography minimally anisotropic conductivity quasiconformal maps Inverse conductivity problem
Three-dimensional dental X-ray imaging by combination of panoramic and projection data
Nuutti Hyvönen Martti Kalke Matti Lassas Henri Setälä Samuli Siltanen
Inverse Problems & Imaging 2010, 4(2): 257-271 doi: 10.3934/ipi.2010.4.257
A novel three-dimensional dental X-ray imaging method is introduced, based on hybrid data collected with a dental panoramic device. Such a device uses geometric movement of the X-ray source and detector around the head of a patient to produce a panoramic image, where all teeth are in sharp focus and details at a distance from the dental arc are blurred. A digital panoramic device is reprogrammed to collect two-dimensional projection radiographs. Two complementary types of data are measured from a region of interest: projection data with a limited angle of view, and a panoramic image. Tikhonov regularization is applied to these data in order to produce three-dimensional reconstructions. The algorithm is tested with simulated data and real-world in vitro measurements from a dry mandible. Reconstructions from limited-angle projection data alone do provide the dentist with three-dimensional information useful for dental implant planning. Furthermore, adding panoramic data to the process improves the reconstruction precision in the direction of the dental arc. The presented imaging modality can be seen as a cost-effective alternative to a full-angle CT scanner.
keywords: limited angle data three-dimensional reconstruction. panoramic image X-ray imaging hybrid data Tikhonov regularization
Iterative time-reversal control for inverse problems
Kenrick Bingham Yaroslav Kurylev Matti Lassas Samuli Siltanen
Inverse Problems & Imaging 2008, 2(1): 63-81 doi: 10.3934/ipi.2008.2.63
A novel method to solve inverse problems for the wave equation is introduced. The method is a combination of the boundary control method and an iterative time reversal scheme, leading to adaptive imaging of coefficient functions of the wave equation using focusing waves in unknown medium. The approach is computationally effective since the iteration lets the medium do most of the processing of the data.
    The iterative time reversal scheme also gives an algorithm for approximating a given wave in a subset of the domain without knowing the coefficients of the wave equation.
keywords: Inverse problems wave equation control time reversal.
Discretization-invariant Bayesian inversion and Besov space priors
Matti Lassas Eero Saksman Samuli Siltanen
Inverse Problems & Imaging 2009, 3(1): 87-122 doi: 10.3934/ipi.2009.3.87
Bayesian solution of an inverse problem for indirect measurement $M = AU + $ε is considered, where $U$ is a function on a domain of $\R^d$. Here $A$ is a smoothing linear operator and ε is Gaussian white noise. The data is a realization $m_k$ of the random variable $M_k = P_kA U+P_k$ε , where $P_k$ is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as $U_n=T_nU$, where $T_n$ is a finite dimensional projection, leading to the computational measurement model $M_{kn}=P_k A U_n + P_k$ε . Bayes formula gives then the posterior distribution

$\pi_{kn}(u_n\|\m_{kn})$~ Π n $(u_n)\exp(-\frac{1}{2}$||$\m_{kn} - P_kA u_n$||$\_2^2)$

in $\R^d$, and the mean $\u_{kn}$:$=\int u_n \ \pi_{kn}(u_n\|\m_k)\ du_n$ is considered as the reconstruction of $U$. We discuss a systematic way of choosing prior distributions Π n for all $n\geq n_0>0$ by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is discretization-invariant in the sense that Π n represent the same a priori information for all $n$ and that the mean $\u_{kn}$ converges to a limit estimate as $k,n$→$\infty$. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with $B^1_11$ prior is related to penalizing the $\l^1$ norm of the wavelet coefficients of $U$.

keywords: wavelet discretization invariance Inverse problem statistical inversion Besov space. Bayesian inversion reconstruction
Reconstruction of a compact manifold from the scattering data of internal sources
Matti Lassas Teemu Saksala Hanming Zhou
Inverse Problems & Imaging 2018, 12(4): 993-1031 doi: 10.3934/ipi.2018042

Given a smooth non-trapping compact manifold with strictly convex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. These data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, the scattering data measured on the boundary determine the Riemannian manifold up to isometry.

keywords: Inverse problem Riemannian geometry geodesics partial differential equations compact manifold with boundary
Lassi Päivärinta Matti Lassas Jackie (Jianhong) Shen
Inverse Problems & Imaging 2007, 1(1): i-iii doi: 10.3934/ipi.2007.1.1i
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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Stability of boundary distance representation and reconstruction of Riemannian manifolds
Atsushi Katsuda Yaroslav Kurylev Matti Lassas
Inverse Problems & Imaging 2007, 1(1): 135-157 doi: 10.3934/ipi.2007.1.135
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.
keywords: Inverse problems Riemannian manifold. Boundary distance functions

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