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: VG2.103 |
| Date: Monday, 04/Sept/2023 | |
| 1:30pm - 3:30pm | MS03 1: Compressed Sensing meets Statistical Inverse Learning Location: VG2.103 Session Chair: Tatiana Alessandra Bubba Session Chair: Luca Ratti Session Chair: Matteo Santacesaria |
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Compressed sensing for the sparse Radon transform 1University of Genoa, Italy; 2Politecnico di Torino, Italy
Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear measurements or subsampled isometries and has found many applications, including accelerated magnetic resonance imaging, which is modeled by the subsampled Fourier transform. In our work, we develop a general theory of infinite-dimensional compressed sensing for abstract inverse problems, possibly ill-posed, involving an arbitrary forward operator. This is achieved by considering a generalized restricted isometry property, and a quasi-diagonalization property of the forward map.
As a notable application, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles $\theta_1,\dots,\theta_m$), which models computed tomography. In the case when the unknown signal is $s$-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove exact recovery under the condition $m\gtrsim s,$ up to logarithmic factors.
[1] G. S. Alberti, A. Felisi, M. Santacesaria, I. Trapasso. Compressed sensing for
inverse problems and the sample complexity of the sparse Radon transform, ArXiv
e-prints, arXiv:2302.03577, 2023.
Regularization for learning from unlabeled data using related labeled data Austrian Academy of Sciences, Austria
We consider the problem of learning from unlabeled target datasets using related labeled source datasets, e.g. learning from an image-dataset from a target medical patient using expert-annotated datasets from related source patients. This problem is complicated by (a) missing target labels, e.g. no target expert-annotations of a tumor, and, (b) possible differences in the source and target data generating distributions, e.g. caused by medical patents’ human variations. The major three methods for this problem, are special cases of multiple or cascade regularization methods, i.e., methods involving simultaneously more than one regularization. This talk is based on [1-3] and reviews non-asymptotic (w.r.t. dataset size) error bounds of the major three methods.
[1] W. Zellinger, N. Shepeleva, M.-C. Dinu, H. Eghbal-zadeh, H. D. Nguyen, B. Nessler, S. V. Pereverzyev, B. Moser. The balancing principle for parameter choice in distance-regularized domain adaptation. Advances in Neural Information Processing Systems (NeurIPS). 34: 20798--20811, 2021.
[2] E.R. Gizewski, L. Mayer, B. Moser, D.H. Nguyen, S. Pereverzyev Jr, S.V. Pereverzyev, N. Shepeleva, and W. Zellinger. On a regularization of unsupervised domain adaptation in RKHS. Appl. Comput. Harmon. Anal. 57: 201--227. 2022. https://doi.org/10.1016/j.acha.2021.12.002
[3] M. Holzleitner, S.V. Pereverzyev, W. Zellinger. Domain Generalization by Functional Regression. arXiv preprint arXiv:2302.04724 (2023). https://doi.org/10.48550/arXiv.2302.04724
Random tree Besov priors for detail detection 1Delft University of Technology, Netherlands, The; 2University of Helsinki, Finland
Besov priors are well fitted for imaging since smooth functions with few local irregularities have a sparse expansion in the wavelet basis which is encouraged by the prior. The edge preservation of Besov priors can be enhanced by introducing a new random variable T that takes values in the space of ‘trees’, and which is chosen so that the realizations have jumps only on a small set. The density of the tree, and so the size of the set of jumps, is controlled by a hyperparameter. In this talk I will show how this hyperparameter can be optimized for the data and what the optimal values tell us about behaviour of the signal or image.
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| 4:00pm - 6:00pm | MS03 2: Compressed Sensing meets Statistical Inverse Learning Location: VG2.103 Session Chair: Tatiana Alessandra Bubba Session Chair: Luca Ratti Session Chair: Matteo Santacesaria |
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SGD for statistical inverse problems LUT University Lappeenranta, Finland
We study a statistical inverse learning problem, where we observe the noisy image of a quantity through an operator at some random design points. We consider the SGD schemes to reconstruct the estimator of the quantity for the ill-posed inverse problem. We develop a theoretical analysis for the minimizer of the regularization scheme using the approach of reproducing kernel Hilbert spaces. We discuss the rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.
Convex regularization in statistical inverse learning problems LUT University, Finland
Statistical inverse learning aims at recovering an unknown function $f$ from randomly scattered and possibly noisy point evaluations of another function $g$, connected to $f$ via an ill-posed mathematical model.
In this talk I blend statistical inverse learning theory with convex regularization strategies and derive convergence rates for the corresponding estimators.
An off-the-grid approach to multi-compartment magnetic fingerprinting University of Bath, United Kingdom
We propose a off-the-grid numerical approach to separate multiple tissue compartments in image voxels and to estimate quantitatively their nuclear magnetic resonance (NMR) properties and mixture fractions, given magnetic resonance fingerprinting (MRF) measurements. One of the challenge is that fine-grid discretisation of the multi-dimensional NMR properties creates large and highly coherent MRF dictionaries that can challenge scalability and precision of the numerical methods for sparse approximation. To overcome this issues, we propose an off-the-grid approach equipped with an extended notion of the sparse group lasso regularisation for sparse approximation using continuous Bloch response models. Through numerical experiments on simulated and in-vivo healthy brain MRF data, we demonstrate the effectiveness of the proposed scheme compared to baseline multi-compartment MRF methods.
This is joint work with Mohammad Golbabaee.
How many Neurons do we need? A refined Analysis. Technische Universität Braunschweig, Germany
We present new results for random feature approximation in kernel methods and discuss the connection to generalization properties of two-layer neural networks in the NTK regime. Here, we aim at improving the results of Nitanda and Suzuki [1] in various directions. More precisely, we aim at overcoming the saturation effect appearing in Nitanda and Suzuki [1] by providing fast rates of convergence for smooth objectives. On our way, we also precisely keep track of the number of hidden neurons required for generalization.
[1] A. Nitanda, T. Suzuki. Functional Gradient Boosting for Learning Residual-like Networks with Statistical Guarantees. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics. 108: 2981--2991. 2020.
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| Date: Tuesday, 05/Sept/2023 | |
| 1:30pm - 3:30pm | MS21 1: Prior Information in Inverse Problems Location: VG2.103 Session Chair: Andreas Horst Session Chair: Jakob Lemvig |
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Reconstructing spatio-temporal, sparse tomographic data using cylindrical shearlets University of Bath, United Kingdom
In this talk, I will present a motion-aware variational approach, based on a new multiscale directional system of functions called cylindrical shearlets, to reconstruct moving objects from sparse dynamic data. Compared to conventional separable representations, cylindrical shearlets are very efficient in representing spatio-temporal data, since they are better suited to handle the geometry of these data.
We test our approach on both simulated and measured data. Numerical results demonstrate the advantages of our novel approach with respect to conventional multiscale methods.
Fractal priors for imaging using random wavelet trees University of Helsinki, Finland
A novel Bayesian prior distribution family is introduced, based on wavelet transforms.The priors correspond to well-defined infinite-dimensional random variables and can be approximated by finite-dimensional models.
The non-zero wavelet coefficients are chosen in a systematic way so that prior draws have a specific fractal behaviour. This paves the way for new types of signal and image processing methods that can either promote certain fractal properties in the underlying data, or serve as smart "fingerprints" for measured object types. Realisations of the new priors take values in Besov spaces and have singularities only on a small set with a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator in the denoising problem.
Sampling from a posterior with Besov prior Technical University of Denmark (DTU), Compute, Denmark
Besov priors for Bayesian inverse problems are interesting since they promote various types of regularity on the unknown, especially non-smooth regularity, depending on the choice of basis and parameters of the prior. Besov priors introduces a $p$-norm into the posterior, which makes effective Gaussian samplers inapplicable. Randomize-Then-Optimize (RTO) is an optimization-based sampling algorithm, that computes exact independent samples from a posterior with Gaussian prior and a linear forward operator. We introduce a prior transformation that transforms a Besov prior into a Gaussian prior, which makes Gaussian samplers like RTO applicable. The caveat of the transformation is that the forward operator becomes non-linear even though it originally was linear. To sample from the transformed posterior we use RTO samples as proposals for the Metropolis-Hastings algorithm. We apply this sampling method to a deconvolution problem where the type of Besov prior is varied, to discover the quality of the method and the posterior dependencies on the choice of Besov prior. Our results validate that the computed samples come from the original posterior with Besov prior and shows that the choice of prior basis and parameters has a significant impact on the posterior.
Regularizing Inverse Problems through Translation Invariant Diagonal Frame Decompositions OTH Regensburg, Germany
We consider the challenge of solving the ill-posed reconstruction problem in computed tomography using a translation-invariant diagonal frame decomposition (TI-DFD). First, we review the concept of diagonal frame decompositions (DFD) and their translation-invariant counterparts for general linear operators. Subsequently, we explain how the filter-based regularization methods can be defined using these frame decompositions. Finally, as an example, we introduce the TI-DFD for the Radon transform on $L^2 (\mathbb{R}^2)$ and provide an exemplary construction using the TI wavelet transform. In numerical results, we demonstrate the advantages of our approach over non-translation invariant counterparts.
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| 4:00pm - 6:00pm | MS21 2: Prior Information in Inverse Problems Location: VG2.103 Session Chair: Andreas Horst Session Chair: Jakob Lemvig |
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Regularized, pretrained and subspace-restricted Deep Image Prior for CT reconstruction 1Department of Computer Science, University College London, United Kingdom; 2Department of Engineering, University of Cambridge, United Kingdom; 3Center for Industrial Mathematics, University of Bremen, Germany; 4Department of Mathematics, The Chinese University of Hong Kong, P. R. China; 5Research Unit of Mathematical Sciences, University of Oulu, Finland
Computed tomography (CT) is an important tool in both medicine and industry. By now, a great variety of deep learning (DL) techniques has been developed for inverse imaging tasks including CT reconstruction. In constrast to most DL approaches, the deep image prior (DIP) is an unsupervised framework that does not rely on a large training dataset, but only on the single degraded observation. The central observation with DIP is that the early-stopped optimization an untrained networks can lead to favorable solutions, thus acting as an implicit prior.
We extend the DIP in several ways. First, we add an explicit prior in the form of a total variation regularization term, which can stabilize and improve the reconstruction. Second, we pretrain on a post-processing task with easy-to-generate synthetic data, which induces prior information, learned from the synthetic image class and the operator-specific degradation, into the subsequent unsupervised DIP optimization. This two-stage procedure of supervised pretraining and unsupervised fine-tuning is called the educated DIP (EDIP) and often requires a significantly shorter optimization time in the fine-tuning stage compared to untrained DIP. Finally, we experiment with restricting the parameter space in the fine-tuning stage of EDIP. Using an affine linear subspace, which is expanded around the pretraining parameters with a sparsified basis obtained from many checkpoints saved during the pretraining, both overfitting behaviour can be reduced and second order optimization methods become feasible, enabling more stable and faster reconstruction.
Monitoring of hemorrhagic stroke using Electrical Impedance Tomography University of Eastern Finland, Finland
In this talk, we present recent progress in development of electrical impedance tomography (EIT) based bedside monitoring of hemorrhagic stroke. We present the practical setup and pipeline for this novel application of EIT and the CT prior informed image reconstruction method we have developed for it. Feasibility of the approach is studied with simulated data from anatomically highly accurate simulation models and experimental phantom data from a laboratory setup.
Edge-preserving inversion with $\alpha$-stable priors 1LUT University, Finland; 2Heriot Watt University
The $\alpha$-stable distributions are a family of heavy-tailed and infinitely divisible distributions that are well-suited to edge-preserving inversion in the context of discretization of infinite-dimensional continuous-time statistical inverse problems. In this talk we discuss some of the technical issues arising from the application of such priors.
Optimal learning of high-dimensional classification problems using deep neural networks Katholische Universität Eichstätt-Ingolstadt, Germany
We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity. We establish universal lower bounds for this estimation problem, for general classes of continuous decision boundaries. For the class of locally Barron-regular decision boundaries, we find that the optimal estimation rates are essentially independent of the underlying dimension and can be realized by empirical risk minimization methods over a suitable class of deep neural networks. These results are based on novel estimates of the $L^1$ and $L^\infty$ entropies of the class of Barron-regular functions.
This is joint work with Philipp Petersen (University of Vienna).
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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | MS30 1: Inverse Problems on Graphs and Machine Learning Location: VG2.103 Session Chair: Emilia Lavie Kyllikki Blåsten Session Chair: Matti Lassas Session Chair: Jinpeng Lu |
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Continuum limit for lattice Hamiltonians University of Tsukuba, Japan
We consider the Hamiltonian perturbed by a potential on an infinite periodic lattice. We are interested in the behavior of the solution for the lattice model to that
for the equation for the continuous model as the mesh size tends to 0. For some lattices such as square, triangular and hexagonal lattices, we show that the scattering solutions (i.e. the soultions associated with the continuous spectrum) converge to the solution to the Shrodedinger equation in the continuous model. For the case of the hexagonal lattice, we can also derive the convergence to the massless Dirac equation.
The idea of the proof relies on the micro-local calculus for lattice Schroedinger operators and the classical method of the limiting absorption principle.
Quantum computing algorithms for inverse problems on graphs 1University of Jyväskylä, Finland; 2University of Helsinki, Finland
Quantum computing is a technology that utilizes quantum mechanical phenomena to do computation faster than is believed to be possible with classical computers. It is a rapidly developing and interdisciplinary field comprising of physics, computer science, and mathematics. It is predicted that in the future quantum computers will enable scientists to solve problems outside the capabilities of classical computers in many fields such as molecular simulations in drug discovery and complex combinatorics problems.
In this talk, we consider a quantum algorithm for an inverse travel time problem on a graph. This problem is a discrete version of the inverse travel time problem encountered in seismic and medical imaging and the boundary rigidity problem studied in Riemannian geometry. We also consider the computational complexity of the inverse problem, and show that the quantum algorithm has a quadratic improvement in computational cost when compared to the standard classical algorithm.
Inverse problems for the graph Laplacian 1LUT University, Finland; 2University of Tsukuba, Japan; 3University of Helsinki, Finland
We study the discrete version of Gel'fand's inverse spectral problem, formulated as follows for a finite weighted graph and the graph Laplacian on it. Suppose we are given a subset $B$ of vertices and the spectral data $(\lambda_j,\phi_j|_B)$, where $\lambda_j$ are the eigenvalues of the graph Laplacian and $\phi_j|_B$ are the values of the corresponding eigenfunctions on $B$.
We consider if these data uniquely determine the graph structure and the weights. In general, this problem is not uniquely solvable without assumptions on the graph or the set $B$ due to counterexamples.
We introduce a so-called Two-Points Condition on graphs (with respect to $B$), and prove that the inverse spectral problem is uniquely solvable under this condition.
We also consider inverse problems for random walks on finite graphs. We show that under the Two-Points Condition, the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on $B$.
Inverse problems on manifolds via graph-based semi-supervised learning 1University of Chicago, United States of America; 2Princeton University, United States of America
In this talk I will introduce graphical representations of stochastic partial differential equations with the goals of approximating Matérn Gaussian fields on manifolds and generalizing the Matérn model to abstract point clouds. I will show that these graph-based prior models can give optimal posterior contraction in semi-supervised learning, and illustrate their use in various inverse problems on manifolds.
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | MS30 2: Inverse Problems on Graphs and Machine Learning Location: VG2.103 Session Chair: Emilia Lavie Kyllikki Blåsten Session Chair: Matti Lassas Session Chair: Jinpeng Lu |
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Deep Invertible Approximation of Topologically Rich Maps between Manifolds 1South Dakota State University, United States of America; 2University of Helsinki; 3University of Basel; 4Rice University
How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? In this talk, we will provide the surprisingly simple answer. By exploiting the topological parallels between locally bilipschitz maps, covering spaces, and local homeomorphisms as well as universal approximation arguments from machine learning, we find that a novel network of the form $p \circ \mathcal{E}$, where $\mathcal{E}$ is a smooth embedding and $p$ a fixed coordinate projection, are universal approximators of local diffeomorphisms between compact smooth submanifolds embedded in $\mathbb{R}^n$. We emphasize the case when the map to be learned changes topology. Further, we find that by constraining the projection $p$, multivalued inversions of our networks can be computed without sacrificing universality. As an application of the problem, we show that the question of learning a group invariant function where the group action is unknown can be naturally reduced to the question of learning local diffeomorphisms when the group action is continuous, finite, and has constant-sized orbits. In this context the novel inversion result permits us to recover orbits of the group action.
Some inverse problems on graphs with internal functionals University of Utah, United States of America
We consider the problem of finding the resistors in a network from knowing the power that they dissipate under loads imposed at a few terminal nodes. This data could be obtained e.g. from thermal imaging of the network. We use a method inspired by Bal [1] to give sufficient conditions under which the linearized problem admits a unique solution. Similar results are shown for a discrete analogue to the Schrödinger equation and for the case of impedances or complex valued conductivities.
[1] Bal, Guillaume. Hybrid inverse problems and redundant systems of partial differential equations, Inverse problems and applications 619: 15-48, 2014.
Imaging water supply pipes using pressure waves 1LUT University, Finland; 2Hong Kong University of Science and Technology, Hong Kong
I will present a collaboration with applied mathematicians and civil engineers from the mathematical point of view. We worked on the problem of imaging water supply pipes for problem detection (is there a problem? where is the problem? how severe is the problem?). I will talk about the one-dimensional setting and also present a reconstruction algorithm for tree networks. The problem is modeled mathematically by a quantum tree graph with fluid pressure and flow, and the pipe's internal cross-sectional area as an unknown. The method is based on a simple time reversal boundary control method originally presented by Sondhi and Gopinath for one dimensional problems and later by Oksanen to higher dimensional manifolds.
Recontructing Interactions from Dynamics University of Basel, Switzerland
Simple interactions between particles, people, or neurons give rise to astoundigly complex dynamics on the underlying interaction graphs. I will describe a class of models for dynamical systems on graphs which seems to provide an accurate description for a variety of phenomena from diverse domains. I will then show how this "deep graph dynamics prior" leads to an algorithm to reconstruct the unknown interaction graph when only the dynamics are observed. Potential applications in physics, publich health, Earth science, and neuroscience are important and numerous.
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| 4:00pm - 6:00pm | MS30 3: Inverse Problems on Graphs and Machine Learning Location: VG2.103 Session Chair: Emilia Lavie Kyllikki Blåsten Session Chair: Matti Lassas Session Chair: Jinpeng Lu |
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Learned Solvers for Forward and Backward Image Flow Schemes 1University College London, United Kingdom; 2University of Oulu, Finland; 3Improbable, United Kingdom
It is increasingly recognised that there is a close relationship between some network architectures and iterative solvers for partial differential equations. In this talk we present a network architecture for forward and inverse problems in non-linear diffusion. By design the architecture is non-linear, learning an anisotropic diffusivity function for each layer from the output of the previous layer. The performed updates are explicit, by which we obtain better interpretability and generalisability compared to classical architectures. Since backward diffusion is unstable, a learned regularisation is implicitly learned to stabilise this process.
We test results on synthetic image data sets that have undergone edge-preserving diffusion and on experimental data of images view through variable density scattering media.
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| Date: Friday, 08/Sept/2023 | |
| 1:30pm - 3:30pm | MS59 1: Advanced Reconstruction and Phase Retrieval in Nano X-ray Tomography Location: VG2.103 Session Chair: Tim Salditt Session Chair: Anne Wald |
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Resolution of reconstruction from discrete Radon transform data University of Central Florida, United States of America
In this talk we overview recent results on the analysis of resolution of reconstruction from discrete Radon transform data. We call our approach Local Resolution Analysis, or LRA. LRA yields simple formulas describing the reconstruction from discrete data in a neighborhood of the singularities of $f$ in a variety of settings. We call these formulas the Discrete Transition Behavior (DTB). The DTB function provides the most direct, fully quantitative link between the data sampling rate and resolution. This link is now established for a wide range of integral transforms, conormal distributions $f$, and reconstruction operators. Recently the LRA was generalized to the reconstruction of objects with rough edges. Numerical experiments demonstrate that the DTB functions are highly accurate even for objects with fractal boundaries.
Deep Learning for Reconstruction in Nano CT 1Saarland University, Germany; 2University of Göttingen; 3University of Stuttgart, Geramany; 4University of Bremen, Germany
Tomographic X-ray imaging on the nano-scale is an important tool to visualise 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.
An iterative reconstruction method called RESESOP-Kaczmarz was introduced in [1] which requires the motion to be estimated. However, since the motion is hard to estimate and its incorporation into the reconstruction process strongly increases the numerical effort, we investigate a learned version of RESESOP-Kaczmarz. Imaging data was programmatically simulated to train a deep network which unrolls the iterative image reconstruction of the original algorithm. The network therefore learns the back-projected image after a fixed number of iterations.
[1] S. E. Blanke et al. Inverse problems with inexact forward operator: iterative regularization and application in dynamic imaging, Inverse Problems 36 124001, 2020.
Learned post-processing approaches for nano-CT reconstruction 1University of Bremen, Germany; 2Saarland University, Germany; 3Institute for Numerical and Applied Mathematics, University of Göttingen, Germany
X-ray computed tomography on the nano-meter scale is a challenging imaging task.
Tiny perturbations, such as environmental vibrations, technology imprecision or a thermal drift over time, lead to considerable deviations in the measured projections for nano-CT.
Reconstruction algorithms must take into account the presence of these deviations in order to avoid strong artifacts.
We study different learned post-processing approaches for nano-CT reconstruction on simulated datasets featuring relative object shifts and rotations.
The initial reconstruction is provided by a classical method (FBP, Kaczmarz) or a deviation-aware method (Dremel method, RESESOP-Kaczmarz).
Neural networks are then trained supervisedly to post-process such initial reconstructions.
We consider (i) a directly trained U-Net post-processing, and (ii) conditional normalizing flows, which learn an invertible mapping between a simple random distribution and the image reconstruction space, conditioned on the initial reconstruction.
Normalizing flows do not only yield a reconstruction, but also an estimate of the posterior density.
As a simple indicator of reconstruction uncertainty one may evaluate the pixel-wise standard deviation over samples from the estimated posterior.
X-ray phase and dark-field retrieval from propagation-based images, via the Fokker-Planck Equation 1School of Physics and Astronomy, Monash University, Australia; 2School of Physical and Chemical Sciences, University of Canterbury, New Zealand
Conventional x-ray imaging, which measures the intensity of the transmitted x-ray wavefield, is extremely useful when imaging strongly-attenuating samples like bone, but of limited use when imaging weakly-attenuating samples like the lungs or brain. In recent years, it has been seen that the phase of the transmitted x-ray wavefield contains useful information about these weakly-attenuating samples, however it is not possible to directly measure x-ray phase. This necessitates the use of mathematical models that relate the observed x-ray intensity, which is measurable, to the x-ray phase. These models can then be solved to retrieve how the sample has changed the x-ray phase; the inverse problem of phase retrieval. A widely adopted example is the use of the Transport of Intensity Equation (TIE) to retrieve x-ray phase from an intensity image collected some distance downstream of the sample, a distance at which sample-induced phase variations have resulted in self-interference of the wave and changed the local observed intensity [1]. The use of a single-exposure ‘propagation-based’ set-up like this, where no optics (gratings, crystals etc.) are required, makes for a robust and simple x-ray imaging set-up, which is also compatible with time-sequence imaging.
In this talk, we present an extension to the TIE, the X-ray Fokker-Planck Equation [2, 3], and associated novel retrieval algorithms for extracting x-ray phase and dark-field [5-7] from propagation-based images.
The TIE describes how x-ray intensity evolves with propagation from the sample to a downstream camera, for a wavefield with given phase and intensity. The X-ray Fokker-Planck Equation adds an additional term that incorporates how dark-field effects from the sample will be seen in the observed intensity [2,3]. X-ray dark-field effects are present when the sample contains microstructures that are not directly resolved, but which scatter the wavefield in such a way as to locally reduce image contrast. Examples of dark-field-inducing microstructure include powders, carbon fibres and the air sacs in the lungs. Until very recently [4], it was considered that crystals or gratings were required optical elements in the experimental set-up in order to capture a dark-field image.
Using the X-ray Fokker-Planck Equation, we have derived several novel algorithms that allow dark-field retrieval from propagation-based images. Because phase and dark-field effects evolve differently with propagation, images captured at two different sample-to-detector distances allow the separation and retrieval of dark-field images and phase images [5]. Alternatively, dark-field and phase images can be retrieved by looking at sample-induced changes in a patterned illumination via a Fokker-Planck approach, either using a single short exposure [6], or by scanning the pattern across the sample to access the full spatial resolution of the detector [7]. Incorporating dark-field effects in the TIE not only allows a dark-field image to be extracted from propagation-based images, but also increases the potential spatial resolution of the retrieved phase image. These propagation-based Fokker-Planck approaches are best suited to small samples (e.g. under 10 cm), so may provide avenues for fast and simple phase and dark-field micro/nano-tomography.
[1] D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins. Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object, Journal of Microscopy 206(1): 33-40, 2002.
[2] K. S. Morgan, D. M. Paganin. Applying the Fokker–Planck equation to grating-based x-ray phase and dark-field imaging, Scientific Reports 9(1): 17465, 2019.
[3] D.M. Paganin, K. S. Morgan. X-ray Fokker–Planck equation for paraxial imaging, Scientific Reports, 9(1): 17537, 2019.
[4] T.E. Gureyev, D.M. Paganin, B. Arhatari, S. T. Taba, S. Lewis, P. C. Brennan, H. M. Quiney. Dark-field signal extraction in propagation-based phase-contrast imaging, Physics in Medicine & Biology, 65(21): 215029, 2020.
[5] T. A. Leatham, D. M. Paganin, K. S. Morgan. X-ray dark-field and phase retrieval without optics, via the Fokker–Planck equation, IEEE Transactions on Medical Imaging (in press), 2023.
[6] M. A. Beltran, D. M. Paganin, M. K. Croughan, K. S. Morgan. Fast implicit diffusive dark-field retrieval for single-exposure, single-mask x-ray imaging, Optica, 10(4): 422-429, 2023.
[7] S. J. Alloo, K. S. Morgan, D. M. Paganin, K. M. Pavlov. Multimodal intrinsic speckle-tracking (MIST) to extract images of rapidly-varying diffuse X-ray dark-field, Scientific Reports, 13(1): 5424, 2023.
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| 4:00pm - 6:00pm | MS59 2: Advanced Reconstruction and Phase Retrieval in Nano X-ray Tomography Location: VG2.103 Session Chair: Tim Salditt Session Chair: Anne Wald |
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Multi-stage Deep Learning Artifact Reduction for Computed Tomography Leiden University, the Netherlands
Computed Tomography (CT) is a challenging inverse problem that involves reconstructing images from projection data. The CT pipeline typically comprises three stages: 1) acquisition of projection images, 2) transposition of projection images into sinogram images, and 3) computation of reconstruction images. In practice, the projection images are often corrupted, resulting in various imaging artifacts such as noise, zinger artifacts, and ring artifacts in the reconstructed images. Although recent deep learning-based methods have shown promise in reducing noise through post-processing of CT images, they struggle to effectively address globally distributed artifacts along with noise.
Classical artifact reduction methods, on the other hand, have demonstrated success in reducing globally distributed artifacts by targeting individual types of artifacts before the reconstruction stage. These methods operate in the natural domain where the artifacts are most prominent. Inspired by that, we propose to reduce artifacts in all projection, sinogram, and reconstruction stages with deep learning. This approach enables accurate reduction of globally distributed artifacts along with noise, leading to improved CT image quality. Experiments on both simulated and real-world datasets validate the effectiveness of our proposed approach.
Deep learning for phase retrieval from Fresnel diffraction patterns 1Univ. Grenoble Alpes, CNRS, UMR 5525, VetAgro Sup, Grenoble INP, TIMC, 38000 Grenoble, France; 2Univ Lyon, INSA-Lyon, Université Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm, CREATIS UMR 5220, U1206, F-69621 Villeurbanne, France
We present our recent developments in phase retrieval from propagation-based X-ray phase contrast images using deep learning-based approaches. Previously, deep convolutional neural networks had been used as post-processing step to a linear phase retrieval algorithm [1]. In a first approach, we investigated the use of deep convolutional neural networks to directly retrieve phase and amplitude from a propagation distance series of phase contrast images [2]. Due to a structure that seems well adapted to the properties of the phase contrast images, taking into account features at several scales and connecting the corresponding feature maps, we chose the mixed-scale dense network (MS-DN) [3] architecture as network structure. We developed a transfer learning approach where the network is trained on simulated phase contrast images generated from projections of random objects with simple geometric shape.
We showed that the use of a simple pre-processing to transform the input to the image domain improved results, providing some support to the hypothesis that including knowledge of the image formation process in the network improves reconstruction quality. Some work has been done in this direction using generative adversarial networks by introducing a model of the image formation in a CycleGAN network [4].
Going one step further, information on how to solve the phase retrieval problem can be introduced into the neural network, algorithm unrolling being one such approach [5]. In algorithm unrolling, parts of an iterative algorithm, usually the regularization part, are replaced by neural networks. The networks learn the steps of the chosen iterative algorithm. These networks can then be applied in a sequential fashion, making the run-time application very efficient, moving the calculation load from the iterative reconstruction to the off-line training of the networks.
Based on this idea, we proposed the Deep Gauss-Newton network (DGN) [6]. Gauss-Newton type algorithms have been successfully used for phase retrieval from Fresnel diffraction patterns [7]. Inspired by this, we developed an unrolling-type algorithm based on a Gauss-Newton iteration. Both the regularization and the inverse Hessian are replaced by neural networks. The same network is used for each iteration, making the method very economical in terms of network weights. An initial reconstruction is not required; the algorithm can be initialized at zero. It can retrieve simultaneously the phase and attenuation sorption from one single diffraction pattern. We applied the DGN to both simulated and experimental data, for which it substantially improved the reconstruction error and the resolution compared both to the standard iterative algorithm and the MSDN-based method.
Future work includes extension of the algorithms to tomographic and time-resolved imaging, as well as to other imaging problems. Code for both algorithms will be made available through the PyPhase package [8] in a future release.
[1] C. Bai, M. Zhou, J. Min, S. Dand, X. Yu, P. Zhang, T. Peng, B. Yao. Robust contrast-transfer-function phase retrieval via flexible deep learning networks, Opt. Lett. 44 (21), 5141–5144, 2019.
[2] K. Mom, B. Sixou, M. Langer. Mixed scale dense convolutional networks for x-ray phase contrast imaging, Appl. Opt. 61, 2497–2505, 2022.
[3] D. M. Pelt, J. A. Sethian, A mixed-scale dense convolutional neural network for image analysis, Proc. Natl.402 Acad. Sci. USA 115, 254–259, 2018.
[4] Y. Zhang, M. A. Noack, P. Vagovic, K. Fezzaa, F. Garcia-Moreno, T. Ritschel, P. Villanueva-Perez. PhaseGAN: a deep-learning phase-retrieval approach for unpaired datasets, Opt. Express 29, 19593–19604, 2021.
[5] V. Monga, Y. Li, Y. C. Eldar. Algorithm unrolling: Interpretable, efficient deep learning for signal and image processing, IEEE Signal Proc. Mag. 38, 18–44, 2021.
[6] K. Mom, M. Langer, B. Sixou. Deep Gauss-Newton for phase retrieval, Opt. Lett. 48, 1136–1139, 2023.
[7] S. Maretzke, M. Bartels, M. Krenkel, T. Salditt, T. Hohage. Regularized newton methods for X-ray phase contrast and general imaging problems, Opt. Express 24, 6490–6506, 2016.
[8] M. Langer, Y. Zhang, D. Figueirinhas, J.-B. Forien, K. Mom, C. Mouton, R. Mokso, P. Villanueva-Perez. PyPhase – a Python package for X-ray phase imaging, J. Synch. Radiat. 28, 1261–1266, 2021.
Time resolved and multi-resolution tomographic reconstruction strategies in practice. 1DTU Physics, Technical University of Denmark, Lyngby, Denmark; 2Advanced Photon Source, Argonne National Laboratory, Lemont, IL, USA
A collimated X-ray beam is the trademark of synchrotron X-ray sources and comes with certain benefits for tomography, namely the simplicity of parallel beam tomographic reconstruction. Building on this a number of new approaches emerges to reconstruct a 3D volume from truncated X-ray projections. I will mainly consider here truncation in the time domain. One specificity of imaging at synchrotron instruments is that individual angular projections are acquired on a sub-ms time-frame and the entire tomographic dataset in a fraction of a second [1,2]. This enables time resolved studies of dynamic processes at the micrometer spatial and sub-second temporal resolution. Despite this fast acquisition the sample is often evolving at a faster rate, giving rise to motion artefact in the reconstructed volume. One possible approach to reconstruct an artifact-free 3D volume from (in the traditional sense) inconsistent projections is to use the concept of compressed sensing in the way that data in the temporal direction is represented by a linear combination of appropriate basis functions [3]. In our approach we perform L1 norm minimization for the gradient in both spatial and temporal variables. The optimal choice of basis functions is case specific and is the matter of further investigation.
Multiresolution acquisition is an attractive tomographic approach, but comes with it’s own challenges. I will discuss an approach to merge high and low resolution datasets of the same sample [4] for the extension of the reconstructed volume.
[1] R. Mokso, D.A. Schwyn, S.M. Walker et al. Four-dimensional in vivo X-ray microscopy with projection guided gating, Scientific Reports 5 (1), 8727, 2015.
[2] F. Garcia-Moreno et al. Using X-ray tomoscopy to explore the dynamics of foaming metal, Nature Communications. 10(1), 3762, 2019.
[3] V. Nikitin, M. Carlsson, F. Andersson, R. Mokso. Four-dimensional tomographic reconstruction by time domain decomposition, IEEE Transaction on Computational Imaging 5(3), 409, 2019.
[4] L. Varga, R. Mokso. Iterative High Resolution Tomography from Combined High-Low Resolution Sinogram Pairs, Proceedings of International Workshop on Combinatorial Image Analysis, 150–163, 2018.
Tomographic Reconstruction in X-ray Near-field Diffractive Imaging: from Laboratory $\mu$CT to Synchrotron Nano-Imaging Georg-August-Universität Göttingen, Germany
X-rays can provide information about the structure of matter, on multiple length scales from bulk materials to nanoscale devices, from organs to organelle, from the organism to macromolecule. Due to the widespread lack of suitable lenses, the majority of investigations are rather indirect – apart from classical shadow radiography perhaps. While diffraction problems have been solved since long, the modern era has brought about lensless coherent imaging with X-rays, down to the nanoscale. How can we address and implement optimized tomography solutions for phase contrast inhouse and synchrotron data, taking into account partial coherence, propagation and cone beam geometry? We show how solutions and algorithms of mathematics of inverse problems [1-3] help us to meet the challenges of phase retrieval, tomographic reconstruction, and more generally image processing of bulky data. We also include illustrative bioimaging projects such as mapping the human brain [4,6] of fighting infectious diseases [6].
References:
[1] T. Salditt, A. Egner, R. D. Luke (Eds.)
Nanoscale Photonic Imaging
Springer Nature, TAP, 134, Open Access Book, 2020.
[2] L. M. Lohse, A.-L. Robisch, M. Töpperwien, S. Maretzke, M. Krenkel, J. Hagemann, T. Salditt
A phase-retrieval toolbox for X-ray holography and tomography
Journal of Synchrotron Radiation, 27, 3, 2020.
[3] S. Huhn, L.M. Lohse, J. Lucht, T. Salditt.
Fast algorithms for nonlinear and constrained phase retrieval in near-field X-ray holography based on Tikhonov regularization - arXiv preprint arXiv:2205.01099, 2022.
[4] M. Eckermann, B. Schmitzer, F. van der Meer, J. Franz, O. Hansen, C. Stadelmann, T. Salditt.
Three-dimensional virtual histology of the human hippocampus based on phase-contrast computed tomography
Proc. Natl. Acad. Sci., 118, 48, e2113835118, 2021.
[5] M. Eckermann, J. Frohn, M. Reichardt, M. Osterhoff, M. Sprung, F. Westermeier, A.Tzankov, C. Werlein, M. Kuehnel, D. Jonigk, T. Salditt.
3d Virtual Patho-Histology of Lung Tissue from Covid-19 Patients based on Phase Contrast X-ray Tomography,
eLife, 9:e60408, 2020.
[6] M. Reichardt, P.M. Jensen, V.A. Dahl, A.B. Dahl, M. Ackermann, H. Shah, F. Länger, C. Werlein, M.P. Kuehnel, D. Jonigk, T. Salditt.
3D virtual histopathology of cardiac tissue from Covid-19 patients based on phase-contrast X-ray tomography,
eLife, 10:e71359, 2021.
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