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: VG1.105 |
| Date: Monday, 04/Sept/2023 | |
| 1:30pm - 3:30pm | MS52 1: Integral geometry, rigidity and geometric inverse problems Location: VG1.105 Session Chair: Francois Sylvain Monard Session Chair: Plamen Stefanov |
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Inverse problem for Yang-Mills-Higgs fields 1University of Helsinki, Finland; 2Fudan University, China; 3University of Cambridge, UK
We show that the Yang-Mills potential and Higgs field are uniquely determined (up to the natural gauge) from source-to-solution type data associated with the classical Yang-Mills-Higgs equations in the Minkowski space. We impose natural non-degeneracy conditions on the representation for the Higgs field and on the Lie algebra of the structure group which are satisfied for the case of the Standard Model. Our approach exploits non-linear interaction of waves to recover a broken non-abelian light ray transform of the Yang-Mills field and a weighted integral transform of the Higgs field.
The Lorentzian scattering rigidity problem and rigidity of stationary metrics Purdue University, United States of America
We study scattering rigidity in Lorentzian geometry: recovery of a Lorentzian metric from the scattering relation $\mathcal{S}$ known on a lateral boundary. We show that, under a non-conjugacy assumption, every defining function $r(x,y)$ of the submanifold of pairs of boundary points which can be connected by a lightlike geodesic plays the role of the boundary distance function in the Riemannian case in the following sense. Its linearization is the light ray transform of tensor fields of order two which are the perturbations of the metric. Next, we study scattering rigidity of stationary metrics in time-space cylinders and show that it can be reduced to boundary rigidity of magnetic systems on the base; a problem studied previously. This implies several scattering rigidity results for stationary metrics.
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| 4:00pm - 6:00pm | MS52 2: Integral geometry, rigidity and geometric inverse problems Location: VG1.105 Session Chair: Francois Sylvain Monard Session Chair: Plamen Stefanov |
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Resonant forms at zero for dissipative Anosov flows 1University of Zurich, Switzerland; 2University of Cambridge, UK
The Ruelle Zeta Function of a chaotic (Anosov) flow is a meromorphic function in the complex plane defined as an infinite product over closed orbits. Its behaviour at zero is expected to carry interesting topological and dynamical information, and is encoded in certain resonant spaces of differential forms for the action of the Lie derivative on suitable spaces with anisotropic regularity. In this talk, I will introduce a new notion of helicity (average self-linking), and explain how this can be used to compute the resonant spaces for any Anosov flow in 3D, with particular emphasis in the dissipative (non volume-preserving) case. A prototype example is given by the geodesic flow of an affine connection with torsion and we shall see that for such a flow the order of vanishing drops by 1 in relation to the usual geodesic flow due to the Sinai-Ruelle-Bowen measure having non-zero winding cycle.
Ray transform problems arising from seismology University of Jyväskylä, Finland
Many different ray transform problems arise from seismology. My examples are periodic ray transform problems in the presence of interfaces, linearized travel time tomography in strong anisotropy, and a partial data problem originating from shear wave splitting. I will discuss the underlying inverse problems and the arising integral geometry problems. This talk is based on joint work with de Hoop, Katsnelson, and Mönkkönen.
X-ray mapping properties and degenerately elliptic operators 1Northwestern University; 2University of California, Santa Cruz; 3Indian Institute of Technology Gandhinagar
We discuss recent results regarding $C^\infty$-isomorphism properties of weighted normal operators of the X-ray transform on manifolds with boundary, in joint work [1] with Francois Monard and Rohit Kumar Mishra. The crux of the result depends on understanding the Singular Value Decomposition of weighted X-ray transforms/backprojection operators, which itself can be obtained via intertwining with certain degenerately elliptic differential operators. We also discuss recent work [2] with Francois Monard on developing tools to study such degenerately elliptic operators even further. Such tools include a scale of Sobolev spaces which take into account behavior up to the boundary, as well as generalizations of Dirichlet and Neumann traces called boundary triplets associated to degenerately elliptic operators which pick out the first and second most singular terms of a function near the boundary.
[1] R. Mishra, F. Monard, Y. Zou. The $C^\infty$-isomorphism property for a class of singularly-weighted X-ray transforms. Inverse Problems 39: 024001, 2023. https://doi.org/10.1088/1361-6420/aca8cb
[2] F. Monard, Y. Zou. Boundary triples for a family of degenerate elliptic operators of Keldysh type, arXiv: 2302.08133, 2023.
The range of the non-Abelian X-ray transform University of Bonn, Germany
We discuss a nonlinear analogue of the Pestov-Uhlmann range characterisation for geodesic X-ray transforms on simple surfaces. The transform under consideration takes as input matrix-valued and possibly direction dependent functions (which may encode magnetic fields or connections on a vector bundle) and outputs their 'scattering data' at the boundary. The range of this transform can be completely described in terms of boundary objects, and this description is reminiscent of the Ward correspondence for anti-self-dual Yang-Mills fields, but without solitonic degrees of freedom. The talk is based on joint work with Gabriel Paternain.
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| Date: Tuesday, 05/Sept/2023 | |
| 1:30pm - 3:30pm | MS52 3: Integral geometry, rigidity and geometric inverse problems Location: VG1.105 Session Chair: Francois Sylvain Monard Session Chair: Plamen Stefanov |
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Marked length spectrum rigidity for Anosov surfaces Sorbonne Université, France
On a closed Riemannian manifold, the marked length spectrum rigidity problem consists in recovering the metric from the knowledge of the lengths of its closed geodesics (marked by the free homotopy classes of the manifold). I will present a solution to this problem for Anosov surfaces namely, surfaces with uniformly hyperbolic geodesic flow (in particular, all negatively-curved surfaces are Anosov). This generalizes to the Anosov setting the celebrated rigidity results by Croke and Otal from the 90s.
Weakly nonlinear geometric optics for the Westervelt equation Leibniz Universität Hannover, Germany
In this talk we will discuss the non-diffusive Westervelt equation, which describes the time evolution of pressure in a medium relative to an equilibrium position. It is a second order quasilinear hyperbolic equation, involving a space dependent parameter which multiplies the nonlinear term. Given a medium with compactly supported but unknown nonlinearity, we would like to recover the latter by probing the medium from different directions with high frequency waves and measuring the exiting wave. To do so, we construct approximate solutions for the forward problem via nonlinear geometric optics and discuss its well posedness. We then explain how the X-ray transform of the nonlinearity can be recovered from the measurements, which allows for it to be reconstructed. Based on joint work with Plamen Stefanov.
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| 4:00pm - 6:00pm | MS54 3: The x-ray transform and its generalizations: Theory, methods, and applications Location: VG1.105 Session Chair: Suman Kumar Sahoo |
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The Calderón problem for space-time fractional parabolic operators with variable coefficients 1TIFR CAM, Bangalore, India; 2RICAM, Austria
We study an inverse problem for variable coefficient fractional parabolic operators of
the form $(\partial_t − \textrm{div}(A(x)\nabla_x))^s + q(x, t)$ for $s \in (0, 1)$ and show the unique recovery of q from exterior measured data. Similar to the fractional elliptic case, we use Runge type approximation argument which is obtained via a global weak unique continuation property. The proof of such a unique continuation result involves a new Carleman estimate for the associated variable coefficient extension
operator. In the latter part of the work, we prove analogous unique determination results for fractional parabolic operators with drift.
[1] A. Banerjee, S. Senapati. The Calderón problem for space-time fractional parabolic operators with variable coefficients, arXiv: 2205.12509, 2022.
Rich tomography reconstruction problems in applications. University of Manchester, United Kingdom
Rich tomography refers generally to problems in which the image has more than just a scalar per voxel, and often the measurement is more than one scalar per source and detector pair. In this talk I will give a number of examples of real problems where the data collection systems exist and I will review their mathematical formulation, what is known and what is yet to be determined about the reconstruction problem (as well as sufficiency of data, range characterization and stability,
Small Angle X-ray Scattering (SAXS) tomography is an example where a diffraction pattern is measured for each ray, and the inverse problem is to determine the `reciprocal space map' a function of three variables at each point.
Several techniques involve the imaging of strain in a crystalline or polycrystalline material. I will show the formulation of the problem where the measurement uses neutrons and electrons. In polycrystalline materials the texture, or distribution of crystal orientations over a given scale, is often an `nuisance variable' but can be of interest in its own right. I will suggest some possible mathematical challenges.
Finally the imaging of magnetic fields is a vector tomography problem and I will contrast the method using polarimetric tomography with neutrons and also a method using electron tomography.
Localized artifacts in medical imaging ETH Zurich, Switzerland
Medical imaging reconstruction is typically regularized with methods that lead to stability in an $L^2$ sense. However, we argue that the $L^2$ norm is not always a good metric with which to assess the quality of image reconstruction. For instance, two objects might be close in $L^2$, while one of them carries a localized, clearly visible artifact, not present in the other. While this issue has been raised for deep learning based algorithms, we show as an example, that the classical regularization method of compressed sensing for MRI is also not protected from such possible instabilities. This is joint work with Giovanni S. Alberti (University of Genoa) and Tandri Gauksson (ETH Zurich).
Explicit inversion of momentum ray transform IISER Bhopal, India
The inversion of the ray transform serves as an important mathematical tool for investigating object properties from external measurements with extensive applications spanning medical imaging to geophysics. However, the inversion of the ray transform on symmetric tensor fields is constrained by the presence of an infinite dimensional null space. One natural question is whether we can utilize supplementary data in the form of higher order moments of the ray transform for the explicit recovery of the entire tensor field. In this talk, we will focus our attention on normal operators associated to momentum ray transforms (the composition of the transform with its formal $L^2$ adjoint), and introduce an approach for the explicit reconstruction of entire symmetric $m$ tensor field from this data.
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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | MS33 1: Quantifying uncertainty for learned Bayesian models Location: VG1.105 Session Chair: Marta Malgorzata Betcke Session Chair: Martin Holler |
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Equivariant Neural Networks for Indirect Measurements University of Bremen, Germany
In the recent years, deep learning techniques have shown great success in various tasks related to inverse problems, where a target quantity of interest can only be observed through indirect measurements of a forward operator. Common approaches apply deep neural networks in a post-processing step to the reconstructions obtained by classical reconstruction methods. However, the latter methods can be computationally expensive and introduce artifacts that are not present in the measured data and, in turn, can deteriorate the performance on the given task.
To overcome these limitations, we propose a class of equivariant neural networks that can be directly applied to the measurements to solve the desired task. To this end, we build appropriate network structures by developing layers that are equivariant with respect to data transformations induced by symmetries in the domain of the forward operator. We rigorously analyze the relation between the measurement operator and the resulting group representations and prove a representer theorem that characterizes the class of linear operators that translate between a given pair of group actions.
Based on this theory, we extend the existing concepts of Lie group equivariant deep learning to inverse problems and introduce new representations that are the results of the involved measurement operations. This allows us to efficiently solve classification, regression or even reconstruction tasks based on indirect measurements also for very sparse data problems, where a classical reconstruction based approach may be hard or even impossible. To illustrate the effectiveness of our approach, we perform numerical experiments on selected inverse problems and compare our results to existing methods.
Bayesian MRI reconstruction with joint uncertainty estimation using diffusion priors 1University Medical Center Göttingen, Germany; 2Institute of Biomedical Imaging, Graz University of Technology, Graz, Austria; 3German Centre for Cardiovascular Research (DZHK), Partner Site Göttingen, Germany; 4Cluster of Excellence "Multiscale Bioimaging: from Molecular Machines to Networks of Excitable Cells'' (MBExC), University of Göttingen, Germany
The application of generative models in MRI reconstruction is shifting researchers' attention from the unrolled reconstruction networks to the probabilistic methods which can be used for unsupervised medical image reconstruction [1-4]. We formulate the image reconstruction problem from the perspective of Bayesian inference, which enables efficient sampling from the learned posterior probability distributions [1-2]. Different from conventional deep learning-based MRI reconstruction techniques, samples are drawn from the posterior distribution given the measured k-space using the Markov chain Monte Carlo (MCMC) method. Because the generative model can be learned from an image database independently from the forward operator, the same pre-trained models can be applied to k-space acquired with different sampling schemes or receive coils. Here, we present additional results in terms of the uncertainty of reconstruction, the transferability of learned information, and the comparison using data from the fastMRI challenge.
[1] G. Luo, M. Heide, M. Uecker. Using data-driven Markov chains for MRI reconstruction with Joint Uncertainty Estimation. Proc. Intl. Soc. Mag. Reson. Med. 30: 0298.
[2] G. Luo, M. Blumenthal, M. Heide, M. Uecker. Bayesian MRI reconstruction with joint uncertainty estimation using diffusion models. Magn. Reson. Med. 90: 295- 311, 2023.
[3] A. Jalal, M. Arvinte, G. Daras, E. Price, A. Dimakis, J. Tamir. Robust Compressed Sensing MRI with Deep Generative Priors. Neural Information Processing Systems 34: 14938–14954, 2021.
[4] H. Chung, C. Ye. Score-based diffusion models for accelerated MRI, Medical Image Analysis 80: 102479, 2022.
Utilizing variational autoencoders in the Bayesian inverse problem of photoacoustic tomography University of Eastern Finland, Finland
Photoacoustic tomography (PAT) is an imaging modality based on the photoacoustic effect. In the inverse problem of PAT, an initial pressure distribution induced by absorption of an externally introduced light is estimated from measured photoacoustic data. In the recent years, utilisation of machine learning in the inverse problem of PAT has gained significant interest. However, many of these machine learning-based methods do not provide information regarding the uncertainty of the reconstructed image.
In this work, we proposed a machine learning-based framework for the Bayesian inverse problem of PAT. The approach is based on the variational autoencoder (VAE) and the recently proposed uncertainty quantification variational autoencoder (UQ-VAE). In the VAE and UQ-VAE, an approximation of the true underlying posterior distribution is estimated by minimizing a divergence between the true and estimated posterior distributions using a neural network. The approach is evaluated using numerical simulations both in full and limited view measurement geometries with multiple levels of measurement noise.
Scalable Bayesian uncertainty quantification with learned convex regularisers 1University College London, United Kingdom; 2Heriot Watt University, United Kingdom
The last decade brought us substantial progress in computational imaging techniques for current and next-generation interferometric telescopes, such as the SKA. Imaging methods have exploited sparsity and more recent deep learning architectures with remarkable results. Despite good reconstruction quality, obtaining reliable uncertainty quantification (UQ) remains a common pitfall of most imaging methods. The UQ problem can be addressed by reformulating the inverse problem in the Bayesian framework. The posterior probability density function provides a comprehensive understanding of the uncertainties. However, computing the posterior in high-dimensional settings is an extremely challenging task. Posterior probabilities are often computed with sampling techniques, but these cannot yet cope with the high-dimensional settings from radio imaging.
This work proposes a method to address uncertainty quantification in radio-interferometric imaging with data-driven (learned) priors for very high-dimensional settings. Our model uses an analytic physically motivated model for the likelihood and exploits a data-driven prior learned from data. The proposed prior can encode complex information learned implicitly from training data and improves results from handcrafted priors (e.g., wavelet-based sparsity-promoting priors). We exploit recent advances in neural-network-based convex regularisers for the prior that allow us to ensure the log-concavity of the posterior while still being expressive. We leverage probability concentration phenomena of log-concave posterior functions that let us obtain information about the posterior avoiding the use of sampling techniques. Our method only requires the maximum-a-posteriori (MAP) estimation and evaluations of the likelihood and prior potentials. We rely on convex optimisation methods to compute the MAP estimation, which are known to be much faster and better scale with dimension than sampling strategies. The proposed method allows us to compute local credible intervals, i.e., Bayesian error bars, and perform hypothesis testing of structure on the reconstructed image. We demonstrate our method by reconstructing simulated radio-interferometric images and carrying out fast and scalable uncertainty quantification.
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | MS33 2: Quantifying uncertainty for learned Bayesian models Location: VG1.105 Session Chair: Marta Malgorzata Betcke Session Chair: Martin Holler |
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Calibration-Based Probabilistic Error Bounds for Inverse Problems in Imaging 1Graz University of Technology, Austria; 2Universität Graz, Austria
Traditional hand-crafted regularizers, such as the total variation, have a profound history in the context of inverse problems.
Typically, they are accompanied by a geometrical interpretation and experts are familiar with (artifacts in) the resulting reconstructions.
Modern, learned regularizers can hardly be interpreted in this way, thus it is important to supply uncertainty maps or error bounds in addition to any reconstruction.
In this talk, we give an overview of calibration-based methods that provide 1) pixel-wise probabilistic error bounds or 2) probabilistic confidence with respect to entire structures in the reconstruction.
We show results on the clinically highly relevant problem of undersampled magnetic resonance reconstruction.
Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging 1Graz University of Technology; 2University of Graz
We present a method for obtaining pixel-wise error bounds in Bayesian regularization of inverse imaging problems. The proposed approach employs estimates of the posterior variance together with techniques from conformal prediction in order to obtain error bounds with coverage guarantees, without making any assumption on the underlying data distribution. It is generally applicable to Bayesian regularization approaches, independent, e.g., of the concrete choice of the prior. Furthermore, the coverage guarantees can also be obtained in case only approximate sampling from the posterior is possible. With this in particular, the proposed framework is able to incorporate any learned prior in a black-box manner.
Such a guaranteed coverage without assumptions on the underlying distributions is only achievable since the magnitude of the error bounds is, in general, unknown in advance. Nevertheless, as we confirm with experiments with multiple regularization approaches, the obtained error bounds are rather tight.
A preprint of this work is available at https://arxiv.org/abs/2212.12499
How to sample from a posterior like you sample from a prior 1DTU, Denmark; 2Emory University, USA
The importance of quantifying uncertainties is rising in many applications of inverse problems. One way to estimate uncertainties is to explore the posterior distribution, e.g. in the context of Bayesian inverse problems. Standard approaches in exploring the posterior, e.g. the Markov Chain Monte Carlo (MCMC) methods, are often inefficient for large-scale and non-linear inverse problems.
In this work, we propose a method that exploits data to construct accelerated sampling from the posterior distributions for goal-oriented inverse problems. We use variational encoder-decoder (VED) to approximate the mapping that relates a measurement vector to the posterior distribution. The output of the VED network is an approximation of the true distribution and can estimate its moment, e.g. using Monte-Carlo methods. This enables real-time uncertainty quantification. The proposed method showcases a promising approach for large-scale inverse problems.
Uncertainty Quantification for Computed Tomography via the Linearised Deep Image Prior University College London, United Kingdom
Existing deep-learning based tomographic image reconstruction methods do not provide accurate estimates of reconstruction uncertainty, hindering their real-world deployment. In this talk we present a method, termed as the linearised deep image prior (DIP) that estimates the uncertainty associated with reconstructions produced by the DIP with total variation regularisation (TV). We discuss how to endow the DIP with conjugate Gaussian-linear model type error-bars computed from a local linearisation of the neural network around its optimised parameters. This approach provides pixel-wise uncertainty estimates and a marginal likelihood objective for hyperparameter optimisation. Throughout the talk we demonstrate the method on synthetic data and real-measured high-resolution 2D $\mu$CT data, and show that it provides superior calibration of uncertainty estimates relative to previous probabilistic formulations of the DIP.
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| 4:00pm - 6:00pm | CT07: Contributed talks Location: VG1.105 Session Chair: Christian Aarset |
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Accelerating MCMC for imaging science by using an implicit Langevin algorithm 1University of Edinburgh, United Kingdom; 2Heriot-Watt University, United Kingdom; 3Maxwell Institute for Mathematical Sciences, United Kingdom
In this work, we present a highly efficient gradient-based Markov chain Monte Carlo methodology to perform Bayesian computation in imaging problems. Like previous Monte Carlo approaches, the proposed method is derived from a discretisation of a Langevin diffusion. However, instead of a conventional explicit discretisation like Euler-Maruyama, here we use an implicit discretisation based on the theta-method. In particular, the proposed sampling algorithm requires to solve an optimization problem in each step. In the case of a log-concave posterior, this optimisation problem is strongly convex and thus can be solved efficiently, leading to effective step sizes that are significantly larger than traditional methods permit. We can show that even for these large step sizes the corresponding Markov Chain has low bias while also preserving the posterior variance. We demonstrate the proposed methodology on a range of problems including non-blind image deconvolution and denoising. Comparisons with state-of-the-art MCMC methods confirm that the Markov chains generated with our method exhibit significantly faster convergence speeds, and produce lower mean square estimation errors at equal computational budget.
Quasi-Monte Carlo methods for Bayesian optimal experimental design problems governed by PDEs Free University of Berlin, Germany
The goal in Bayesian optimal experimental design is to maximize the expected information gain for the reconstruction of unknown quantities when there is a limited budget for collecting measurement data. We consider Bayesian inverse problems governed by partial differential equations. This leads us to consider an optimization problem, where the objective functional involves nested high-dimensional integrals which we approximate by using tailored rank-1 lattice quasi-Monte Carlo (QMC) rules. We show that these QMC rules achieve faster-than-Monte Carlo convergence rates. Numerical experiments are presented to assess the theoretical results.
Maximum marginal likelihood estimation of regularisation parameters in Plug & Play Bayesian estimation: Application to non-blind and semi-blind image deconvolution 1Heriot-Watt University, United Kingdom; 2Heriot-Watt University, United Kingdom; 3Université de Bordeaux
Bayesian Plug & Play (PnP) priors are widely acknowledged as a powerful framework for solving a variety of inverse problems in imaging. This Bayesian PnP framework has made tremendous advances in recent years, resulting in state-of-the-art methods. Although PnP methods have been distinguished by their ability to regularize Bayesian inverse problems through a denoising algorithm, setting the amount of regularity enforced by the prior, determined by the noise level parameter of the denoiser, has been an issue for several reasons. This talk aims to present an empirical Bayesian extension of an existing Plug & Play (PnP) Bayesian inference method. The main novelty of this work is that we estimate the regularisation parameter directly from the observed data by maximum marginal likelihood estimation (MMLE). However, noticing that the MMLE problem is computationally and analytically intractable, we incorporate a Markov kernel within a stochastic approximation proximal gradient scheme to address this difficulty. The resulting method calibrates a regularisation parameter by MMLE while generating samples asymptotically distributed according to the empirical Bayes (pseudo-) posterior distribution of interest. Additionally, the proposed method can estimate other unknown parameters of the model using MMLE; such as the noise level of the observation model, and the parameters of the forward operator simultaneously. The proposed method has been demonstrated with a range of non-blind and semi-blind image deconvolution problems, as well as compared to state-of-the-art methods.
Choosing observations to mitigate model error in Bayesian inverse problems 1Eindhoven University of Technology, Netherlands, The; 2Universität Potsdam, Germany
In inverse problems, one often assumes a model for how the data is generated from the underlying parameter of interest. In experimental design, the goal is to choose observations to reduce uncertainty in the parameter. When the true model is unknown or expensive, an approximate model is used that has nonzero `model error' with respect to the true data-generating model. Model error can lead to biased parameter estimates. If the bias is large, uncertainty reduction around the estimate is undesirable. This raises the need for experimental design that takes model error into account. We present a framework for model error-aware experimental design in Bayesian inverse problems. Our framework is based on Lipschitz stability results for the posterior with respect to model perturbations. We use our framework to show how one can combine experimental design with models of the model error in order to improve the results of inference.
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| Date: Friday, 08/Sept/2023 | |
| 1:30pm - 3:30pm | MS50 1: Mathematics and Magnetic Resonance Imaging Location: VG1.105 Session Chair: Kristian Bredies Session Chair: Christian Clason Session Chair: Martin Uecker |
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Deep learning MR image reconstruction and task-based evaluation Department Artificial Intelligence in Biomedical Engineering, FAU Erlangen-Nuremberg, Germany
The inverse problem of reconstructing MR images $u$ from Fourier ($k$-) space data $f$ takes the form of the optimization problem:
$$\min \| Au - f \|_2^2 + \lambda \mathcal{R}(u).$$
$A=\mathcal{F}_\Omega C$ is the forward operator that describes the MR encoding process. It consists of a Fourier transform $\mathcal{F}_\Omega$ that maps from image space to Fourier ($k$-) space coefficients at the coordinates $\Omega$ and a diagonal matrix $C$ that contains the sensitivity profiles of the receiver coils of the MR system. $\mathcal{R}$ is a regularizer that separates between true image content and artifacts introduced by an accelerated acquisition. It has been demonstrated that deep learning methods that map the image reconstruction optimization problem onto unrolled neural networks and learn a regularizer from training data [1] achieve state of the art performance in public research challenges [2].
In this work, we will present an update on the performance of learned image reconstruction for a range of clinically relevant applications and discuss the issue of missing-, as well as artificially hallucinated fine-detail image features [3]. We will present results for cardiac, oncological and neuroimaging applications, and will also introduce a novel task-based evaluation for the quality of the reconstructed images using the fastMRI+ dataset [4].
[1] K. Hammernik, T. Klatzer, E. Kobler, M. P. Recht, D. K. Sodickson, T. Pock, F. Knoll. Learning a Variational Network for Reconstruction of Accelerated MRI Data, Magnetic Resonance in Medicine 79: 3055–3071, 2018. https://doi.org/10.1002/mrm.26977
[2] M. J. Muckley, B. Riemenschneider, A. Radmanesh, S. Kim, G. Jeong, J. Ko, Y. Jun, H. Shin, D. Hwang, M. Mostapha, S. Arberet, D. Nickel, Z. Ramzi, P. Ciuciu, J.-L. Starck, J. Teuwen, D. Karkalousos, C. Zhang, A. Sriram, Z. Huang, N. Yakubova, Y. W. Lui, F. Knoll. Results of the 2020 fastMRI Challenge for Machine Learning MR Image Reconstruction, IEEE Transactions on Medical Imaging 40: 2306–2317, 2021. https://doi.org/10.1109/TMI.2021.3075856
[3] A. Radmanesh, M. J. Muckley, T. Murrell, E. Lindsey, A. Sriram, F. Knoll, D. K. Sodickson, Y.W. Lui. Exploring the Acceleration Limits of Deep Learning Variational Network–based Two-dimensional Brain MRI, Radiology: Artificial Intelligence 4, 2022. https://doi.org/10.1148/ryai.210313
[4] R. Zhao, B. Yaman, Y. Zhang, R. Stewart, A. Dixon, F. Knoll, Z. Huang, Y. W. Lui, M. S. Hansen, M. P. Lungren. fastMRI+: Clinical Pathology Annotations for Knee and Brain Fully Sampled Multi-Coil MRI Data, Scientific Data 2022 9: 1–6, 2022. https://doi.org/10.1038/s41597-022-01255-z
Learning Fourier sampling schemes for MRI by density optimization 1Institut de Mathématiques de Toulouse, France; 2University of Toulouse; 3INSA Toulouse; 4Centre de Biologie Intégrative (CBI), Laboratoire MCD
An MRI scanner roughly allows measuring the Fourier transform of the image representing a volume at user-specified locations. Finding an optimal sampling pattern and reconstruction algorithm is a longstanding issue. While Shannon and compressed sensing theories dominated the field over the last decade, a recent trend is to optimize the sampling scheme for a specific dataset.
Early works investigated algorithms that find the best subset among a set of feasible trajectories.
More recently, some works proposed to optimize the positions of the sampling locations continuously [3].
In this talk, we will first show that this optimization problem usually possesses a combinatorial number of spurious minimizers [1].
This effect can however be mitigated by using large datasets of signals and specific preconditioning techniques.
Unfortunately, the dataset size, the costly reconstruction processes and the computation of the non-uniform Fourier transform makes the problem computationally challenging.
By optimizing the sampling density rather than the points locations, we show that the problem can be solved significantly faster while preserving competitive results [2].
[1] A. Gossard, F. de Gournay, P. Weiss. Spurious minimizers in non uniform Fourier sampling optimization, Inverse Problems 38: 105003, 2022.
[2] A. Gossard, F. de Gournay, P. Weiss. Bayesian Optimization of Sampling Densities in MRI, arXiv: 2209.07170, 2022.
[3] G. Wang, T. Luo, J.-F. Nielsen, D. C Noll, J. A Fessler. B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI, IEEE Transactions on Medical Imaging 41: 2318--2330, 2022.
Acceleration strategies for Magnetic Resonance Spin Tomography in Time-Domain (MR‐STAT) reconstructions Computational Imaging Group for MRI Therapy & Diagnostics, Department of Radiotherapy, University Medical Center Utrecht, Utrecht, Netherlands
Magnetic Resonance Spin Tomography in Time‐Domain (MR-STAT) is an emerging quantitative magnetic resonance imaging technique which aims at obtaining multi-parametric tissue parameter maps (T1, T2, proton density, etc) from short scans. It describes the relationship between the spatial-domain tissue parameters and the time-domain measured signal by using a comprehensive, volumetric forward model. The MR-STAT reconstruction is cast as a large-scale, ODE constrained, nonlinear inversion problem, which is very challenging in terms of both computing time and memory.
In this presentation, I’ll talk about recent progresses about the acceleration strategies for MR-STAT reconstructions, for example, using a neural network model for the solution of the underlying differential equation model, applying alternating direction method of multipliers (ADMM) etc.
Learning Spatio-Temporal Regularization Parameter-Maps for Total Variation-Minimization Reconstruction in Dynamic Cardiac MRI 1Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany, Germany; 2Humboldt-Universit ̈at zu Berlin, Department of Mathematics, Berlin, Germany; 3echnische Universit ̈at Berlin, Institute of Mathematics, Berlin, Germany; 4Finden Ltd, Rutherford Appleton Laboratory, Harwell Campus, Didcot, United Kingdom; 5Science and Technology Facilities Council, Harwell Campus, Didcot, United Kingdom; 6Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany; 7School of Mathematical Sciences, Queen Mary University of London, United Kingdom
In dynamic cardiac Magnetic Resonance Imaging (MRI), one is interested in the assessment of the cardiac function based on a series of images which show the beating heart. Because the measurements typically take place during a breathhold of the patients, it is desirable to accelerate the scan by undersampling the data which yields an ill-posed inverse problem which requires the use of regularization methods.
A prominent and successful example of regularization method is the well-known total variation (TV)-minimization approach which imposes sparsity of the image in its gradient domain. Thereby, the choice of the regularization parameter which balances between the data-fidelity term and the TV-term plays a crucial role. Moreover, having only a scalar regularization parameter which globally dictates the strength of the regularization seems to be sub-optimal for various reasons. Intuitively speaking, the strength of the TV-term should be locally dependent based on the content of the image. However, obtaining entire regularization parameter-maps for dynamic problems can be a challenging task.
In this work, we propose a simple yet efficient approach for estimating patient-dependent spatio-temporal regularization parameter-maps for dynamic MRI based on TV-minimization. The overall approach is based on recent developments on algorithm unrolling using deep Neural Networks (NNs). A first NN estimates a spatio-temporal regularization parameter-map from an input image which is then fixed and used to formulate a reconstruction problem which a second network – an unrolled scheme using the primal dual hybrid gradient method – approximately solves. The approach combines NNs with a well-established model-based variational method and yields an entirely interpretable and convergent reconstruction scheme which can be used to improve over TV with merely scalar regularization parameters.
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| 4:00pm - 6:00pm | MS50 2: Mathematics and Magnetic Resonance Imaging Location: VG1.105 Session Chair: Kristian Bredies Session Chair: Christian Clason Session Chair: Martin Uecker |
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MRI Pulse Design via discrete-valued optimal control University of Graz, Austria
Magnetic Resonance Imaging (MRI) is an active imaging methodology that uses radio frequency excitation and response of magnetic spin ensembles under a strong static external magnetic field to measure the distribution of hydrogen atoms in a sample. This distribution correlates with different tissues in a human body, allowing non-invasive medical imaging without ionizing radiation. The mathematical model for the behavior of magnetic spin ensembles under magnetic fields is the so-called Bloch equation, which is a bilinear differential equation. The problem of generating optimal excitation pulses for imaging purposes can thus be formulated and solved as an optimal control problem. We present the basic setup and methods, show practical examples, and discuss how to incorporate structural constraints on the optimal pulses.
Null Space Networks for undersampled Fourier data Universität Innsbruck, Austria
Preserving data consistency is a key property of learned image reconstruction. This can be achieved either by specific network architecture or by subsequent projection of the network reconstruction. In this talk, we analyze null-space networks for undersampled image reconstruction. We numerically compare image reconstruction from undersampled Fourier data and investigate the effect integrating data consistency in the network architecture
Deep Learning Approaches for Non-Linear Inverse Problems in MRI Reconstruction 1Institute of Biomedical Imaging, Graz University of Technology, Graz, Austria; 2Institute for Diagnostic and Interventional Radiology of the University Medical Center Göttingen, Germany
MRI is an important tool for clinical diagnosis. Although recognized for being non-invasive and producing images of high quality and excellent soft tissue contrast, its long acquisition times and high cost are problematic. Recently, deep learning techniques have been developed to help solve these issues by improving acquisition speed and image quality.
The multi-coil measurement process is modeled by a linear operator, the SENSE encoding model
$$
\begin{aligned}
A:\mathbb{C}^{N_x\times N_y}&\to \mathbb{C}^{N_S \times N_C}\\
x &\mapsto y=\mathcal{PFC}x.
\end{aligned}
$$
The discretized image $x$ corresponds to the complex-valued transversal magnetization in the tissue. In the encoding process, it is first weighted with the coil-sensitivity maps of the $N_C$ receive $\mathcal{C}$oils, then $\mathcal{F}$ourier transformed and finally projected to the $N_S$ sample points of the acquired sampling $\mathcal{P}$attern. Unrolled model-based deep learning approaches are motivated by classical optimization algorithm of the linear inverse problem and integrate learned prior knowledge by learned regularization terms. Typical examples of end-2-end trained networks from the field of MRI are the Variational Network [1] or MoDL [2].
Despite MRI reconstruction often being treated linearly, there are many applications that require non-linear approaches. For instance, the estimation of coil-sensitivity maps can be challenging. An alternative to the use of calibration measurements or pre-estimation of the sensitivity maps from fully-sampled auto-calibration regions is to integrate the estimation into the reconstruction problem. This results in a non-linear - in fact, bilinear - forward model of the form
$$
\begin{aligned}
F:\mathbb{C}^{N_x\times N_y}\times \mathbb{C}^{N_x\times N_y\times N_c}&\to \mathbb{C}^{N_S \times N_C}\\
x = \begin{pmatrix}
x_{\mathrm{img}}\\x_{\mathrm{col}}
\end{pmatrix} &\mapsto y=\mathcal{PF}\left(x_{\mathrm{img}}\odot x_{\mathrm{col}}\right)\,.
\end{aligned}
$$
A possible approach to solve the corresponding inverse problem is the iteratively regularized Gauss-Newton method (IRGNM) [3], which can in turn be combined with deep-learning based regularization [4] similarly to MoDL.
Another source of non-linearity in the reconstruction is the temporal evolution of transverse magnetization. The magnetization follows the Bloch equations, which are parametrized by tissue-specific relaxation parameters $T_1$ and $T_2$. In quantitative (q)MRI, parameter maps $x_{\mathrm{par}}$ are estimated instead of qualitative images $x_{\mathrm{img}}$ of transverse magnetization. In model-based qMRI, physical models that map the parameter maps $x_{\mathrm{par}}$ to the transverse magnetization are combined with encoding models to create non-linear forward models of the form [5]:
$$
\begin{aligned}
F:\mathbb{C}^{N_x\times N_y\times N_p}\times \mathbb{C}^{N_x\times N_y\times N_c}&\to \mathbb{C}^{N_S \times N_C}\\
x = \begin{pmatrix}
x_{\mathrm{par}}\\x_{\mathrm{col}}
\end{pmatrix} &\mapsto y=\mathcal{PF}\left(\mathcal{M}(x_{\mathrm{par}})\odot x_{\mathrm{col}}\right)
\end{aligned}
$$
An efficient way to solve a particular class of such non-linear inverse problems is the approximation of the non-linear signal model in linear subspaces, which in turn can be well combined with deep-learning based regularization [6].
This talk will cover deep-learning based approaches to solve the non-linear inverse problems defined above.
[1] K. Hammernik, T. Klatzer, E. Kobler, M. P. Recht, D. K. Sodickson, T. Pock, F. Knoll. Learning a variational network for reconstruction of accelerated MRI data, Magn. Reson. Med. 79: 3055-3071, 2018.
[2] H. K. Aggarwal, M. P. Mani, M. Jacob. MoDL: Model-Based Deep Learning Architecture for Inverse Problems, IEEE Trans. Med. Imaging. 38: 394-405, 2019.
[3] M. Uecker, T. Hohage, K. T. Block, J. Frahm. Image reconstruction by regularized nonlinear inversion—Joint estimation of coil sensitivities and image content, Magn. Reson. Med. 60: 674-682, 2018.
[4] M. Blumenthal, G. Luo, M. Schilling, M. Haltmeier, M. Uecker, NLINV-Net: Self-Supervised End-2-End Learning for Reconstructing Undersampled Radial Cardiac Real-Time Data, Proc. Intl. Soc. Mag. Reson. Med. 28: 0499, 2022
[5] X. Wang et al. Physics-based reconstruction methods for magnetic resonance imaging. Phil. Trans. R. Soc. A. 379: 20200196, 2021
[6] M. Blumenthal et al. Deep Subspace Learning for Improved T1 Mapping using Single-shot Inversion-Recovery Radial FLASH. Proc. Intl. Soc. Mag. Reson. Med. 28: 0241, 2022
Mathematical Methods in Parallel MRI University of Göttingen, Germany
Magnetic Resonance Imaging (MRI) is an important technique in medical imaging. In the subfield of Parallel MRI, multiple receive coils are used to reconstruct tomographic images with fewer data acquisition steps compared to ordinary MRI.
In this talk we will take a deeper look into the mathematical background of some of the prominent reconstruction methods. We will show that, in the course of these methods, implicit assumptions on the structure of the signals and the sensitivity profiles that are associated to the receive coils are made. In order to get a better understanding of the methods at hand and possible improvements, we aim to make these assumptions explicit.
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