11th Applied Inverse Problems Conference
September 4 - 8, 2023 | Göttingen, Germany
Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
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Session Overview | |
| Location: VG3.101 |
| Date: Monday, 04/Sept/2023 | |
| 1:30pm - 3:30pm | MS55 1: Nonlinear Inverse Scattering and Related Topics Location: VG3.101 Session Chair: Yang Yang |
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An Inverse Problem for Nonlinear Time-dependent Schrodinger Equations with Partial Data 浙江大学, China, People's Republic of
In this talk, I will present some recent results on solving inverse boundary value problems for nonlinear PDEs, especially for a time-dependent Schrodinger equation with time-dependent potentials with partial boundary Dirichlet-to-Neumann map. After a higher order linearization step, the problem will be reduced to implementing special geometrical optics (GO) solutions to prove the uniqueness and stability of the reconstruction. This is a joint work with my PhD student Xuezhu Lu and Prof. Ru-Yu Lai.
Supercomputing-based inverse modeling of high-resolution atmospheric contaminant source intensity distribution using remote sensing data 1School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou 510006, China; 2Guangdong Province Key Laboratory of Computational Science, Guangzhou 510006, China; 3JülichSupercomputing Centre, Forschungszentrum Jülich, Jülich 52425, Germany; 4Numerical Mathematics, RWTH Aachen University, Aachen 52056, Germany
Atmospheric pollution prevention and control is an important global issue[1]. Observing the emission of harmful trace gases and their atmospheric transport dynamics on a global scale is of great significance for in-depth study of major problems, such as climate change and ecological and environmental change[2, 3]. In recent years, the inverse problem of atmospheric contaminant source intensity distribution has attracted more and more attention from researchers[4]. Due to its mathematical ill-posedness and high computational costs, it is necessary to develop new computational tools[5]. Accurate, rapid, and stable inverse analysis of atmospheric contaminant source intensity distribution, and subsequently using high-resolution numerical simulation methods to predict the local or large-scale, short-term or long-term atmospheric environmental impacts caused by major sudden natural disasters and industrial pollution events, has important scientific significance and practical value.
By using limited information of satellite observation data obtained through NIS (Nonlinear Inverse Scattering) technology[7, 8, 9], we have established a high-throughput parallel computing framework for solving the mathematical and physical inverse problem of high-resolution spatiotemporal atmospheric contaminant sources distribution[10, 11, 12]. A real-time inverse analysis and transport simulation of atmospheric contaminant source intensity distribution with high resolution, stability, and reliability are realized. And thus the high-resolution reanalysis data and prediction information on a global scale are available, which can not be directly obtained by current satellite or optical radar measurement technologies. Considering the influence of complex physical and chemical processes, such as the transport of particles in wind fields and the scattering of particles due to light irradiation, the relationship between unknown source parameters and observed contaminant concentrations is usually nonlinear[2, 5, 7, 13, 14, 15]. Therefore, we comprehensively use numerical simulation, optimization methods, and statistical inference techniques[6, 16, 17]. Taking volcanic eruption and forest fire as examples, based on remote sensing data, we use the jointly developed Lagrangian transport model MPTRAC (Massive Parallel Trajectory Calculation) for forward simulation[11]. Combined with heuristic methods such as segmented strong constraint "product rule" proposed by us, the computational bottleneck of traditional serial regularization methods for solving such inverse problems is overcome[10, 12]. And a million-core supercomputing-based inverse calculation strategy is developed, which greatly reduces time costs while ensuring accuracy and reliability, meeting the needs of future real-time prediction tools.
This study provides practical application scenarios for NIS technology, and plays an important theoretical and practical role in ensuring aviation safety, exploring the mechanism of pollutant degradation, and revealing the causes of global climate change.
[1] N.-N. Zhang et al. Spatiotemporal trends in PM2. 5 levels from 2013 to 2017 and regional demarcations for joint prevention and control of atmospheric pollution in China, Chemosphere 210: 1176-1184, 2018.
[2] S. Huang et al. Inverse problems in atmospheric science and their application, Journal of Physics: Conference Series, IOP Publishing, 2005.
[3] W. Freeden et al. Handbook of geomathematics, 2nd edition. Springer Berlin Heidelberg, 2015.
[4] M. S. Zhdanov. Inverse Theory and Applications in Geophysics, 2nd edition. Elsevier Science, 2015.
[5] J. L. Mueller, S. Siltanen. Linear and nonlinear inverse problems with practical applications, SIAM, 2012.
[6] Y. Bai et al. Computational methods for applied inverse problems, Walter de Gruyter, 2012.
[7] D. Efremenko, A. Kokhanovsky. Foundations of Atmospheric Remote Sensing, Springer, 2021.
[8] W. C. Chew et al. Nonlinear diffraction tomography: The use of inverse scattering for imaging, Int J Imaging Syst Technol. 7(1):16-24, 1996.
[9] T. Hasegawa, T. Iwasaki. Microwave imaging by quasi-inverse scattering, Electron Comm Jpn Pt I. 87(5):52-61, 2004.
[10] Y. Heng et al. Inverse transport modeling of volcanic sulfur dioxide emissions using large-scale simulations, Geoscientific Model Development 9(4): 1627-1645, 2016.
[11] L. Hoffmann et al. Massive-Parallel Trajectory Calculations version 2.2 (MPTRAC-2.2): Lagrangian transport simulations on graphics processing units (GPUs), Geoscientific Model Development 15(7): 2731-2762, 2022.
[12] M. Liu et al. High-Resolution Source Estimation of Volcanic Sulfur Dioxide Emissions Using Large-Scale Transport Simulations, Computational Science – ICCS 2020: 60-73, 2020.
[13] K. Chadan et al. An introduction to inverse scattering and inverse spectral problems, Society for Industrial and Applied Mathematics, 1997.
[14] D. P. Winebrenner, J. Sylvester. Linear and nonlinear inverse scattering, SIAM Journal on Applied Mathematics, 59(2): 669-699, 1998.
[15] V. Isakov. Inverse problems for partial differential equations, Springer, 2006.
[16] E. Haber et al. On optimization techniques for solving nonlinear inverse problems, Inverse problems, 6(5): 1263, 2000.
[17] R. C. Aster et al. Parameter estimation and inverse problems, Elsevier, 2018.
Some progresses in Carleman estimates and their applications in inverse problems for stochastic partial differential equations University of Electronic Science and Technology of China, P. R. China
This talk studies Carleman estimates and their applications for inverse problems of stochastic partial differential equations. By establishing new Carleman estimates for the stochastic parabolic and hyperbolic systems, conditional stability for inverse problems of these systems are proven. Based on the idea of Tikhonov method, regularized solutions are proposed. The analysis of the existence and uniqueness of the regularized solutions, and proofs for error estimate under an a priori assumption are presented. Numerical verification of the regularization, based on the idea of kernel- based learning method, including numerical algorithms and examples are also illustrated.
Inverse scattering problems with incomplete data Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China, People's Republic of
Inverse scattering problems aim to determine unknown scatterers with wave fields measured around the scatterers. However, from the practical point of views, we have only limited information, e.g., limited aperture data phaseless data and sparse data. In this talk, we introduce some data retrieval techniques and the applications in the inverse scattering problems. The theoretical and numerical methods for inverse scattering problems with multi-frequency spase measurements will also be mentioned.
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| 4:00pm - 6:00pm | MS55 2: Nonlinear Inverse Scattering and Related Topics Location: VG3.101 Session Chair: Yang Yang |
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Imaging with two-photon absorption optics Auburn University, United States of America
In this talk, we briefly talk about the inverse boundary/medium problems with the semilinear transport model which naturally rises from two-photon absorption optics. The model can be formally derived from a paraxial setting of a nonlinear absorption wave model. For the related inverse problems, we consider two cases. For the inverse boundary problem, we adopted the usual linearization technique and prove the uniqueness of reconstruction. For the inverse medium problems, we consider the problem from photoacoustic imaging specifically in static and time-dependent settings and prove the uniqueness of the reconstruction for the absorption coefficients.
Hopf lemma for fractional diffusion equations and application to inverse problem Ningbo University, China, People's Republic of
In this talk, we will discuss an inverse problem of determining the spatially dependent source term and the Robin boundary coefficient in a time-fractional diffusion equation, with the aid of extra measurement data at a subdomain near the accessible boundary. Firstly, the spatially varying source is uniquely determined in view of the unique continuation principle and Duhamel principle for the fractional diffusion equation. The Hopf lemma for a homogeneous time-fractional diffusion equation is proved and then used to prove the uniqueness of recovering the Robin boundary coefficient. Numerically, based on the theoretical uniqueness result, we apply the classical Tikhonov regularization method to transform the inverse problem into a minimization problem, which is solved by an iterative thresholding algorithm. Finally, several numerical examples are presented to show the accuracy and effectiveness of the proposed algorithm.
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| Date: Tuesday, 05/Sept/2023 | |
| 1:30pm - 3:30pm | MS47 1: Scattering and spectral imaging: inverse problems and algorithms Location: VG3.101 Session Chair: Eric Todd Quinto Session Chair: Gael Rigaud |
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Microlocal properties and injectivity for Ellipsoidal and hyperbolic Radon transforms 1Tufts University, United States of America; 2University of Manchester, England; 3Brigham and Women's Hospital, United States of America
We present novel microlocal results for generalized ellipsoid and hyperboloid Radon transforms in Euclidean Space and we apply our results to Ultrasound Reflection Tomography (URT). We introduce a new Radon transform, $R$, which integrates compactly supported distributions over ellipsoids and hyperboloids with centers on a smooth nypersurface, $S$ in $\mathbb{R}^n$. $R$ is shown to be a Fourier Integral Operator and in our main theorem we prove that $R$ satisfies the Bolker condition if and only if the support of the function is in a connected set that is not intersected by any plane tangent to $S$. In this case, backprojection type reconstruction operators such as the normal operator $R^* R$ do not add artifacts to the reconstruction.
We apply our results to a cylindrical geometry that could be used in URT. We prove injectivity results and investigate the visible singularities in this modality. In addition, we present reconstructions of image phantoms in two dimensions that illustrate our microlocal theory.
Motion detection in diffraction tomography 1TU Berlin, Germany; 2University of Vienna, Austria; 3RICAM, Linz, Austria; 4Christian Doppler Laboratory MaMSi, Vienna, Austria
We study the mathematical imaging problem of optical diffraction tomography (ODT) for the scenario of a rigid particle rotating in a trap created by acoustic or optical forces. Under the influence of the inhomogeneous forces, the particle carries out a time-dependent smooth, but irregular motion. The rotation axis is not fixed, but continuously undergoes some variations, and the rotation angles are not equally spaced, which is in contrast to standard tomographic reconstruction assumptions. Once the time-dependent motion parameters are known, the particle’s scattering potential can be reconstructed based on the Fourier diffraction theorem, considering it is compatible with making the first order Born or Rytov approximation.
The aim of this presentation is twofold: We first need to detect the motion parameters from the tomographic data by detecting common circles in the Fourier-transformed data. This can be seen as analogue to method of common lines from cryogenic electron microscopy (cryo-EM), which is based on the assumption that the assumption that the light travels along straight lines. Then we can reconstruct the scattering potential of the object utilizing non-uniform Fourier methods.
[1] M. Quellmalz, P. Elbau, O. Scherzer, G. Steidl. Motion Detection in Diffraction Tomography by Common Circle Methods 2022.
https://arxiv.org/abs/2209.08086
Deep learning to tackle model inexactness and motion in Compton Scattering Tomography University of Stuttgart, Germany
Modelling the Compton scattering effect leads to many challenges such as non-linearity of the forward model, multiple scattering and high level of noise for moving targets. While the non-linearity is addressed by a necessary linear approximation of the first-order scattering with respect to the sought-for electron density, the multiple-order scattering stands for a substantial and unavoidable part of the spectral data which is difficult to handle due to highly complex forward models. However, the smoothness properties of the operators modelling the different scattering orders suggests that differential operators can be used to reduce the level of multiple scattering. Last but not least, the stochastic nature of the Compton effect may involve a large measurement noise, in particular when the object under study is subject to motion, and therefore time must be taken into account. To tackle these different issues, we discuss in this talk a Bayesian method based on the generalized Golub-Kahan bidiagonalization and explore the possibilities to mimic and improve the stochastic approach with deep neural networks.
Diffusion based regularization for multi-energy CT with limited data Universität Stuttgart, Germany
As shown by the rise of data-driven and learning techniques, the use of specific
features in datasets is essential to build satisfactory solutions to ill-posed
inverse problems suffering limitations, sparsity and large level of noise.
The well-known total-variation regularization has become a standard approach due
to producing good results without any a priori knowledge.
Providing additional information, it is possible to improve the reconstruction
using forward/backward diffusions. An example is the so-called Perona-Malik
functional which is based on a priori on the global contrast.
Such a construction of regularizers is particularly relevant with machine
learning techniques in which a database can provide natural features and
informations such as contrast and edges.
We propose to study this approach and to validate its potential on multi-energy
CT (computerized tomography) subject to limitations such as sparsitiy and
limited angles.
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| 4:00pm - 6:00pm | MS47 2: Scattering and spectral imaging: inverse problems and algorithms Location: VG3.101 Session Chair: Eric Todd Quinto Session Chair: Gael Rigaud |
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V-line tensor tomography 1University of Texas at Arlington, United States of America; 2Dartmouth College, United States of America; 3Indian Institute of Technology Gandhinagar, India
The V-line transform (VLT) maps a function to its integrals along V-shaped trajectories with a vertex inside the support of the function. This transform and its various generalizations appear in mathematical models of several imaging techniques utilizing scattered particles. The talk presents recent results on inversion of generalized VLTs defined on vector fields and symmetric 2-tensor fields in the plane.
Optimal parameter design for spectral CT 1NC State University, United States of America; 2University of Chicago, USA
Spectral CT is an x-ray transmission imaging technique that uses the energy dependence of x-ray photon attenuation to determine elemental composition of an object of interest. Mathematically, forward spectral CT measurements are modeled by a nonlinear integral transform for which no analytical inversion is available. In this talk, I will present some of our recent results on the global uniqueness and the stability of spectral CT reconstructions. These analyses are useful for designing optimal scan parameters, which will be demonstrated using numerical simulations. This is joint work with G. Bal and E. Sidky.
Gamma ray imaging with bidirectional Compton cameras 1FAU Erlangen-Nürnberg; 2Deutsches Elektronen-Synchrotron DESY; 3Universität Hamburg
For in-situ gamma ray spectrometry, Compton cameras are an efficient imaging tool that operate without collimation and therefore attain large sensitivities. Conventionally, Compton cameras are built with separated scattering and absorbing layers. This setup allows detector materials to be tailored to maximize sensitivity and have good energetic or spatial resolution, but often sacrifices the camera's ability to produce spatially resolved images in the whole $4\pi$ field of view resulting in a de facto collimation. We propose the mathematical model of a Compton camera whose detectors are all considered to both scatter and absorb the incoming gamma rays. Since the measurements of the camera do not give any information about the direction of a coincidence of scattering and absorption, we talk of a bidirectional Compton camera. The additional uncertainty is reflected in the operator describing the forward model, which is the weighted sum of two conical Radon transforms. We demonstrate the ability of the system to efficiently image gamma radiation by numerical results on simulated and measured data.
A hybrid algorithm for material decomposition in multi-energy CT Universität Innsbruck, Austria
The aim of multi-energy CT is to reconstruct the distribution of a known set of substances inside a sample by performing CT measurements at different energies. The measurements can be achieved either by using different tube voltages at the source or by means of energy sensitive detectors (e.g. photon counting detectors). In any case the energy dependent absorption of the materials under consideration is used to distinguish the substances in the sample which leads to a nonlinear reconstruction problem. The majority of reconstruction algorithms can be divided into those performing the material decomposition in the sinogram domain and those decomposing the image after inversion of the Radon transform for each energy bin. Both types of algorithms can be implemented very efficiently but also suffer from specific artefacts. More recently one-step algorithms performing decomposition and inversion in one pass have become an active research area. While they eliminate most of the problems of two-step approaches, they are usually computationally costy because they are iterative in nature and the relative similarity of absorption coefficients often leads to poor convergence. We present a method that combines preconditioning in the sinogram domain and an efficient numerical method for the nonlinear problem with a simple and thus fast iteration for the linear part of the problem. Our hybrid method does not suffer from systematic problems like beam hardening or difficulties with not perfectly aligned images for different energy bins. It is iterative but convergence is fast and the computational cost of each iteration is modest.
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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | MS47 3: Scattering and spectral imaging: inverse problems and algorithms Location: VG3.101 Session Chair: Eric Todd Quinto Session Chair: Gael Rigaud |
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Analytic and Deep learning-based Inversions in Circular Compton Scattering Tomography 1CY Cergy Paris University, France; 2Maxwell Institute for Mathematical Sciences, Bayes Center, University of Edinburgh, Edinburgh, EH8 9BT, United Kingdom and the School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom; 3Centro de Matema ́tica Aplicada, Universidad Nacional de San Mart ́ın, Buenos Aires, Argentina; 4CY Cergy Paris University, France
Circular Compton scattering tomography (CCST) where a fixed radiation source and a number of regularly spaced detectors are positioned on a fixed circular frame is recently proposed [1]. It has multiple advantages such as compact and motion-free system, possibility of combination with classic fan-beam CT (computed tomography) as a bi-imaging system, capacity of scanning both small and large objects.
In the case where the detectors are collimated to split up scattered photons coming from two opposite sides of the source-detector segment, the modelling of CCST’s data acquisition leads to a Radon transform on a family of arcs of circles passing through a fixed point (the point source). The analytical inversion of this Radon transform is derived from Cormack’s earlier works.
In the case of non-collimated detectors, the corresponding Radon transform is defined on a specific family of double circular arcs and named DCART (double circular arc Radon transform). The exact inverse formula for this new Radon transform on pair of circles is not available presently. Recently, deep learning-based techniques appear as promising alternatives to solve the ill-posed inverse problems in CT reconstruction from limited-angle and sparse-view projection data. In our work we propose a neural network architecture acting on both image and data domains. The particularity of this architecture lies in its capability to map the projection (Radon domain) to image domain at different scales of the data while extracting important image features used at reconstruction. The obtained results suggest that removing the collimator at detectors in CCST is feasible thanks to deep learning-based techniques.
Another way to by-pass the collimator at detectors is to design a CST with a single detector rotating around a fixed source. The corresponding Radon transform and its inverse are established in [2,3] but the CST is no longer a motion-free system.
[1] C. Tarpau, J. Cebeiro, M. A. Morvidone, M. K. Nguyen. A new concept of Compton scattering tomography and the development of the corresponding circular Radon transform, IEEE Transactions on Radiation and Plasma Medical Sciences (IEEE-TRPMS) 4.4: 433-440, 2020.
https://doi.org/10.1109/TRPMS.2019.2943555
[2] C. Tarpau, J. Cebeiro, M. K. Nguyen, G. Rollet, M. A. Morvidone. Analytic inversion of a Radon transform on double circular arcs with applications in Compton Scattering Tomography, IEEE Transactions on Computational Imaging (IEEE-TCI) 6: 958-967, 2020.
https://doi.org/10.1109/TCI.2020.2999672
[3] J. Cebeiro, C. Tarpau, M. A. Morvidone, D. Rubio, M. K. Nguyen, On a three-dimensional Compton scattering tomography system with fixed source, Inverse Problems, Special issue on Modern Challenges in Imaging 37: 054001, 2021.
https://doi.org/10.1088/1361-6420/abf0f0.
On a cylindrical scanning modality in three-dimensional Compton scatter tomography Brigham and Women's Hospital, United States of America
We present injectivity and microlocal analyses of a new generalized Radon transform, R, which has applications to a novel scanner design in 3-D Compton Scattering Tomography (CST), which we also introduce here. Using Fourier decomposition and Volterra equation theory, we prove that R is injective and show that the image solution is unique. Using microlocal analysis, we prove that R satisfies the Bolker condition (sometimes called the “Bolker assumption”), and we investigate the edge detection capabilities of R. This has important implications regarding the stability of inversion and the amplification of measurement noise. This paper provides the theoretical groundwork for 3-D CST using the proposed scanner design.
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | MS36 1: Advances in limited-data X-ray tomography Location: VG3.101 Session Chair: Jakob Sauer Jørgensen Session Chair: Samuli Siltanen |
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Learned proximal operators meets unrolling: a deeply learned regularisation for limited angle tomography University of Bath, United Kingdom
In recent years, limited angle CT has become a challenging testing ground for several theoretical and numerical studies, where both variational regularisation and data-driven techniques have been investigated extensively. In this talk, I will present a hybrid reconstruction framework where the proximal operator of an accelerated unrolled scheme is learned to ensure suitable theoretical guarantees. The recipe relays on the interplay between sparse regularization theory, harmonic analysis, microlocal analysis and Plug and Play methods.
The numerical results show that these approaches significantly surpasses both pure model- and more data-based reconstruction methods.
A new variational appraoch for limited data reconstructions in x-ray tomography OTH Regensburg, Germany
It is well known that image reconstructions from limited tomographic data often suffer from significant artifacts and missing features. To remove these artifacts and to compensate for the missing information, reconstruction methods have to incorporate additional information about the objects of interest. An important example of such methods is TV reconstruction. It is well known that this technique can efficiently compensate for missing information and reduce reconstruction artifacts. At the same time, however, tomographic data are also contaminated by noise, which poses an additional challenge. The use of a single penalty term (regularizer) within a variational regularization framework must therefore account for both the missing data and the noise. However, it is known that a single regularizer does not work perfectly for both tasks. In this talk, we introduce a new variational formulation that combines the advantages of two different regularizers, one aimed at accurate reconstruction in the presence of noise and the other aimed at selecting a solution with reduced artifacts. Both reconstructions are linked by a data consistency condition that makes them close to each other in the data domain. We demonstrate the proposed method for the limited angle CT problem using a combined curvelet and TV approach.
Material Decomposition Techniques for Spectral Computed Tomography 1University of Bologna, Italy; 2Technical University of Denmark, Denmark; 3University of Naples Federico II, Italy
Spectral computed tomography is an evolving technique which exploits the property of materials
to attenuate X-rays in different ways depending on the specific energy. Compared to conventional CT, spectral CT employs a photon-counting detector that records the energy of individual
photons and produce a fine grid of discrete energy-dependent data. In this way it is easier to
distinguish materials that have similar attenuation coefficients in an energy range, but different
in others. The material decomposition process allows to not only reconstruct the object, but
also to estimate the concentration of the materials that compose it.
Different strategies to reconstruct material-specific images have been developed in the last years,
but many improvements have yet to be made especially for low-dose cases and few projections.
This setup is justified by the slowness and flux limit of the high energy resolution photon counting detectors, but leads to noisier data, especially across the energy channels, and less spatial
information. The study of the noise distribution, together with the usage of suitable regularizers
and the selection of their parameters become crucial to obtain a good quality reconstruction and
material decomposition. The talk will address all these issues by focusing on the case study of
materials that have high atomic number with similar attenuation coefficients and K-edges in the
considered energy range.
Bayesian approach to limited-data CT reconstruction for inspection of subsea pipes Technical University of Denmark
In subsea pipe inspection using X-ray computed tomography (CT), obtaining data is time-consuming and costly due to the challenging underwater conditions. We propose an efficient Bayesian CT reconstruction method with a new class of structural Gaussian priors incorporating known material properties to enhance quality from limited data. Experiments with real and synthetic data demonstrate artifact reduction, increased contrast, and enhanced reconstruction certainty compared to conventional reconstruction methods.
Authhors:
Silja L. Christensen, Nicolai A. B. Riis, Felipe Uribe and Jakob S. Jørgensen
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| 4:00pm - 6:00pm | MS36 2: Advances in limited-data X-ray tomography Location: VG3.101 Session Chair: Jakob Sauer Jørgensen Session Chair: Samuli Siltanen |
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Approaches to the Helsinki Tomography Challenge 2022 ZeTeM (Universität Bremen), Germany
In 2022, the Finnish Inverse Problems Society organized the Helsinki Tomography Challenge (HTC) 2022 with the aim of reconstructing an image using only limited-angle measurements. As part of this challenge, we implemented two methods, namely an Edge Inpainting method and a Learned Primal-Dual (LPD) reconstruction. The Edge Inpainting method consists of several successive steps: A classical reconstruction using Perona-Malik, extraction of visible edges, inpainting of invisible edges using a U-Net and a final segmentation using a U-Net. The Learned Primal-Dual approach adapts the classical LPD in two ways, namely replacing the adjoint with the generalized inverse (FBP) and using large U-Nets in the primal update. For the training of the networks we generated a synthetic dataset since only five samples were provided in the challenge. The results of the challenge showed that the Edge Inpainting Method was competitive for a viewing range up to 70 degrees. In contrast, the Learned Primal Dual approach performed well on all viewing ranges of the challenge and scored second best.
Directional regularization with the Core Imaging Library for limited-angle CT in the Helsinki Tomography Challenge 2022 1Science and Technology Facilities Council, United Kingdom; 2Technical University of Denmark; 3Finden Ltd
The Core Imaging Library (CIL) is a software for Computed Tomography (CT) and other inverse problems. It provides processing algorithms for CT data and tools to write optimisation problems with near math syntax. Last year the Finnish Inverse Problems Society organized the “Helsinki Tomography Challenge 2022” (HTC2022) – an open competition for researchers to submit reconstruction algorithms for a challenging series of real-data limited-angle computed tomography problems. The HTC2022 provided the perfect grounds to test the capabilities of CIL in limited angle CT.
The algorithm we submitted consists of multiple stages: first, pre-processing including beam-hardening correction and data normalization; second a purpose-built directional regularization method exploiting prior knowledge of the scanned object; and finally, a multi-Otsu segmentation method. The algorithm was fully implemented using the optimization prototyping capabilities of CIL and its performance assessed and optimized on the provided training data ahead of submission. The algorithm performed well on limited-angle data down to an angular range of 50 degrees, and in the competition was beaten only by two machine learning based strategies involving generation of very large sets of synthetic training data.
In the spirit of open science, all the data sets are available from the challenge website, https://fips.fi/HTC2022.php, and the submitted algorithm code from https://github.com/TomographicImaging/CIL-HTC2022-Algo2.
VAEs with structured image covariance as priors to inverse imaging problems STFC - UKRI, Scientific Computing, UK
This talk explores how generative models, trained on ground-truth images, can be used as priors for inverse problems, penalizing reconstructions far from images the generator can produce. We utilize variational autoencoders that generate not only an image but also a covariance uncertainty matrix for each image. The covariance can model changing uncertainty dependencies caused by structure in the image, such as edges or objects, and provides a new distance metric from the manifold of learned images. We evaluate these novel generative regularizers on retrospectively sub-sampled real-valued MRI measurements from the fastMRI dataset.
Authors:
Margaret A G Duff (Science and Technology Facilities Council (UKRI))
Ivor J A Simpson (University of Sussex)
Matthias J Ehrhardt (University of Bath)
Neill D F Campbell (University of Bath)
Limited-Angle Tomography Reconstruction via Deep Learning on Synthetic Data 1Heinrich Heine University Düsseldorf, Germany; 2Technical University of Dortmund, Germany
Computed tomography (CT) has become an essential part of modern science. A CT scanner consists of an X-ray source that is spun around an object of interest. On the opposite end of the X-ray source a detector captures X-rays that are not absorbed by the object. The reconstruction of an image is a linear inverse problem which is usually solved by the filtered back projection algorithm. However, when the number of measurements is too small the reconstruction problem is highly ill-posed. This is for example the case when the X-ray source is not spun completely around the object, but rather irradiates the object only from a limited angle. To tackle this problem, we present a deep neural network that performs limited-angle tomography reconstruction. The model is trained on a large amount of carefully-crafted synthetic data. Our approach won the first place in the Helsinki Tomography Challenge 2022.
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| Date: Friday, 08/Sept/2023 | |
| 1:30pm - 3:30pm | MS41 1: Geomathematics Location: VG3.101 Session Chair: Joonas Ilmavirta |
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Geodesic X-ray tomography on manifolds of low regularity University of Jyväskylä, Finland
Geodesic X-ray tomography arises in geomathematics as the linearized travel time problem of planets. Planets have non-smooth geometry as the sound speed is generally non-smooth, can have jump discontinuities and other extreme behavior. In this talk we consider the question: How non-smooth can Riemannian geometry be for the X-ray transform of scalar functions (and tensor fields) to remain injective? We prove that the X-ray transform is (solenoidally) injective on Lipschitz functions (tensor fields) when the Riemannian geometry is simple $C^{1,1}$. The class $C^{1,1}$ is the natural lower bound on regularity to have a well-defined X-ray transform. Our proofs are based on energy estimates derived from a Pestov identity, which lives on the non-smooth unit sphere bundle of the manifold. The talk is based on joint work with Joonas Ilmavirta.
Invariance of the elastic wave equation in the context of Finsler geometry University of Jyväskylä, Finland
In this talk we address the Euclidean elastic wave equation under change of variables and extend this to Riemannian geometry. This is inspired by previous research that has concerned the principal behavior of the Euclidean elastic wave equation under coordinate transformations. Further research has concerned how the density normalized stiffness tensor gives rise to a Finsler metric. With this in mind we will touch upon what one can say about the density and stiffness tensor fields that give rise to the same Finsler metric. In this context we will talk about how this will affect the full elastic wave equation and not only the principal behavior.
Geometrization of inverse problems in seismology University of Jyväskylä, Finland
Seismic waves can be modeled by the elastic wave equation, which has two material parameters: the stiffness tensor and the density. The inverse problem is to reconstruct these two fields from boundary data, and the stiffness tensor can be anisotropic. I will discuss how this problem can be tackled by geometric methods and how that leads to geometric inverse problems in Finsler geometry. This talk is related several other talks in the same minisymposium.
Reconstruction of anisotropic stiffness tensors using algebraic geometry 1Rice University, United States of America; 2University of Jyväskylä, Finland; 3University of Helsinki
Stiffness tensors serve as a fingerprint of a material. We describe how to harness anisotropy, using standard tools from algebraic geometry (e.g., generic geometric integrality, upper-semicontinuity of some standard functions, and Gröbner bases) to uniquely reconstruct the stiffness tensor of a general anisotropic material from an analytically small neighborhood of its corresponding slowness surface.
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| 4:00pm - 6:00pm | MS41 2: Geomathematics Location: VG3.101 Session Chair: Joonas Ilmavirta |
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Inverse scattering: Regularized Lanczos method for the Lippmann-Schwinger equation 1University of Utah, U.S.A.; 2Worcester Polytechnic Institute, U.S.A.; 3Drexel University, U.S.A.; 4Southern Methodist University, U.S.A.
Inverse scattering techniques have broad applicability in geophysics, medical imaging,
and remote sensing. This talk presents a robust direct reduced-order model method for solving inverse scattering problems. The
approach is based on a Lippmann-Schwinger-Lanczos (LSL) algorithm in the frequency domain with two levels of regularization. Numerical experiments for Helmholtz and
Schrödinger problems show that the proposed regularization
scheme significantly improves the performance of the LSL algorithm, allowing
for good reconstructions with noisy data.
Travel Time Tomography in Transversely Isotropic Elasticity via Microlocal Analysis Northwestern University
We will discuss recent results of the author regarding the travel time tomography problem in the context of transversely isotropic elasticity. The works build on previous works regarding X-ray and (elastic) travel time tomography and boundary rigidity problems studied by de Hoop, Stefanov, Uhlmann, Vasy, et al., which reduce the inverse problems to the microlocal analysis of certain operators obtained from a pseudolinearization argument. We will discuss the additional analytic complications in this situation, due to the degenerating ellipticity of certain key operators obtained in the pseudolinearization argument, as well as the machinery developed to tackle these additional complications.
An inverse source problem for the elasto-gravitational equations 1Università di Firenze, Italy; 2Rice University, USA
We study an inverse source problem for a system of elastic-gravitational equations, describing the oscillations of the Earth due to an earthquake.
The aim is to determine the seismic-moment tensor and the position of the point source by using only measurements of the disturbance in the gravity field induced by the earthquake, for an arbitrarily small time window.
The problem is inspired by the recently discovered speed-of-light prompt elasto-gravity signals (PEGS), which can prove beneficial for earthquake early warning systems (EEWS).
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