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Session Overview | |
| Location: VG2.107 |
| Date: Tuesday, 05/Sept/2023 | |
| 1:30pm - 3:30pm | MS40: Dynamic Imaging Location: VG2.107 Session Chair: Peter Elbau |
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Iterative and data-driven motion compensation in tomography University of Stuttgart, Germany
Most tomographic modalities record the data sequentially, i.e. temporal changes of the object lead
to inconsistent measurements. Consequently, suitable models and algorithms have to be developed
in order to provide artefact free images. In this talk, we provide an overview of different strategies, including a data-driven approach to extract explicit motion maps which can then be incorporated within direct or iterative reconstruction methods for the underlying dynamic inverse problem. Our methods are illustrated by numerical results from real as well as simulated data of different imaging modalities.
Sparse optimization algorithms for dynamic imaging University of Hull, United Kingdom
In this talk we introduce a Frank-Wolfe-type algorithm for sparse optimization in Banach spaces. The functional we want to optimize consist of the sum of a smooth fidelity term and of a convex
one-homogeneous regularizer. We exploit the sparse structure of the variational problem by designing iterates as linear combinations of extremal points of the unit ball of the regularizer. For such iterates we prove global sublinear convergence of the algorithm. Then, under additional structural assumptions, we prove a local linear convergence rate. We apply this algorithm to the problem of particles tracking from heavily undersampled dynamic MRI data. This talk is based on the works cited below.
[1] K.Bredies, M.Carioni, S.Fanzon, D.Walter. Asymptotic linear convergence of Fully-Corrective Generalized Conditional Gradient methods. Mathematical Programming, 2023.
[2] K.Bredies, S.Fanzon. An optimal transport approach for solving dynamic inverse problems in spaces of measures. ESAIM:M2AN, 54(6): 2351-2382, 2020.
[3] K.Bredies, M.Carioni, S.Fanzon, F.Romero. A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization. Found Comput Math, 2022
[4] K.Bredies, M.Carioni, S.Fanzon. On the extremal points of the ball of the Benamou–Brenier energy. Bull. London Math. Soc., 53: 1436-1452, 2021.
[5] K.Bredies, M.Carioni, S.Fanzon. A superposition principle for the inhomogeneous continuity equation with Hellinger–Kantorovich-regular coefficients. Communications in Partial Differential Equations, 47(10): 2023-2069, 2022.
New approaches for reconstruction in dynamic nano-CT imaging 1Georg-August-University Göttingen, Germany; 2Saarland University Saarbrücken, Germany; 3University of Stuttgart, Germany
Tomographic X-ray imaging on the nano-scale is an important tool to visualize the structure of materials such as alloys or biological tissue. Due to the small scale on which the data acquisition takes place, small perturbances caused by the environment become significant and cause a motion of the object relative to the scanner during the scan. Since this motion is hard to estimate and its incorporation into the reconstruction process strongly increases the numerical effort, we aim at a different approach for a stable reconstruction: We interpret the object motion as a modelling inexactness in comparison to the model in the static case. This inexactness is estimated and included in an iterative regularization scheme called sequential subspace optimization. Data-driven techniques are investigated to estimate the modelling error and to improve the obtained reconstructions.
Artifact reduction for time dependent image reconstruction in magnetic particle imaging Universität Hamburg, Germany
Magnetic particle imaging (MPI) is a preclinical imaging modality exploiting the nonlinear magnetization response of magnetic nanoparticles to applied dynamic magnetic fields. We focus on MPI using a field-free line for spatial encoding because under ideal assumptions such as static objects, ideal magnetic fields and sequential line rotation, the MPI data are obtained by Radon transformed particle distributions. In practice, field imperfections and moving objects occur such that we have to adapt the Radon transform and jointly reconstruct time dependent particle distributions and adapted Radon data by means of total variation regularization.
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| 4:00pm - 6:00pm | MS57 3: Inverse Problems in Time-Domain Imaging at the Small Scales Location: VG2.107 Session Chair: Eric Bonnetier Session Chair: Xinlin Cao Session Chair: Mourad Sini |
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The electromagnetic waves generated by a cluster of nanoparticles with high refractive indices and corresponding effective medium theory. RICAM, Austrian Academy of Science, Austria
We estimate the electromagnetic fields generated by a cluster of dielectric nanoparticles which are small scaled but enjoy high contrast of their relative permittivity, embedded into a background made of a vacuum. Under certain ratio between their size and contrast, these nanoparticles generate resonances, called dielectric resonances. We first characterize the dominant field generated by a cluster of such dielectric-resonating nanoparticles. In this point-interaction approximation, the nanoparticles can be distributed to occupy volume-like domains or low dimensional hypersurfaces where periodicity is not required. Then we investigate the corresponding effective electromagnetic medium with periodic distribution under some mild assumptions. We show that even though the dielectric nanoparticles are merely generated by the contrasts of their permittivity (and not their permeability), the effective medium is a perturbation of the permeability and not the permittivity. Both of the cases for the effective permeability being positive and negative are studied.
Galerkin Foldy-Lax asymptotic models for time-domain scattering by small particles INRIA, France
Foldy-Lax models are asymptotic models of wave scattering
by multiple obstacles, in the regime when their characteristic size tends to
zero. While frequency-domain Foldy-Lax models are now fairly well-studied
(see e.g. [1,2,3]), their time-domain
counterparts were considered only very recently, see [4,5].
This talk is dedicated to the derivation, stability and convergence analysis
of a time-domain Foldy-Lax model for sound-soft scattering by small obstacles. We start with the analysis of the time-domain counterpart of the respective frequency-domain model for circular scatterers from [6] and show that it is unstable for
some geometric configurations. To stabilize it, we propose its reinterpretation
as a perturbed Galerkin discretization of a single layer boundary integral equation.
Its unperturbed Galerkin discretization is then automatically stable due
to a coercivity-like property of the underlying operator and thus serves as a
basis to derive the stabilized model.
Let us remark that this reinterpretation provides us with an alternative way to derive asymptotic models as Galerkin discretizations of boundary integral formulations with well-chosen basis functions.
We will present the convergence analysis of the new model,
discuss its numerical implementation, and illustrate our findings with numerical experiments.
[1] P. Martin. Multiple scattering, vol. 107 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2006.
[2] D.P. Challa, M. Sini. On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes, Multiscale Model. Simul., 12 ,
pp. 55–108, 2014.
[3] D.P. Challa, M. Sini. The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé system, Math. Nachr., 288, pp. 1834–1872, 2015.
[4] H. Barucq, J. Diaz, V. Mattesi, S. Tordeux. Asymptotic behavior of acoustic waves scattered by very small obstacles, ESAIM Math. Model. Numer. Anal., 55, pp. S705–S731, 2021.
[5] M. Sini, H. Wang, Q. Yao. Analysis of the acoustic waves reflected by a cluster of small holes in the time-domain and the equivalent mass density, Multiscale Model. Simul., 19, pp. 1083–1114, 2021.
[6] M. Cassier, C. Hazard. Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the Foldy-Lax model, Wave Motion, 50, pp. 18–28, 2013.
Electromagnetic waves generated by a moving dielectric under the special relativity assumptions Indian Institute of Science Education and Research Bhopal, India
In this talk, we will consider the electromagntic waves generated by an in-
clusion moving at a uniformly constant speed. We will discuss the direct and inverse scattering problem for the corresponding transmission problem. We
first show that the scattering electromagnetic fields satisfy a related Lippmann-Schwinger system of equations and the solutions of this system of integral equations can be written in terms of the Neumann series under certain assumption on the ratio between the speed of the moving object and the speed of light in
the vaccum. Finally, as an application of this result, we will prove that the
far-field map uniquely determines the unknown moving object.
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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | CT03: Contributed talks Location: VG2.107 Session Chair: Martin Halla |
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The Foldy-Lax approximation of scattered field by many small inclusions near the resonating frequencies for Lam\`e system 1IIT Tirupati, India; 2Radon Institute (RICAM), Austria
We are concerned with the time harmonic elastic scattering in the presence of multiple small-scaled inclusions. The main property we use in this work is the local enhancement of scattering, which occurs at a specific incident frequency when the medium is perturbed with highly contrasted small inhomogeneities; for instance, one can consider contrast on the mass density. Such highly contrasting inclusions generate few local spots at their locations. These spots are generated as possible body waves related to elastic resonances. A family of these resonances is related to the eigenvalues of the elastic Newtonian operator.
\par Our goal is to derive the approximation of elastic scattered field for incident frequencies near to elastic resonances with suitable sufficient conditions. The dominating field generated due to the multiple interactions between a cluster of small inhomogeneities, of sub-wavelength size, is the Foldy-Lax field. The derived result has several applications, to mention a few, firstly in the theory of effective medium to design the materials with desired properties and, secondly, in elastic imaging to solve the inverse problem of recovering the properties of the background medium.
Microlocal Analysis of Multistatic Synthetic Aperture Radar Imaging University of Limerick, Ireland
We consider Synthetic Aperture Radar (SAR) in which scattered waves, simultaneously emitted from a pair of stationary emitters, are measured along a flight track traversed by an aircraft. A linearized mathematical model of scattering is obtained using a Fourier integral operator. This model can then be used to form an image of the ground terrain using backprojection together with a carefully designed data acquisition geometry.
The data is composed of two parts, corresponding to the received signals from each emitter. A backprojection operator can be easily chosen that correctly reconstructs the singularities in the wave speed using just one emitter. One would expect this to lead to a reasonable image of the terrain. However, we expect that application of this backprojection operator to the data from the other emitter will lead to unwanted artifacts in the image. We analyse the operators associated with this situation, and use microlocal analysis to determine configurations of flight path and emitter locations so that we may mitigate the artifacts associated to such “cross talk” between the two emitters.
An inverse problem for the Riemannian minimal surface equation University of Jyväskylä, Finland
In this work we study an inverse problem for the Riemannian minimal surface equation, which is a quasilinear elliptic PDE. Consider a Riemannian manifold $(M,g)$ where $M=\mathbb{R}^n$ and the metric is a so called conformally transversally anisotropic metric i.e. $g=c(\hat{g}\oplus e)$, where $\hat{g}$ is a metric on $\mathbb{R}^{n-1}$. Let $u\colon\Omega\subset\mathbb{R}^{n-1}\to \mathbb{R}$ be a smooth function that satisfies the minimal surface equation. Assume that we can make boundary measurements on the graph of $u$, that is we know the Dirichlet-to-Neumann (DN) map which maps the boundary value $u|_{\partial\Omega}=f$ to the normal derivative $\partial_{\nu}u|_{\partial\Omega}=\hat{g}^{ij}\partial_{x_i}u\nu_j|_{\partial\Omega}$. The Dirichlet data $f$ is the height of minimal surface on the boundary. The normal derivative $\partial_{\nu}u|_{\partial\Omega}$ can be thought of as tension on the boundary caused by the minimal surface. In this talk we show that if we have knowledge of two DN maps corresponding to two different metrics in the same conformal class, then we can deduce that the metrics have the same Taylor series up to a constant multiplier.
This work connects some aspects of two previous articles, that is [1] and [2]. We use the technique of higher order linearization (see for example [3]) that has received increasing attention lately.
[1] J. Nurminen. An inverse problem for the minimal surface equation. Nonlinear Anal.
227,113163:19. 2023
[2] C. I. Cârstea, M. Lassas, T. Liimatainen, L. Oksanen. An inverse problems for
the riemannian minimal surface equation, arXiv: 2203.09262:1–18. 2022
[3] M. Lassas, T. Liimatainen, Y.-H. Lin, M. Salo. Inverse problems for elliptic
equations with power type nonlinearities. J. Math. Pures Appl. (9) 145: 44–
82. 2021
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | CT06: Contributed talks Location: VG2.107 Session Chair: Milad Karimi |
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$L^1$-data fitting for Inverse Problems with subexponentially-tailed data Julius-Maximilians-Universität Würzburg, Germany
Outgoing from [1] and [2] we analyze variational regularization with $L^1$ data fidelity. We investigate discrete models with regular data in the sense that the tails decay subexponentiallly. Therefore, error bounds are provided and numerical simulations of convergence rates are presented.
[1] T. Hohage, F. Werner, Convergence rates for inverse problems with impulsive noise, SIAM J. Numer. Anal., 52: 1203-1221, 2014.
[2] C.König, F. Werner, T. Hohage, Convergence rates for exponentially ill-posed inverse problems with impulsive noise, SIAM J. Numer. Anal., 54: 341-360, 2016.
Globally Convergent Convexification Method for Coefficient Inverse Problems for Three Equations University of North Carolina at Charlotte, United States of America
Three Coefficient Inverse Problems (CIP) will be considered. Respectively, three versions of the globally convergent convexification numerical method will be presented. Global convergence theorems will be outlined and numerical results will be presented. Results outlined below are the first ones for each considered CIP. These CIPs are:
1. CIP for the radiative transport equation with euclidian propagation of particles [1].
2. CIP for the Riemannian radiative transport equation. In this case, particles propagate along geodesic lines between their scattering events [2].
3. Travel Time Tomography Problem in the 3-d case [3]. This is a CIP for the eikonal equation in the 3-d case. First numerical results in 3-d for this CIP will be presented,
[1] M.V. Klibanov, J. Li, L.H. Nguyen, Z. Yang, Convexification numerical method for a coefficient inverse problem for the radiative transport equation, SIAM J. Imaging Sciences, 16;35-63, 2023.
[2] M.V. Klibanov, J. Li, L.H. Nguyen, V.G. Romanov, Z. Yang, Convexification numerical method for a coefficient inverse problem for the Riemannian radiative transfer equation, arxiv: 2212.12593, 2022.
[3] M.V. Klibanov, J. Li, W. Zhang, Numerical solution of the 3-D travel time tomography problem, Journal of Computational Physics, 476:111910, 2023. published online https://doi.org/10.1016/j.jcp.2023.111910.
On modeling and regularization of piezoelectric inverse problems using all-at-once and reduced approaches Humboldt-Universität zu Berlin, Germany
Piezoelectric materials are an essential component for a wide range of electrical devices. Consequently, the range of possible applications for piezoelectric materials is expansive, encompassing, for example, electronic toothbrushes and microphones, as well as ultrasound imaging and sonar devices.
Simplified, the behaviour in the small signal range can be described by a linearly coupled PDE system for mechanical displacement and electrical potential, which can then be extended by a non-linear PDE system to consider piezoelectric material in high signal range.
Since many applications require high precision and also due to the transition to lead-free piezoceramics, a consistent and reproducible characterization of the material parameter set is of very high importance to properly determine the material properties, as the material data provided by the manufacturers often deviate significantly from the real data and are difficult and expensive to measure.
Therefore, this talk will focus on the parameter identification problem for the piezoelectric partial differential equation based on a measured and simulated quantity of the sample. Hence, we will derive the forward operator of this inverse problem generally. Then we will consider this inverse problem using regularization techniques based on all-at-once and reduced iterative methods, and further discuss the connection between the adjoint operators of the all-at-once approach and the adjoint differential equations of the reduced approach. Since several applications exhibit nonlinear material behaviour, the all-at-once approach is of particular interest, especially with respect to computational aspects. Thus, modeling, analysis and the solution of these inverse problems in these different settings by fitting simulated data is the main focus.
Finally, numerical examples are provided.
Solution of the fractional order differential equation for Laplace transform of a boundary functional of a semi-Markov process using inverse Laplace transform Institute of Control Systems of the Ministry of Science and Education of the Republic of Azerbaijan, Azerbaijan
Let $\left\{\xi _{k} \right\}_{k=1}^{\infty } ,$ and $\left\{\zeta _{k} \right\}_{k=1}^{\infty } $ be two independent sequences of random variables defined on any probability space $(\Omega ,\, F,P)$, such that the random variables in each sequence are independent, positive and identically distributed. Now we can construct the stochastic process $X_{1} \left(t\right)$ as follows
$X_{1} \left(t\right)=z-t+\sum _{i=0}^{k-1}\zeta _{i} $, if $\sum _{i=0}^{k-1}\xi _{i} \le t<\sum _{i=0}^{k}\xi _{i} $ ,
where $\xi _{0} =\zeta _{0} =0$. The process $X_{1} \left(t\right)$ is called the semi-Markov random walk process with negative drift, positive jumps. Let this process is delayed by a barrier zero:
\[X(t)=X_{1} \left(t\right)-\mathop{\inf }\limits_{0\le s\le t} \left\{0,X_{1} (s)\right\}\]
Now, we introduce the random variable $\tau _{0} =\inf \left\{t:\, \, X(t)=0\right\}$. We set $\tau _{0} =\infty $ if $X(t)>0$for all $t$. It is obvious that the random variable $\tau _{0} $ is the time of the first crossing of the process $X(t)$ into the delaying barrier at zero level. $\tau _{0} $ is called the boundary functional of the semi-Markov random walk process with negative drift, positive jumps.
The aim of the present work is to determine the Laplace transform of the conditional distribution of the random variable $\tau _{0} $. Laplace transform of the conditional distribution of the random variable $\tau _{0} $. by
\[L(\theta \left|z\right. )=E\left[e^{-\theta \tau _{0} } \left|X(0)=z\right. \right],\, \, \, \, \theta >0,\, \, z\ge 0.\]
Let us denote the conditional distribution of random variable of $\tau _{0} $ and the Laplace transform of the conditional distribution with
\[N(t\left|z\right. )=P\left[\tau _{0} >t\left|X(0)=z\right. \right],\]
and
\[\tilde{N}(\theta \left|z\right. )=\int _{t=0}^{\infty }e^{-\theta t} N(t\left|z\right. )dt ,\]
respectively.
Thus, we can easily obtain that
\[\tilde{N}(\theta \left|z\right. )=\frac{1-L(\theta \left|z\right. )}{\theta } \]
or, equivalently,
\[L(\theta \left|z\right. )=1-\theta \tilde{N}(\theta \left|z\right. ).\]
We construct an integral equation for the $\tilde{N}(\theta \left|z\right. )$. In particular, constructed integral equation reduced to the fractional order differential equation in the class of gamma distributions. The fractional derivatives are described in the Riemann-Liouville sense. Finally , Laplace transform of $\tilde{N}(\theta \left|z\right. )$ is obtained in the form of a threefold sum.
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| 4:00pm - 6:00pm | CT09: Contributed talks Location: VG2.107 Session Chair: Tram Nguyen |
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Diffraction Tomography: Elastic parameters reconstructions 1Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria; 2Faculty of Mathematics, University of Vienna, Austria; 3Christian Doppler Laboratory for Mathematical Modeling and Simulation of Next Generations of Ultrasound Devices (MaMSi), Institute of Mathematics, Austria
In this talk, we introduce an elastic imaging method where elastic properties (i.e. mass density and Lamé parameters) of a weakly scatterer are reconstructed from full-field data of scattered waves. We linearise the inverse scattering problem under consideration using Born's, Rytov's or Kirchhoff's approximation. Primarily, one appeal to the Fourier diffraction theorem developed in our previous work [1] for the pressure-pressure mode (i.e. generating Pressure incident plane waves and measuring the Pressure part of the scattered data). Then, we reconstruct the inverse Fourier transform of the pressure-pressure scattering potential using the inverse $\textit{nonequispaced discrete Fourier transform}$ for 2D transmission acquisition experiments. Finally, we quantify the elastic parameter distributions with different plane wave excitations.
[1] B. Mejri, O. Scherzer. A new inversion scheme for elastic diffraction tomography. arXiv:2212.02798, 2022.
Photoacoustic and Ultrasonic Tomography for Breast Imaging Centrum Wiskunde & Informatica, Computational Imaging, Netherlands
New high-resolution, three-dimensional imaging techniques are being developed that probe the breast without delivering harmful radiation. In particular, photoacoustic tomography (PAT) and ultrasound tomography (UST) promise to give access to high-quality images of tissue parameters with important diagnostic value. However, the involved inverse problems are very challenging from an experimental, mathematical and computational perspective. In this talk, we want to give an overview of these challenges and illustrate them with data from an ongoing clinical feasibility study that uses a prototype scanner for combined PAT and UST.
One-step estimation of spectral optical parameters in quantitative photoacoustic tomography University of Eastern Finland, Finland
In quantitative photoacoustic tomography, information about a target tissue is obtained by estimating its optical parameters. In this work, we propose a one-step methodology for estimating spectral optical parameters directly from photoacoustic time-series data. This is carried out by representing the optical parameters with their spectral models and by combining the models of light and ultrasound propagation. The inverse problem is approached in the framework of Bayesian inverse problems. Concentrations of four chromophores, two scattering related parameters, and the Grüneisen parameter are estimated simultaneously. The methodology is evaluated using numerical simulations.
Stable reconstruction of anisotropic conductivity in magneto-acoustic tomography with magnetic induction University of Limerick, Ireland
We study the issues of stability and reconstruction of the anisotropic conductivity $\sigma$ of a biological medium $\Omega\subset\mathbb{R}^3$ by the hybrid inverse problem of Magneto-Acoustic Tomography with Magnetic Induction (MAT-MI). More specifically, we consider a class of anisotropic conductivities given by the symmetric and uniformly positive definite matrix-valued functions $A(x,\gamma(x))$, $x\in\Omega$, where the one-parameter family $t\mapsto A(x, t)$, $t\in[\lambda^{-1}, \lambda]$, is assumed to be $\textit{a-priori}$ known. Under suitable conditions that include $A(\cdot, \gamma(\cdot))\in C^{1,\beta}(\Omega)$, with $0<\beta\leq 1$, we obtain a Lipschitz type stability estimate of the scalar function $\gamma$ in the $L^2(\Omega)$ norm in terms of an internal functional that can be physically measured in the MAT-MI experiment. We demonstrate the effectiveness of our theoretical framework in several numerical experiments, where $\gamma$ is reconstructed in terms of the internal functional. Our result extends previous results in MAT-MI where the conductivity $\sigma$ was either isotropic or of the simpler anisotropic form $\gamma D$, with $D$ an $\textit{a priori}$ known matrix-valued function in $\Omega$. In particular, the more general type of anisotropic conductivity considered here allows for the anisotropic structure to depend non-linearly on the unknown scalar parameter $\gamma$ to be reconstructed. This is joint work with Romina Gaburro, Shari Moskow and Isaac Woods
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