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Speaker: Gaik Ambartsoumian Topic: Generalized Radon transforms in tomography Abstract: The mathematical models of many modern imaging modalities are based on Radon-type transforms that integrate an unknown (image) function along various curves or surfaces. The problem of image reconstruction in these cases requires the inversion of such transforms, as well as the study of their properties. The latter include the description of their injectivity sets and ranges, microlocal regularity and stability of their inversion, numerical implementation of the inversion formulas and algorithms, etc. The lectures will discuss the mathematical techniques, known results and open problems related to the spherical Radon transform arising in thermo- and photo-acoustic tomographies, elliptical Radon transform arising in bi-static setups of ultrasound imaging and Radar, and V-line transform arising in the single scattering optical tomography. |
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Speaker: Jan Boman Topic: Microlocal analysis of generalized Radon transforms Abstract: As is well known the concept of wave front set has revolutionized the theory of partial differential equations. In tomography the wave front set is useful for understanding how edges and contours in the object will be shown in the image and also why artefacts sometimes occur. The analytic wave front set has been used for proving uniqueness theorems by means of arguments from Hörmander's proof of Holmgren's uniqueness theorem. In the lectures I will explain the definition and basic properties of the (analytic) wave front set for distributions. I will describe how linear operators, especially the Radon transform, map singularities as described by the wave front set and how the analytic wave front set can be used for proving uniqueness theorems. |
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Speaker: Kim Knudsen Topic: Numerical methods in inverse problems Abstract: Complex Geometrical Optics (CGO) solutions to partial differential equations provide a powerful framework for the mathematical analysis of certain inverse problems such as the Calderón problem. Not only do CGO solutions provide uniqueness proofs, they also allow for reconstruction algorithms that in principle solves the full non-linear inverse problems at once. In 2D the algorithm is known as the Dbar method (due to complex conjugate differentiation entering the process); a similar approach is possible in 3D. In these lectures we will discuss these reconstruction algorithms in the context of the Calderón problem in 2D and 3D, give the theory behind and show numerical implementations and examples. In addition we consider the partial data Calderón problem, discuss uniqueness results and a possible reconstruction algorithm. |
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Speaker: Peter Kuchment Topic: Hybrid imaging problems Abstract: The talks will focus on the following topics: 1. A brief survey of the standard tomographic techniques. Smoothing properties, (in)stability, and embedding theorems.
2. Types of PDEs involved and (in)stability. Microlocal properties. Hybrid (coupled physics) techniques and their tentative classification. Most prominent examples of coupled physics techniques.
3. Why are the hybrid methods stabler? problems with interior information. Infinitesimal microlocal analysis.
4. Reconstructions techniques.
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Speaker: Slava Kurylev Topic: Inverse problems associated with Riemannian and Lorentz geometries Abstract: In these lectures we discuss two inverse problems which have a distinguished geometric flavour. The first one is the inverse problems of the recovering of a compact Riemannian manifold (N, g) from the data collected on the its boundary ∂N. These data are related to the Laplace operator ΔN of (N, g) with some classical boundary condition and, therefore, is called the boundary spectral data. By means of the boundary Control (BC) method, the principal ideas of which we intend to explain, we show how the boundary spectral data determine an “image”, R(N) ∈ L∞(∂N) of N. Next we explain how it is possible to extract the topology and geometry of N from R(N).
Our second inverse problem is the one of the reconstruction of (a part of) a Lorentian manifold, (M, g), satisfying some additional conditions, from light observations. We assume that an open submanifold Ω ⊂ N is filled with light-sources. The light from these sources is registered in another subdomain U ⊂ M covered with time-like geodesic along which we do register the light (this corresponds to the astrophysical observations). We show that these data make it possible to determine the topology and, upto a conformal factor, geometry of Ω. It turns out that there is a deep analogy of how the distance functions from R(N) determine (N, g) and how the light observations determine (Ω, g).
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Speaker: Cliff Nolan Topic: Inverse problems in seismic and radar imaging Abstract: We review the basic microlocal analysis needed for RADAR/seismic imaging and develop a model of scattered waves (suitable for both RADAR/seismic imaging). Following Rakesh, we show how under mild assumptions, the scattering operator (which maps the unknown high-frequency component of the wave speed to the recorded high-frequency scattered waves) is a Fourier integral operator. We then consider various configurations of source/receiver pairs for the emitted/recorded waves and examine the backprojected image that results from applying a (weighted) backprojection operator to the recorded scattered waves. This image is a graph of the high-frequency component of the wave speed at various locations on the earth's surface (RADAR)/subsurface (seismic). A particular focus of the talks will be to examine the various artifacts that can be present in the backprojected image and the strength of such artifacts, compared to the expected strength of the true image. |
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Speaker: Rakesh Topic: Inverse problems for hyperbolic PDEs Abstract: We introduce Inverse Problems by describing three inverse problems arising from an elliptic PDE, a hyperbolic PDE, and a transport equation, and state some of the results. We then focus on some formally determined inverse problems for hyperbolic PDEs, introduce Carleman estimates (useful for studying non-well posed Cauchy problems), and show how it was adapted to solve a formally determined inverse problem for a multidimensional hyperbolic PDE. |
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Speaker: Bastian von Harrach Topic: Introduction to the numerical solution of inverse problems and shape detection in electrical impedance tomography Abstract: In the first lecture we will give an introduction to inverse problems and study the problem of ill-posedness. The second lecture shows how to construct convergent numerical solvers for inverse problems at the example of Tikhonov regularization. The last two lectures concentrate on recent research on the prominent non-linear inverse problem of Electrical Impedance Tomography (EIT). We will show how to detect conductivity anomalies in EIT using the Factorization and the Monotonicity Method. |