Browsing by Subject "Inverse problems"

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  • Helin, T.; Lassas, M.; Oksanen, L.; Saksala, T. (2018)
    Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information is available. In this paper we study the stochastic wave equation partial derivative(2)(t)u- Delta(g)u = chi W, where W is a random variable with the white noise statistics on R1+n, n >= 3, chi is a smooth function vanishing for negative times and outside a compact set in space, and Delta(g) is the Laplace Beltrami operator associated to a smooth non-trapping Riemannian metric tensor g on R-n. The metric tensor g models the medium to be imaged, and we assume that it coincides with the Euclidean metric outside a compact set. We consider the empirical correlations on an open set chi subset of R-n, C-T(t(1), x(1), t(2), x(2)) = 1/T integral(T)(0) u(t(1) s, x(1))u(t(2) s, x(2))ds, t(1), t(2) > 0, x(1), x(2) is an element of chi, for T > 0. Supposing that chi is non-zero on chi and constant in time after t > 1, we show that in the limit T -> infinity, the data C-T becomes statistically stable, that is, independent of the realization of W. Our main result is that, with probability one, this limit determines the Riemannian manifold (R-n, g) up to an isometry. (C) 2018 Elsevier Masson SAS. All rights reserved.
  • Chen, Xi; Lassas, Matti; Oksanen, Lauri; Paternain, Gabriel P. (2022)
    We consider the geometric non-linear inverse problem of recovering a Hermitian connection A from the source-to-solution map of the cubic wave equation square(A)phi + kappa vertical bar phi vertical bar(2) phi = f, where kappa not equal 0 and square(A) is the connection wave operator in the Minkowski space R1+3. The equation arises naturally when considering the Yang-Mills-Higgs equations with Mexican hat type potentials. Our proof exploits the microlocal analysis of non-linear wave interactions, but instead of employing information contained in the geometry of the wave front sets as in previous literature, we study the principal symbols of waves generated by suitable interactions. Moreover, our approach relies on inversion of a novel non-abelian broken light ray transform, a result interesting in its own right.
  • Lassas, Matti; Saksala, Teemu (2019)
    Let (N, g) be a Riemannian manifold with the distance function d(x, y) and an open subset M subset of N. For x is an element of M we denote by D-x the distance difference function D-x:F x F -> R, given by D-x(z(1), z(2)) = d(x, z(1)) - d(x, z(2)), z(1), z(2) is an element of F = N \ M. We consider the inverse problem of determining the topological and the differentiable structure of the manifold M and the metric g vertical bar M on it when we are given the distance difference data, that is, the set F, the metric g vertical bar F, and the collection D(M) = {D-x; x is an element of M}. Moreover, we consider the embedded image D(M) of the manifold M, in the vector space C(F x F), as a representation of manifold M. The inverse problem of determining (M, g) from D(M) arises e.g. in the study of the wave equation on R x N when we observe in F the waves produced by spontaneous point sources at unknown points (t, x) is an element of R x M. Then D-x (z(1), z(2)) is the difference of the times when one observes at points z(1) and z(2) the wave produced by a point source at x that goes off at an unknown time. The problem has applications in hybrid inverse problems and in geophysical imaging.
  • Koskinen, Anssi (Helsingin yliopisto, 2020)
    The applied mathematical field of inverse problems studies how to recover unknown function from a set of possibly incomplete and noisy observations. One example of real-life inverse problem is image destriping, which is the process of removing stripes from images. The stripe noise is a very common phenomenon in various of fields such as satellite remote sensing or in dental x-ray imaging. In this thesis we study methods to remove the stripe noise from dental x-ray images. The stripes in the images are consequence of the geometry of our measurement and the sensor. In the x-ray imaging, the x-rays are sent on certain intensity through the measurable object and then the remaining intensity is measured using the x-ray detector. The detectors used in this thesis convert the remaining x-rays directly into electrical signals, which are then measured and finally processed into an image. We notice that the gained values behave according to an exponential model and use this knowledge to transform this into a nonlinear fitting problem. We study two linearization methods and three iterative methods. We examine the performance of the correction algorithms with both simulated and real stripe images. The results of the experiments show that although some of the fitting methods give better results in the least squares sense, the exponential prior leaves some visible line artefacts. This suggests that the methods can be further improved by applying suitable regularization method. We believe that this study is a good baseline for a better correction method.
  • Lehtonen, Akseli; Correia, Carlos M.; Helin, Tapio (2018)
    SLODAR (SLOpe Detection And Ranging) methods recover the atmospheric turbulence profile from cross-correlations of wavefront sensor (WFS) measurements, based on known turbulence models. Our work grows out of several experiments showing that turbulence statistics can deviate significantly from the classical Kolmogorov/ von Kármán models, especially close to the ground. We present a novel SLODAR-type method which simultaneously recovers both the turbulence profile in the atmosphere and the turbulence statistics at the ground layer - namely the slope of the spatial frequency power law. We consider its application to outer scale (L0)- reconstruction and investigate the limits of the joint estimation of such parameters.
  • Krupchyk, Katya; Liimatainen, Tony; Salo, Mikko (2022)
    In this article we study the linearized anisotropic Calderon problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calderon problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial ingredients in the proof of our result are the construction of Gaussian beam quasimodes on the transversal manifold, with exponentially small errors, as well as the FBI transform characterization of the analytic wave front set. (c) 2022 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
  • Tukiainen, Simo (Finnish Meteorological Institute, 2016)
    Finnish Meteorological Institute Contributions 123
    Measurements of the Earth's atmosphere are crucial for understanding the behavior of the atmosphere and the underlying chemical and dynamical processes. Adequate monitoring of stratospheric ozone and greenhouse gases, for example, requires continuous global observations. Although expensive to build and complicated to operate, satellite instruments provide the best means for the global monitoring. Satellite data are often supplemented by ground-based measurements, which have limited coverage but typically provide more accurate data. Many atmospheric processes are altitude-dependent. Hence, the most useful atmospheric measurements provide information about the vertical distribution of the trace gases. Satellite instruments that observe Earth's limb are especially suitable for measuring atmospheric profiles. Satellite instruments looking down from the orbit, and remote sensing instruments looking up from the ground, generally provide considerably less information about the vertical distribution. Remote sensing measurements are indirect. The instruments observe electromagnetic radiation, but it is ozone, for example, that we are interested in. Interpreting the measured data requires a forward model that contains physical laws governing the measurement. Furthermore, to infer meaningful information from the data, we have to solve the corresponding inverse problem. Atmospheric inverse problems are typically nonlinear and ill-posed, requiring numerical treatment and prior assumptions. In this work, we developed inversion methods for the retrieval of atmospheric profiles. We used measurements by Optical Spectrograph and InfraRed Imager System (OSIRIS) on board the Odin satellite, Global Ozone Monitoring by Occultation of Stars (GOMOS) on board the Envisat satellite, and ground-based Fourier transform spectrometer (FTS) at Sodankylä, Finland. For OSIRIS and GOMOS, we developed an onion peeling inversion method and retrieved ozone, aerosol, and neutral air profiles. From the OSIRIS data, we also retrieved NO2 profiles. For the FTS data, we developed a dimension reduction inversion method and used Markov chain Monte Carlo (MCMC) statistical estimation to retrieve methane profiles. Main contributions of this work are the retrieved OSIRIS and GOMOS satellite data sets, and the novel retrieval method applied to the FTS data. Long satellite data records are useful for trends studies and for distinguishing between anthropogenic effects and natural variations. Before this work, GOMOS daytime ozone profiles were missing from scientific studies because the operational GOMOS daytime occultation product contains large biases. The GOMOS bright limb ozone product vastly improves the stratospheric part of the GOMOS daytime ozone. On the other hand, the dimension reduction method is a promising new technique for the retrieval of atmospheric profiles, especially when the measurement contains little information about the vertical distribution of gases.
  • Chung, Francis J.; Ola, Petri; Salo, Mikko; Tzou, Leo (2018)
    In this article we consider an inverse boundary value problem for the time-harmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of Bukhgeim-Uhlmann and Kenig-Sjostrand-Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system. (C) 2017 Elsevier Masson SAS. All rights reserved.
  • Lehtonen, Jonatan; Helin, Tapio (2019)
    A number of experiments have demonstrated that turbulence profiles can change very rapidly. However, turbulence profiling techniques typically rely on estimating correlations from up to a few minutes of data, delivering accurate profiles at the cost of a time delay. We present a novel method for tracking the turbulence profile in real time using AO telemetry. This method can provide complementary information without requiring any additional instruments, and may provide new insights into turbulence dynamics.
  • Fefferman, Charles; Ivanov, Sergei; Kurylev, Yaroslav; Lassas, Matti; Narayanan, Hariharan (2020)
    We study the geometric Whitney problem on how a Riemannian manifold (M, g) can be constructed to approximate a metric space (X, d(X)). This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold S subset of R-m, m > n needs to be constructed to approximate a point cloud in Rm. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov-Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results, we give a new characterization of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterize the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalization of the Whitney embedding construction where approximative coordinate charts are embedded in R-m and interpolated to a smooth submanifold.
  • Kian, Yavar; Kurylev, Yaroslav; Lassas, Matti; Oksanen, Lauri (2019)
    We consider a restricted Dirichlet-to-Neumann map Lambda(T)(S, R) associated with the operator partial derivative(2)(t) - Delta(g) + A + q where Delta(g) is the Laplace-Beltrami operator of a Riemannian manifold (M, g), and A and q are a vector field and a function on M. The restriction Lambda(T)(S, R) corresponds to the case where the Dirichlet traces are supported on (0, T) x S and the Neumann traces are restricted on (0, T) x R. Here S and R are open sets, which may be disjoint, on the boundary of M. We show that Lambda(T)(S, R) determines uniquely, up the natural gauge invariance, the lower order terms A and q in a neighborhood of the set R assuming that R is strictly convex and that the wave equation is exactly controllable from S in time T/2. We give also a global result under a convex foliation condition. The main novelty is the recovery of A and q when the sets R and S are disjoint. We allow A and q to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy. Crown Copyright (C) 2019 Published by Elsevier Inc. All rights reserved.