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: VG1.101 |
| Date: Monday, 04/Sept/2023 | |
| 1:30pm - 3:30pm | MS54 1: The x-ray transform and its generalizations: Theory, methods, and applications Location: VG1.101 Session Chair: Suman Kumar Sahoo |
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Unique continuation for the momentum ray transform 1University of Jyväskylä, Finland; 2National Chengchi University, Taiwan
We will explain the relation between momentum ray transform and the fractional Laplacian. As a consequence of unique continuation property of the fractional Laplacian, one can discuss similar properties for momentum ray transform (of integer and fractional order). In addition, we will also explain the relation between the weighted ray transform and the cone transform, which appeared in different imaging approaches (for example, Compton camera).
This talk is prepared based on the work [1].
[1] J. Ilmavirta, P.-Z. Kow, S. K. Sahoo. Unique continuation for the momentum ray transform, arXiv: 2304.00327, 2023.
The linearized Calderon problem for polyharmonic operators University of Jyvaskyla, Finland
The density of products of solutions to several types of partial differential equations, such as elliptic, parabolic, and hyperbolic equations, plays a significant role in solving various inverse problems, dating back to Calderon's fundamental work. In this talk, we will discuss some density properties of solutions to polyharmonic operators on the space of symmetric tensor fields. These density questions are closely related to the linearized Calderon problem for polyharmonic operators. This is joint work with Mikko Salo.
The star transform and its links to various areas of mathematics 1University of Texas at Arlington, United States of America; 2Dartmouth College, United States of America
The divergent beam transform maps a function $f$ in $\mathbb{R}^n$ to an $n$-dimensional family of its integrals along rays, emanating from a variable vertex location inside the support of $f$. The star transform is defined as a linear combination of divergent beam transforms with known coefficients. The talk presents some recent results about the properties of the star transform, its inversion, and a few interesting connections to different areas of mathematics.
Linearized Calderon problem for biharmonic operator with partial data 1Tata Institute of Fundamental Research (TIFR), India; 2University of Jyvaskyla, Finland
The product of solutions of the Laplace equation vanishing on an arbitrary closed subset of the boundary is dense in the space of integrable functions ([1]). In this talk, we will discuss a similar problem with the biharmonic operator replacing the Laplace operator. More precisely, we show that the integral identity
$$
\int \left [a \Delta u v+ \sum a^1_i \partial_i u v + a^0 u v \right ]\, \mathrm{d}x = 0
$$
for all biharmonic functions vanishing on an proper closed subset of the boundary implies that $(a, a^1, a^0)$ vanish identically. This work is a collaboration with R. S. Jaiswal and S. K. Sahoo.
[1] D. Ferreira, C. E. Kenig, J. Sjostrand, G. Uhlmann. On the linearized local Calderon problem. Math. Res. Lett. 16: 955-970, 2009.
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| 4:00pm - 6:00pm | MS54 2: The x-ray transform and its generalizations: Theory, methods, and applications Location: VG1.101 Session Chair: Suman Kumar Sahoo |
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Inversion of the momenta X-ray transform of symmetric tensor fields in the plane Johann Radon Institute (RICAM), Austria
The X-ray transform of symmetric tensor fields recovers the tensor field only up to a potential field. In 1994, V. Sharafutdinov showed that augmenting the X-ray data with several momentum-ray transforms establishes uniqueness, with a most recent work showing stability of the inversion. In this talk, I will present a different approach to stably reconstruct symmetric tensor fields compactly supported in the plane.
The method is based on the extension of Bukhgeim's theory to a system of $A$-analytic maps.
This is joint work with H. Fujiwara, D. Omogbhe and A. Tamasan.
Simultaneous recovery of attenuation and source density in SPECT and multibang regularisation University of Manchester, United Kingdom
I will discuss results about simultaneous recovery of the attenuation and source density in the SPECT inverse problem, which is given mathematically by the attenuated ray transform. Assuming the attenuation is piecewise constant and the source density piecewise smooth we show that, provided certain conditions are satisfied, it is possible to uniquely determine both. I will also discuss a numerical algorithm that allows for determination of both parameters in the case when the range of the piecewise constant attenuation is known and look at some synthetic numerical examples. This is based on joint work with Philip Richardson.
Inversion of a restricted transverse ray transform on symmetric $m$-tensor fields in $\mathbb{R}^3$ Indian Institute of Technology Gandhinagar, India
In this work, we study a restricted transverse ray transform on symmetric $m$-tensor fields in $\mathbb{R}^3$ and provide an explicit inversion algorithm to recover the unknown $m$-tensor field. We restrict the transverse ray transform to all lines passing through a fixed curve $\gamma$ satisfying the Kirillov-Tuy condition. This restricted data is used to find the weighted Radon transform of components of the unknown tensor field, which we use to recover components of the tensor field explicitly.
Inverse problems, unique continuation and the fractional Laplacian University of Cambridge, United Kingdom
The Calderón problem is a famous nonlinear model inverse problem: Do voltage and current measurements on the boundary of an object determine its electric conductivity uniquely? X-ray computed tomography is a famous linear model inverse problem studied via Radon transforms. We discuss how the fractional Laplacians pop up in the analysis of Radon transforms. We then discuss recent results on the unique continuation of the fractional Laplacians and the related Caffarelli-Silvestre extension problem for $L^p$ functions. We explain some of the implications to the analysis of Radon transforms with partial data and its further generalizations. Finally, we discuss the role of unique continuation in recent mathematical studies of the Calderón problem to nonlocal equations.
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| Date: Tuesday, 05/Sept/2023 | |
| 1:30pm - 3:30pm | MS22 1: Imaging with Non-Linear Measurements: Tomography and Reconstruction from Phaseless or Folded Data Location: VG1.101 Session Chair: Matthias Beckmann Session Chair: Robert Beinert Session Chair: Michael Quellmalz |
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Gradient Methods for Blind Ptychography Helmholtz Center Munich, Germany
Ptychography is an imaging technique, the goal of which is to reconstruct the object of interest from a set of diffraction patterns obtained by illuminating its small regions. When the distribution of light within the region is known, the recovery of the object from ptychographic measurements becomes a special case of the phase retrieval problem. In the other case, also known as blind ptychography, the recovery of both the object and the distribution is necessary.
One of the well-known reconstruction methods for blind ptychography is extended Ptychographic Iterative Engine. Despite its popularity, there was no known analysis of its performance. Based on the fact that it is a stochastic gradient descent method, we derive its convergence guarantees if the step sizes are chosen sufficiently small. The second considered method is a generalization of the Amplitude Flow algorithm for phase retrieval, a gradient descent scheme for the minimization of the amplitude-based squared loss. By applying an alternating minimization procedure, the blind ptychography is reduced to phase retrieval subproblems, which can be solved by performing a few steps of Amplitude Flow. The resulting procedure converges to a critical point at a sublinear rate.
Inversion of the Modulo Radon Transform via Orthogonal Matching Pursuit University of Bremen, Germany
In the recent years, the topic of high dynamic range (HDR) tomography has started to gather attention due to recent advances in the hardware technology. The issue is that registering high-intensity projections that exceed the dynamic range of the detector cause sensor saturation, which, in turn, leads to a loss of information. Inspired by the multi-exposure fusion strategy in computational photography, a common approach is to acquire multiple Radon Transform projections at different exposure levels that are algorithmically fused to facilitate HDR reconstructions.
As opposed to this, a single-shot alternative to the multi-exposure fusion approach has been proposed in our recent line of work which is based on the Modulo Radon Transform, a novel generalization of the conventional Radon transform. In this case, Radon Transform projections are folded via a modulo non-linearity, which allows HDR values to be mapped into the dynamic range of the sensor and, thus, avoids saturation or clipping. The folded measurements are then mapped back to their ambient range using reconstruction algorithms.
In this talk we introduce a novel Fourier domain recovery method, namely the OMP-FBP method, which is based on the Orthogonal Matching Pursuit (OMP) algorithm and Filtered Back Projection (FBP) formula. The proposed OMP-FBP method offers several advantages; it is agnostic to the modulo threshold or the number of folds, can handle much lower sampling rates than previous approaches and is empirically stable to noise and outliers. The effectivity of the OMP-FBP recovery method is illustrated by numerical experiments.
This talk is based on joint work with Ayush Bhandari (Imperial College London).
Phaseless sampling of the short-time Fourier transform University of Vienna, Austria
We discuss recent advances in phaseless sampling of the short-time Fourier transform (STFT). The main focus of the talk lies in the question if phaseless samples of the STFT contain enough information to recover signals belonging to infinite-dimensional function spaces. It turns out, that this problem differs from ordinary sampling in a rather fundamental and subtle way: if the sampling set is a lattice then uniqueness is unachievable, independent of the choice of the window function and the density of the lattice. Based on this discretization barrier, we present possible ways to still achieve unique recoverability from samples: lattice-uniqueness is possible if the signal class gets restricted to certain proper subspaces of $L^2(\mathbb R)$, such as the class of compactly-supported functions or shift-invariant spaces. Finally, we highlight that sampling on so-called square-root lattices achieves uniqueness in $L^2(\mathbb R)$.
Phase Retrieval in Optical Diffraction Tomography TU Berlin, Germany
Optical diffraction tomography (ODT) consists in the recovery of the three-dimensional scattering potential of a microscopic object rotating around its center from a series of illuminations with coherent light. Standard reconstruction algorithms such as the filtered backpropagation require knowledge of the complex-valued wave at the measurement plane. In common physical measurement setups, the collected data only consists in intensities; so only phaseless measurements are available. To overcome the loss of the required phases, we propose a new reconstruction approach for ODT based on three key ingredients. First, the light propagation is modeled using Born's approximation enabling us to use the Fourier diffraction theorem. Second, we stabilize the inversion of the non-uniform discrete Fourier transform via total variation regularization utilizing a primal-dual iteration, which also yields a novel numerical inversion formula for ODT with known phase. The third ingredient is a hybrid input-output scheme. We achieve convincing numerical results showing that the computed 2D and 3D reconstructions are even comparable to the ones obtained with known phase. This indicate that ODT with phaseless data is possible.
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| 4:00pm - 6:00pm | MS22 2: Imaging with Non-Linear Measurements: Tomography and Reconstruction from Phaseless or Folded Data Location: VG1.101 Session Chair: Matthias Beckmann Session Chair: Robert Beinert Session Chair: Michael Quellmalz |
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Uniqueness theory for 3D phase retrieval and unwrapping UC Davis, United States of America
We present general measurement schemes with which unique conversion of diffraction patterns into the phase projection for a strong phase object (respectively, the projection for a weak phase object) in each direction can be guaranteed without the knowledge of relative orientations and locations.
This approach has the practical implication of enabling classification and alignment, when relative orientations are unknown, to be carried out in terms of (phase) projections, instead of diffraction patterns.
We also discuss conditions for unique determination of a strong phase object from its phase projection data, including, but not limited to, random tilt schemes densely sampled from a spherical triangle of vertexes in three orthogonal directions and other deterministic tilt schemes.
Interaction Models in Ptychography Helmholtz Munich, Germany
Over the recent years, ptychography became a standard technique for high resolution scanning transmission electron microscopy.
To achieve better and better resolutions, the mathematical model had to be refined several times.
In the simplest approach the measurements can be understood as a discrete, phaseless short-time Fourier transform
$$ I(s) = \left| \mathcal F [\phi \cdot \tau_s w]\right|^2. $$
Here, $\tau_s w$ is an (often unknown) window function, shifted by $s$, $\phi$ the object we would like to recover and $I(s)$
the measured intensity at position $s$. Typically, a few thousands of such measurements are recorded, where $s$ lies on a regular grid. However, for specimens thicker than a few nanometers, this approximation already breaks down.
A more sophisticated interaction model $M(\phi, \tau_s w)$ is needed.
Furthermore, the incoherence of the microscope is a crucial limit and has to be considered as well.
We end up with a model like
$$ I(s) = \sum_j \left| \mathcal F [M( \phi, \tau_s w_j)]\right|^2. $$
In this talk we give an overview over these approaches and discuss their challenges.
We also show reconstructions of experimental data.
Tackling noise in multiple dimensions via hysteresis modulo sampling Imperial College London, United Kingdom
Mapping a multi-dimensional function in a predefined bounded amplitude range can be achieved via a transformation known as the modulo nonlinearity. The recovery of the function from the modulo samples was addressed for the one-dimensional case as part of the Unlimited Sensing Framework (USF) based on uniform samples [1-3], but also based on neuroscience inspired time encoded samples [4]. Alternative analyses implemented de-noising of modulo data in one and multiple dimensions [5]. Extensions of the recovery to multi-dimensional inputs typically amount to a line-by-line analysis of the data on one-dimensional slices. Apart from enabling the reconstruction of a wider class of inputs, this approach does not show an inherent need to apply modulo for high dimensional inputs.
In this talk, we present a modulo sampling operator specifically tailored to multiple dimensional inputs, called multi-dimensional modulo-hysteresis [6]. It is shown that the model can use dimensions two and above to generate redundancy that can be exploited for robust input recovery. This redundancy is particularly made possible by the hysteresis parameter of the operator. A few properties of the new operator are proven, followed by a guaranteed input recovery approach. We demonstrate theoretically and via numerical examples that when the input is corrupted by Gaussian noise the reconstruction error drops asymptotically to 0 for high enough sampling rates, which was not possible for the one-dimensional scenario. We additionally extend the recovery guarantees to classes of non-bandlimited inputs from shift-invariant spaces and include additional simulations with different noise distributions. This work enables extensions to multi-dimensional inputs for neuroscience inspired sampling schemes [4], inherently known for their noisy characteristics.
[1] Bhandari, F. Krahmer, R. Raskar. On unlimited sampling and reconstruction. IEEE Trans. Sig. Proc. 69 (2020) 3827–3839. doi:10.1109/tsp.2020.3041955
[2] Bhandari, F. Krahmer, T. Poskitt. Unlimited sampling from theory to practice: Fourier-Prony recovery and prototype ADC. IEEE Trans. Sig. Proc. (2021) 1131-1141. doi:10.1109/TSP.2021.3113497.
[3] D. Florescu, F. Krahmer, A. Bhandari. The surprising benefits of hysteresis in unlimited sampling: Theory, algorithms and experiments. IEEE Trans. Sig. Proc. 70 (2022) 616–630. doi:10.1109/tsp.2022.3142507
[4] D. Florescu, A. Bhandari. Time encoding via unlimited sampling: theory, algorithms and hardware validation. IEEE Trans. Sig. Proc. 70 (2022) 4912-4924.
[5] H. Tyagi. Error analysis for denoising smooth modulo signals on a graph. Applied and Computational Harmonic Analysis 57 (2022) 151–184.
[6] D. Florescu, A. Bhandari. Multi-Dimensional Unlimited Sampling and Robust Reconstruction. arXiv preprint 2002 arXiv:2209.06426.
Multi-window STFT phase retrieval University of Vienna, Austria
We consider the problem of recovering a function $f\in L^2(\mathbb{R})$ (up to a multiplicative phase factor) from phase-less samples of its short-time Fourier transform $V_g f$, where
$$
V_g f(x,y) =\int_\mathbb{R} f(t) \overline{g(t-x)} e^{-2\pi i y t}\,dt,
$$
with $g\in L^2(\mathbb{R})$ a window function. Recently established dicretization barriers state that in general $f$ is not uniquely determined given $|V_g f(\Lambda)|:=\{|V_g f(\lambda)|, \lambda\in\Lambda\}$ if $\Lambda\subseteq \mathbb{R}^2$ is a lattice (irrespectively of the choice of the window $g$ and the density of the lattice $\Lambda$).
We show that these discretization barriers can be overcome by employing multiple window functions. More precisely, we prove that
$$
\{|V_{g_1} f(\Lambda)|, |V_{g_2} f(\Lambda)|, |V_{g_3} f(\Lambda)|, |V_{g_4} f(\Lambda)|\}
$$
uniquely determines $f\sim e^{i\theta}f$ when $g_1,\ldots,g_4$ are suitably chosen windows provided that $\Lambda$ has sufficient density.
Joint work with Philipp Grohs and Lukas Liehr.
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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | MS22 3: Imaging with Non-Linear Measurements: Tomography and Reconstruction from Phaseless or Folded Data Location: VG1.101 Session Chair: Matthias Beckmann Session Chair: Robert Beinert Session Chair: Michael Quellmalz |
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Phase retrieval from time-frequency structured data ETH Zurich, Switzerland
Certain imaging and audio processing applications require the reconstruction of an image or signal from its phaseless time-frequency or time-scale representation, such as the magnitude of its Gabor or wavelet transform.
Such problems are inherently unstable, however, we formulate a relaxed notion of solution, meaningful for audio processing applications, under which stability can be restored.
The question of uniqueness becomes particularly delicate in the sampled setting. There, we show the first result evidencing the fundamental non-uniqueness property of phase retrieval from Gabor transform measurements. By restricting to appropriate function classes, positive results on the uniqueness can be obtained.
Furthermore, we present our most recent result which establishes uniqueness of phase retrieval from sampled wavelet transform measurements, without restricting the function class, when 3 wavelets are employed.
Computational Imaging from Structured Noise Imperial College London, United Kingdom
Almost all modern day imaging systems rely on digital capture of information. To this end, hardware and consumer technologies strive for high resolution quantization based acquisition. Antithetical to folk wisdom, we show that sampling quantization noise results in unconventional advantages in computational sensing and imaging. In particular, this leads to a novel, single-shot, high-dynamic-range imaging approach. Application areas include consumer and scientific imaging, computed tomography, sensor array imaging and time-resolved 3D imaging. In each case, we present a mathematically guaranteed recovery algorithm and also demonstrate a first hardware prototype for basic digital acquisition of quantization noise.
Phase retrieval framework for direct reconstruction of the projected refractive index applied to ptychography and holography 1CXNS — Center for X-ray and Nano Science CXNS, Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany; 2Current address: NERSC, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA; 3Department Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
The interaction of an object with a coherent (x-ray) probe often encodes its properties in a complex-valued function, which is then detected in an intensity-only measurement. Phase retrieval methods commonly infer this complex-valued function from the intensity. However, the decoding of the object from the complex-valued function often involves some ambiguity in the phase, e.g., when the phase shift in the object exceeds $2\pi$. Here, we present a phase retrieval framework to directly recover the amplitude and phase of the object. This refractive framework is straightforward to integrate into existing algorithms.
As examples, we introduce refractive algorithms for ptychography and near-field holography and demonstrate this method using measured data.
Zero-optics X-ray dark-field imaging using dual energies 1Monash University, Australia; 2University of Canterbury, New Zealand; 3University of New England, Australia
Traditional X-ray imaging achieves contrast using the attenuation of photons, making differentiation of materials of a similar density difficult. Improvements in the coherence of X-ray sources opened the way for phase changes in a material to be measured in an intensity image. In addition, the scattered component of the X-ray beam has been probed in X-ray dark-field imaging. Novel dark-field imaging techniques show promise in the detection and assessment of samples with significant micro-scale porosity, such as human lungs. Advanced dark-field imaging techniques rely on measuring sample-induced deviations on a patterned and interferometrically probed beam, requiring a highly-stable set-up and multiple exposures. Propagation-based imaging (PBI) is an experimentally-simple phase-contrast imaging technique, which relies on the downstream interference of refracted and diffracted coherent X-rays to reconstruct sample phase. Recently, PBI has been extended to dark-field reconstruction by modelling the downstream intensity using an X-ray imaging version of the Fokker-Planck diffusion equation [1, 2]. Separating the effects of refraction and diffusion on the beam requires multiple measurements, which was first achieved by imaging the sample at multiple propagation distances [3]. A multi-energy beam creates another possibility; the recent proliferation of energy-discriminating photon-counting detectors has led to an increased interest in spectral methods of coherent X-ray imaging [4]. In this talk we present the first results of inverting and solving the Fokker-Planck equation using spectral information, under assumption of a single-material sample. A linearised model is used to reconstruct sample projected thickness and dark-field in simulated and measured images. Strong attenuation energy-dependence presents challenges in reconstruction when deviating from strict single-material samples. We discuss Fokker-Planck dark-field reconstruction, and present a hybrid approach to the inverse problem, based on treating the post-sample wavefront as pseudo-patterned intensity, which improves stability in multi-material samples. Exploiting spectral dependence to reconstruct phase and dark-field would allow for imaging without having to move any part of the set-up, and would enable single-exposure imaging when combining a polychromatic source with an energy-discriminating detector. This would avoid registration issues, reduce the required dose, and open the door for time-resolved propagation-based dark-field imaging and fast CT.
[1] 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.
[2] D. M. Paganin, K. S. Morgan. X-ray Fokker-Planck equation for paraxial imaging, Scientific Reports 9(1): 17537, 2019.
[3] 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, 2023.
[4] F. Schaff, K. S. Morgan, J. A. Pollock, L. C. P. Croton, S. B. Hooper, & M. J. Kitchen. Material Decomposition Using Spectral Propagation-Based Phase-Contrast X-Ray Imaging, IEEE Transactions on Medical Imaging 39(12): 3891–3899, 2020.
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | MS24 1: Learned Regularization for Solving Inverse Problems Location: VG1.101 Session Chair: Johannes Hertrich Session Chair: Sebastian Neumayer |
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The Power of Patches for Training Normalizing Flows 1Humboldt-University Berlin, Germany; 2Technical University Berlin, Germany; 3University Bremen, Germany
We introduce two kinds of data-driven patch priors learned from very few images: First, the Wasserstein patch prior penalizes the Wasserstein-2 distance between the patch distribution of the reconstruction and a possibly small reference image. Such a reference image is available for instance when working with materials’ microstructures or textures. The second regularizer learns the patch distribution using a normalizing flow. Since already a small image contains a large number of patches, this enables us to train the regularizer based on very few training images. For both regularizers, we show that they induce indeed a probability distribution such that they can be used within a Bayesian setting. We demonstrate the performance of patch priors for MAP estimation and posterior sampling within Bayesian inverse problems. For both approaches, we observe numerically that only very few clean reference images are required to achieve high-quality results and to obtain stability with respect to small perturbations of the problem.
Trust your source: quantifying source condition elements for variational regularisation methods 1Queen Mary University of London, United Kingdom; 2University of Bath, United Kingdom; 3University of Genoa, Italy; 4The Alan Turing Institute, United Kingdom
Source conditions are a key tool in variational regularisation to derive error estimates and convergence rates for ill-posed inverse problems. In this paper, we provide a recipe to practically compute source condition elements as the solution of convex minimisation problems that can be solved with first-order algorithms. We demonstrate the validity of our approach by testing it for two inverse problem case studies in machine learning and image processing: sparse coefficient estimation of a polynomial via LASSO regression and recovery of an image from a subset of the coefficients of its Fourier transform. We further demonstrate that the proposed approach can easily be modified to solve the learning task of identifying the optimal sampling pattern in the Fourier domain for given image and variational regularisation method, which has applications in the context of sparsity promoting reconstruction from magnetic resonance imaging data. We conclude by presenting a methodology with which data-driven regularisations with quantitative error estimates can be designed and trained.
Plug-and-Play image reconstruction is a convergent regularization method University of Innsbruck, Austria
Non-uniqueness and instability are characteristic features of image reconstruction processes. As a result, it is necessary to develop regularization methods that
can be used to compute reliable approximate solutions. A regularization method
provides of a family of stable reconstructions that converge to an exact solution
of the noise-free problem as the noise level tends to zero. The standard regularization technique is defined by variational image reconstruction, which minimizes
a data discrepancy augmented by a regularizer. The actual numerical implementation makes use of iterative methods, often involving proximal mappings of the
regularizer. In recent years, plug-and-play image reconstruction (PnP) has been developed as a new powerful generalization of variational methods based on replacing
proximal mappings by more general image denoisers. While PnP iterations yield
excellent results, neither stability nor convergence in the sense of regularization has
been studied so far. In this work, we extend the idea of PnP by considering families
of PnP iterations, each being accompanied by its own denoiser. As our main theoretical result, we show that such PnP reconstructions lead to stable and convergent
regularization methods. This shows for the first time that PnP is mathematically
equally justified for robust image reconstruction as variational methods.
Provably Convergent Plug-and-Play Quasi-Newton Methods 1University of Cambridge, United Kingdom; 2University of Bath, United Kingdom; 3University of Birmingham, United Kingdom
Plug-and-Play (PnP) methods are a class of efficient data-driven methods for solving imaging inverse problems, wherein one incorporates an off-the-shelf denoiser inside iterative optimization schemes such as proximal gradient descent and ADMM. PnP methods have been shown to yield excellent reconstruction performance and are also supported by convergence guarantees. However, existing provable PnP methods impose heavy restrictions on the denoiser (such as nonexpansiveness) or the fidelity term (such as strict convexity). In this work, we propose a provable PnP method that imposes relatively light conditions based on proximal denoisers and introduce a quasi-Newton step to greatly accelerate convergence. By specially parameterizing the deep denoiser as a gradient step, we further characterize the fixed points of the resulting quasi-Newton PnP algorithm.
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| 4:00pm - 6:00pm | MS24 2: Learned Regularization for Solving Inverse Problems Location: VG1.101 Session Chair: Johannes Hertrich Session Chair: Sebastian Neumayer |
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Learning Sparsifying Regularisers EPFL Lausanne, Switzerland
Solving inverse problems is possible, for example, by using variational models. First, we discuss a convex regularizer based on a one-hidden-layer neural network with (almost) free-form activation functions. Our numerical experiments have shown that this simple architecture already achieves state-of-the-art performance in the convex regime. This is very different from the non-convex case, where more complex models usually result in better performance. Inspired by this observation, we discuss an extension of our approach within the convex non-convex framework. Here, the regularizer can be non-convex, but the overall objective has to remain convex. This maintains the nice optimization properties while allowing to significantly boost the performance. Our numerical results show that this convex-energy-based approach is indeed able to outperform the popular BM3D denoiser on the BSD68 test set for various noise scales.
Shared Prior Learning of Energy-Based Models for Image Reconstruction 1University of Bonn, Germany; 2Technical University of Graz, Austria
In this talk, we propose a novel learning-based framework for image reconstruction particularly designed for training without ground truth data, which has three major building blocks: energy-based learning, a patch-based Wasserstein loss functional, and shared prior learning. In energy-based learning, the parameters of an energy functional composed of a learned data fidelity term and a data-driven regularizer are computed in a mean-field optimal control problem. In the absence of ground truth data, we change the loss functional to a patch-based Wasserstein functional, in which local statistics of the output images are compared to uncorrupted reference patches. Finally, in shared prior learning, both aforementioned optimal control problems are optimized simultaneously with shared learned parameters of the regularizer to further enhance unsupervised image reconstruction. We derive several time discretization schemes of the gradient flow and verify their consistency in terms of Mosco convergence. In numerous numerical experiments, we demonstrate that the proposed method generates state-of-the-art results for various image reconstruction applications - even if no ground truth images are available for training.
Gradient Step and Proximal denoisers for convergent plug-and-play image restoration. University of Bordeaux, France
Plug-and-Play (PnP) methods constitute a class of iterative algorithms for imaging problems where regularization is performed by an off-the-shelf denoiser. Specifically, given an image dataset, optimizing a function (e.g. a neural network) to remove Gaussian noise is equivalent to approximating the gradient or the proximal operator of the log prior of the training dataset. Therefore, any off-the-shelf denoiser can be used as an implicit prior and inserted into an optimization scheme to restore images. The PnP and Regularization by Denoising (RED) frameworks provide a basis for this approach, for which various convergence analyses have been proposed in the literature. However, most existing results require either unverifiable or suboptimal hypotheses on the denoiser, or assume restrictive conditions on the parameters of the inverse problem. We will introduce the Gradient Step and Proximal denoisers, and their variants, recently proposed to restore RED and PnP algorithms to their original form as (nonconvex) real proximal splitting algorithms. These new algorithms are shown to converge towards stationary points of an explicit functional and to perform state-of-the-art image restoration, both quantitatively and qualitatively.
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| Date: Friday, 08/Sept/2023 | |
| 1:30pm - 3:30pm | MS07 1: Regularization for Learning from Limited Data: From Theory to Medical Applications Location: VG1.101 Session Chair: Markus Holzleitner Session Chair: Sergei Pereverzyev Session Chair: Werner Zellinger |
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Regularized Radon - Nikodym differentiation and some of its applications Johann Radon Institute for Computational and Applied Mathematics, Austria
We discuss the problem of estimation of Radon-Nikodym derivatives. This problem appears in various applications, such as covariate shift adaptation, likelihood-ratio testing, mutual information, and conditional probability estimation. To address the above problem, we employ the general regularization scheme in reproducing kernel Hilbert spaces. The convergence rate of the corresponding regularized learning algorithm is established by taking into account both the smoothness of the derivative and the capacity of the space in which it is estimated. It is done in terms of general source conditions and the regularized Christoffel functions. The theoretical results are illustrated by numerical simulations.
Explicit error rate results in the context of domain generalization JKU Linz, Austria
Given labeled data from different source distributions, the problem of domain generalization is to learn a model that is expected to generalize well on new target distributions for which you only have unlabeled samples. We frame domain generalization as a problem of functional regression. This concept leads to a new algorithm for learning a linear operator from marginal distributions of inputs to the corresponding conditional distributions of outputs given inputs. Our algorithm allows a source distribution-dependent construction of reproducing kernel Hilbert spaces for prediction and satisfies non-asymptotic error bounds for the idealized risk. We intend to give a short overview on the required mathematical concepts and proof techinques, and illustrate our approach by a numerical example. The talk is based on [1].
[1] M. Holzleitner, S. V. Pereverzyev, W. Zellinger. Domain Generalization by Functional Regression. arXiv:2302.04724, 2023.
Addressing Parameter Choice Issues in Unsupervised Domain Adaptation by Aggregation JKU Linz, Austria
We study the problem of choosing algorithm hyper-parameters in unsupervised domain adaptation, i.e., with labeled data in a source domain and unlabeled data in a target domain, drawn from a different input distribution. We follow the strategy to compute several models using different hyper-parameters, and, to subsequently compute a linear aggregation of the models. While several heuristics exist that follow this strategy, methods are still missing that rely on thorough theories for bounding the target error. In this turn, we propose a method that extends weighted least squares to vector-valued functions, e.g., deep neural networks. We show that the target error of the proposed algorithm is asymptotically not worse than twice the error of the unknown optimal aggregation. We also perform a large scale empirical comparative study on several datasets, including text, images, electroencephalogram, body sensor signals and signals from mobile phones. Our method outperforms deep embedded validation (DEV) and importance weighted validation (IWV) on all datasets, setting a new state-of-the-art performance for solving parameter choice issues in unsupervised domain adaptation with theoretical error guarantees. We further study several competitive heuristics, all outperforming IWV and DEV on at least five datasets. However, our method outperforms each heuristic on at least five of seven datasets. This talk is based on [1].
[1] M.-C. Dinu, M. Holzleitner, M. Beck, H. D. Nguyen, A. Huber, H. Eghbal-zadeh, B. A. Moser, S. Pereverzyev, S. Hochreiter, W. Zellinger. Addressing Parameter Choice Issues in Unsupervised Domain Adaptation by Aggregation. The Eleventh International Conference on Learning Representations (ICLR), 2023.
Convex regularization in statistical inverse learning problems 1Università degli Studi di Bologna, Italy; 2University of Bath, UK; 3Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany; 4Lappeenranta-Lahti University of Technology, Finland
We consider a problem at the crossroad between inverse problems and statistical learning: namely, the estimation of an unknown function from noisy and indirect measurements, which are only evaluated at randomly distributed design points. This occurs in many contexts in modern science and engineering, where massive data sets arise in large-scale problems from poorly controllable experimental conditions. When tackling this task, a common ground between inverse problems and statistical learning is represented by regularization theory, although with slightly different perspectives.
In this talk, I will present a unified approach, leading to convergence estimates of the regularized solution to the ground truth, both as the noise on the data reduces and as the number of evaluation points increases. I will mainly focus on a class of convex, $p$-homogeneous regularization functionals ($p$ being between $1$ and $2$), which allow moving from the classical Tikhonov regularization towards sparsity-promoting techniques. Particular attention is given to the case of Besov norm regularization, which represents a case of interest for wavelet-based regularization. The most prominent application I will discuss is X-ray tomography with randomly sampled angular views. I will finally sketch some connections with recent extensions of our approach, including a more general family of sparsifying transforms and dynamical inverse problems.
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| 4:00pm - 6:00pm | MS07 2: Regularization for Learning from Limited Data: From Theory to Medical Applications Location: VG1.101 Session Chair: Markus Holzleitner Session Chair: Sergei Pereverzyev Session Chair: Werner Zellinger |
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Imbalanced data sets in a magnetic resonance imaging case study of preterm neonates: a strategy for identifying informative variables Medical University of Innsbruck, Austria
Background and objective: Variable selection is the process of identifying relevant
data characteristics (features, biomarkers) that are predictive of future outcomes.
There is an arsenal of methods addressing the variable selection problem,
but the available techniques may not work on the so-called imbalanced data sets
containing mainly examples of the same outcome. Retrospective clinical data
often exhibit such imbalanceness. This is the case for neuroimaging data derived
from the magnetic resonance images of prematurely born infants used in
attempt to identify prognostic biomarkers of their possible neurodevelopmental
delays, which is the main objective of the present study.
Methods: The variable selection algorithm used in our study scores the combinations
of variables according to the performance of prediction functions involving
these variables. The considered functions are constructed by kernel
ridge regression with various input variables as regressors. As regression kernels
we used universal Gaussian kernels and the kernels adjusted for underlying data
manifolds. The prediction functions have been trained using data that were randomly
extracted from available clinical data sets. The prediction performance
has been measured in terms of area under the Receiver Operating Characteristic Curve, and maximum performance exhibited by prediction functions has
been averaged over simulations. The resultant average value is then assigned
as the performance index associated with the considered combination of input
variables. The variables allowing the largest index value are selected as the
informative ones.
Results: The proposed variable selection strategy has been applied to two
retrospective clinical datasets containing data of preterm infants who received
magnetic resonance imaging of the brain at the term equivalent age and at
around 12 months corrected age with the developmental evaluation. The first
dataset contains data of 94 infants, with 13 of them being later classified as delayed
in motor skills. The second set contains data of 95 infants, with 7 of them
being later classified as cognitively delayed. The application of the proposed
strategy clearly indicates 2 metabolite ratios and 6 diffusion tensor imaging parameters
as being predictive of motor outcome, as well as 2 metabolite ratios and
2 diffusion tensor imaging parameters as being predictive of cognitive outcome.
Conclusion: The proposed strategy demonstrates its ability to extract the
meaningful variables from the imbalanced clinical datasets. The application of
the strategy provides independent evidence supporting several previous studies
separately suggesting different biomarkers. The application also shows that the
predictor involving several informative variables can exhibit better performance
than single variable predictors.
On Approximation for Multi-Source Domain Adaptation in the Space of Copulas 1Institute for Mathematical Methods in Medicine and Data Based Modeling, Johannes Kepler University Linz, Linz, Austria; 2Software Competence Center Hagenberg, Hagenberg, Austria; 3Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria
The set of $d$-copulas $(d \geq 2)$, denoted by $\mathcal{C}_d$ is a compact subspace of $(\Xi(\mathbb{I}^d), d_{\infty})$, the space of all continuous functions with domain $\mathbb{I}^d$; where $\mathbb{I}$ is the unit interval, $d_{\infty}(f_1,f_2)=\underset{u \in \mathbb{I}^d} {\text{sup}}|f_1(\textbf{u})-f_2(\textbf{u})|$ and the function $C:\mathbb{I}^d\to \mathbb{I}$ is a $d$-copula if, and only if, the following conditions hold:
(i) $C(u_1,..,u_d)=0$ whenever $u_j=0$ for at least one index $j\in\{1,...,d\}$,
(ii) when all the arguments of $C$ are equal to $1$, but possibly for the $j$-th one, then
$$C(1,..,1,u_j,1,..,1)=u_j$$
(iii) $C$ is $d$-increasing i.e., $\forall~ ]\mathbf{a}, \mathbf{b}] \subseteq \mathbb{I}^d, V_C(]\mathbf{a},\mathbf{b}]):=\underset{{\mathbf{v}} \in \text{ver}(]\mathbf{a},\mathbf{b}])}{\sum}\text{sign}(\mathbf{v})C(\mathbf{v}) \geq 0 $ where $\text{sign}(\mathbf{v})=1$, if $v_j=a_j$ for an even number of indices, and $\text{sign}(\mathbf{v})=-1$, if $ v_j=a_j$ for an odd number of indices.
Note that every copula $C\in \mathcal{C}_d$ induces a $d$-fold stochastic measure $\mu_{C}$ on $(\mathbb{I}^d, \mathcal{B}(\mathbb{I})^d)$ defined on the rectangles $R = ]\mathbf{a}, \mathbf{b}]$ contained in $\mathbb{I}^d$, by
$$\mu_{C}(R):=V_{C}(]\mathbf{a}, \mathbf{b}]).$$
We will focus on specific copulas whose support is possibly a fractal set and discuss the uniform convergence of empirical copulas induced by orbits of the so-called chaos game (a Markov process induced by transformation matrices $\mathcal{T}$, compare [4]). We aim at learning, i.e., approximating an unknown function $f$ (see also [5]), from random samples based on the examples of patterns, namely the so-called chaos game.
Further details on copulas can be found in the monographs [1,2,3].
In this talk, we will first investigate the problem of learning in a relevant function space for an individual domain with the chaos game representation. Within this framework, we further formulate the problem of domain adaptation with multiple sources [6], where we discuss the method of aggregating the already obtained approximated functions in each domain to derive a function with a small error with respect to the target domain.
Acknowledgement:
This research was carried out under the Austrian COMET program (project S3AI with FFG no. 872172, www.S3AI.at, at SCCH, www.scch.at), which is funded by the Austrian ministries BMK, BMDW, and the province of Upper Austria.
[1] F. Durante, C. Sempi. Principles of copula theory. CRC Press, 2016.
[2] R. B. Nelsen. An introduction to copulas. Springer Series in Statistics. Springer, second edition, 2006.
[3] C. Alsina, M. J. Frank, B. Schweizer. Associative functions. Triangular norms and copulas.World Scientific Publishing Co. Pte. Ltd.,2006.
[4] W. Trutschnig, J.F. Sanchez. Copulas with continuous, strictly increasing singular conditional distribution functions. J. Math. Anal. Appl. 410(2): 1014–1027, 2014.
[5] F. Cucker, S. Smale. On the mathematical foundations of learning. Bull. Amer. Math. Soc. (N.S.) 39(1): 1–49, 2002.
[6] Y. Mansour, M. Mohri, A. Rostamizadeh. Domain adaptation with multiple sources. Advances in neural information processing systems 21, 2008.
Learning segmentation on unlabeled MRI data using labeled CT data University of Vienna, Austria
The goal of supervised learning is that of deducing a classifier from a given labeled data set. In several concrete applications, such as medical imagery, one however often operates in the setup of domain adaptation.
Here, a classifier is learnt from a source labeled data set and generalised to a target unlabeled data set, with the two data sets moreover belonging to different domains (e.g. different patients, different machine setups etc.).
In our work, we use the SIFA framework [1] as a basis for medical image segmentation for a cross-modality adaptation between MRI and CT images. We have combined the SIFA algorithm with linear aggregation as well as importance-weighted validation of those trained models to remove the arbitrariness in the choice of parameters.
This presentation shall give an overview of domain adaptation and show the latest version of our experiments.
[1] C. Chen, Q. Dou, H. Chen, J. Qin, P. Heng. Unsupervised Bidirectional Cross-Modality Adaptation via Deeply Synergistic Image and Feature Alignment for Medical Image Segmentation. IEEE Transactions on Medical Imaging 39: 2494-2505, 2020.
Parameter choice in distance-regularized domain adaptation Austrian Academy of Sciences, Austria
We address the unsolved algorithm design problem of choosing a justified regularization parameter in unsupervised domain adaptation, the problem of learning from unlabeled data using labeled data from a different distribution. Our approach starts with the observation that the widely-used method of minimizing the source error, penalized by a distance measure between source and target feature representations, shares characteristics with penalized regularization methods. This observation allows us to extend Lepskii’s balancing principle, and it’s related error bound, to unsupervised domain adaptation. This talk is partially based on [1].
[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, 2021.
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