Browsing by Subject "inverse problems"

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  • Lai, Ru-Yu; Shankar, Ravi; Spirn, Daniel; Uhlmann, Gunther (2017)
    We consider the problem of reconstructing the features of a weak anisotropic background potential by the trajectories of vortex dipoles in a nonlinear Gross-Pitaevskii equation. At leading order, the dynamics of vortex dipoles are given by a Hamiltonian system. If the background potential is sufficiently smooth and flat, the background can be reconstructed using ideas from the boundary and the lens rigidity problems. We prove that reconstructions are unique, derive an approximate reconstruction formula, and present numerical examples.
  • Agnelli, J. P.; Çöl, A.; Lassas, M.; Murthy, R.; Santacesaria, M.; Siltanen, S. (2020)
    Electrical impedance tomography (EIT) is an emerging non-invasive medical imaging modality. It is based on feeding electrical currents into the patient, measuring the resulting voltages at the skin, and recovering the internal conductivity distribution. The mathematical task of EIT image reconstruction is a nonlinear and ill-posed inverse problem. Therefore any EIT image reconstruction method needs to be regularized, typically resulting in blurred images. One promising application is stroke-EIT, or classification of stroke into either ischemic or hemorrhagic. Ischemic stroke involves a blood clot, preventing blood flow to a part of the brain causing a low-conductivity region. Hemorrhagic stroke means bleeding in the brain causing a high-conductivity region. In both cases the symptoms are identical, so a cost-effective and portable classification device is needed. Typical EIT images are not optimal for stroke-EIT because of blurriness. This paper explores the possibilities of machine learning in improving the classification results. Two paradigms are compared: (a) learning from the EIT data, that is Dirichlet-to-Neumann maps and (b) extracting robust features from data and learning from them. The features of choice are virtual hybrid edge detection (VHED) functions (Greenleaf et al 2018 Anal. PDE 11) that have a geometric interpretation and whose computation from EIT data does not involve calculating a full image of the conductivity. We report the measures of accuracy, sensitivity and specificity of the networks trained with EIT data and VHED functions separately. Computational evidence based on simulated noisy EIT data suggests that the regularized grey-box paradigm (b) leads to significantly better classification results than the black-box paradigm (a).
  • Kirpichnikova, Anna; Korpela, Jussi; Lassas, Matti J.; Oksanen, Lauri (2021)
    We study the wave equation on a bounded domain of Rm and on a compact Riemannian manifold M with boundary. We assume that the coefficients of the wave equation are unknown but that we are given the hyperbolic Neumann-to-Dirichlet map. that corresponds to the physical measurements on the boundary. Using the knowledge of. we construct a sequence of Neumann boundary values so that at a time T the corresponding waves converge to zero while the time derivative of the waves converge to a delta distribution. The limit of such waves can be considered as a wave produced by an artificial point source. The convergence of the wave takes place in the function spaces naturally related to the energy of the wave. We apply the results for inverse problems and demonstrate the focusing of the waves numerically in the one-dimensional case.
  • Caro, Pedro; Helin, Tapio; Kujanpää, Antti; Lassas, Matti (2019)
    Scattering from a non-smooth random field on the time domain is studied for plane waves that propagate simultaneously through the potential in variable angles. We first derive sufficient conditions for stochastic moments of the field to be recovered from empirical correlations between amplitude measurements of the leading singularities, detected in the exterior of a region where the potential is almost surely supported. The result is then applied to show that if two sufficiently regular random fields yield the same correlations, they have identical laws as function-valued random variables.
  • Beretta, Elena; Cavaterra, Cecilia; Ratti, Luca (2020)
    In this paper we consider the monodomain model of cardiac electrophysiology. After an analysis of the well-posedness of the model we determine an asymptotic expansion of the perturbed potential due to the presence of small conductivity inhomogeneities (modelling small ischemic regions in the cardiac tissue) and use it to detect the anomalies from partial boundary measurements. This is done by determining the topological gradient of a suitable boundary misfit functional. The robustness of the algorithm is confirmed by several numerical experiments.
  • Feizmohammadi, Ali; Ilmavirta, Joonas; Kian, Yavar; Oksanen, Lauri (2021)
    We study uniqueness of the recovery of a time-dependent magnetic vectorvalued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet-to-Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold.
  • Suonperä, Ensio (Helsingin yliopisto, 2019)
    The motivation for the methods developed in this thesis rises from solving the severely ill-posed inverse problem of limited angle computed tomography. Breast tomosynthesis provides an example where the inner structure of the breast should be reconstructed from a very limited measurement angle. Some parts of the boundaries of the structure can be recovered from the X-ray measurements and others can not. These are referred to as visible and invisible boundaries. For parallel beam measurement geometry directions of visible and invisible boundaries can be deduced from the measurement angles. This motivates the usage of the concept of wavefront set. Roughly speaking, a wavefront set contains boundary points and their directions. The definition of wavefront set is based on Fourier analysis, but its characterization with the decay properties of functions called shearlets is used in this thesis. Shearlets are functions based on changing resolution, orientation, and position of certain generating functions. The theoretical part of this thesis focuses on studying this connection between shearlets and wavefront sets. This thesis applies neural networks to the limited angle CT problem since neural networks have become state-of-the-art in many computer vision tasks and achieved impressive performance in inverse problems related to imaging. Neural networks are compositions of multiple simple functions, typically alternating linear functions and some element-wise non-linearities. They are trained to learn values for a huge amount of parameters to approximate the desired relation between input and output spaces. Neural networks are very flexible function approximators, but high dimensional optimization of parameters from data makes them hard to interpret. Convolutional neural networks (CNN) are the ones that succeed in tasks with image-like inputs. U-Net is a CNN architecture with very good properties, like learning useful parameters form considerably small data sets. This thesis provides two U-Net based CNN methods for solving limited angle CT problems. The main focus is on method projecting model-based reconstructions such that the projections have the desired wavefront sets. The guiding principle of this projector network is that it should not change reconstruction already projected to the given wavefront set. Another network estimates the invisible part of the wavefront set from the visible one. Few different data sets are simulated to train and evaluate these methods and performance on real data is also tested. A combination of the wavefront set estimator and the projector networks were used to postprocess model-based reconstructions. The fact this postprocessing has two steps increases the interpretability and the control over the processes performed by neural networks. This postprocessing increased the quality of reconstructions significantly and quality was even better when the true wavefront set was given for the projector as a prior.
  • Tamminen, J.; Tarvainen, T.; Siltanen, S. (2017)
    The D-bar method at negative energy is numerically implemented. Using the method, we are able to numerically reconstruct potentials and investigate exceptional points at negative energy. Subsequently, applying the method to diffuse optical tomography, a new way of reconstructing the diffusion coefficient from the associated Complex Geometrics Optics solution is suggested and numerically validated.