arXiv subjects

Andreas Hauptmann

Publications and source records attributed to Andreas Hauptmann.

At least 19 recordsLinked to original sources

arXiv2026

Gradient Descent on Point Clouds and Applications in Learned Operator Correction

We consider the problem of minimising an energy over an unknown manifold that is given implicitly by a point cloud. For a known manifold one can define a gradient descent scheme analogously to the classical construction in Euclidean spaces. However, when the manifold is not known one has to simultaneously estimate the manifold whilst minimising the energy. We define a gradient descent scheme which remains in a neighbourhood of the manifold and, under suitable stability and sampling assumptions, converges to a neighbourhood of a local minimiser whose size vanishes as the time step and sampling errors vanish. As an example we show the application of the methodology to learning operator corrections in inverse problems.

math.NA↗

arXiv2026

Enabling self-supervised learned primal dual with Noise2Inverse

X-ray computed tomography reconstruction is an ill-posed inverse problem, particularly in low-dose and sparse-angle settings where measurements are noisy and incomplete. While learned reconstruction methods such as the Learned Primal-Dual algorithm achieve strong performance, they typically rely on supervised training with access to ground-truth data, which is often unavailable in practice. In this work, we propose a self-supervised reconstruction method by extending the Noise2Inverse framework to the Learned Primal-Dual algorithm. The resulting approach, called Noise2Inverse Learned Primal-Dual (N2I-LPD), enables training of a learned iterative reconstruction operator without ground-truth images by exploiting the statistical independence of noise in distinct measurements with respect to angular rotation of the CT-scan. We compare the proposed method with classical reconstruction methods, as well as neural network-based approaches such as a U-Net trained within the same N2I framework. The results demonstrate that N2I-LPD achieves improved reconstruction quality, highlighting the potential of combining learned reconstruction operators with self-supervised training strategies for practical CT imaging scenarios where ground-truth data is unavailable.

eess.IV↗

arXiv2026

QVaR: a Quantum Variational Regularization method for Linear Inverse Problems

We present a tailored framework for solving regularized linear inverse problems using quantum optimization methods. By discretizing the solution space and encoding data fidelity and regularization terms into quadratic unconstrained binary optimization (QUBO) models, we formulate both Tikhonov- and sparsity-promoting regularized inverse problems within a unified quantum optimization framework. We further introduce a notion of quantum sensitivity that characterizes the effect of perturbations arising from approximate quantum evolution and discretization. We derive bounds relating these perturbations to stability properties of the underlying inverse problem, thereby establishing a theoretical connection between quantum solution variability and classical notions of ill-posedness. The framework is extended to variational inverse problems through wavelet-based representations and complemented by reduced-order modeling strategies in both parameter and Hamiltonian spaces to mitigate current hardware limitations. Numerical experiments on simulated and physical quantum hardware indicate that the resulting low-energy solution distributions retain information about the underlying inverse problem, while also revealing the limitations imposed by finite-time evolution, discretization, and hardware noise.

math.NA↗

arXiv2026

Deep Image Prior for photoacoustic tomography can mitigate limited-view artifacts

We study the deep image prior (DIP) framework applied to photoacoustic tomography (PAT) as an unsupervised reconstruction approach to mitigate limited-view artifacts and noise commonly encountered in experimental settings. Efficient implementation is achieved by employing recently published fast forward and adjoint algorithms for circular measurement geometries. Initialization via a fast inverse and total variation (TV) regularization are applied to further suppress noise and mitigate overfitting. For comparison, we compute a classical TV reconstruction. Our experiments comprise simulated PAT measurements under limited-view geometries and varying levels of added noise as well as experimental measurements together with using a digital twin for quality assessment. Our findings suggest that DIP framework provides an effective unsupervised strategy for robust PAT reconstruction even in the challenging case of a limited view geometry providing improvement in several quantitative measures over total variation reconstructions.

eess.IV↗

arXiv2026

Utilising a learned forward operator in the inverse problem of photoacoustic tomography

We study the use of a learned forward operator in the inverse problem of photoacoustic tomography. The Fourier neural operator to approximate the photoacoustic wave propagation is used. Further, the inverse problem is solved using a gradient-based approach with automatic differentiation. The methodology is evaluated using numerical simulations, and the results are compared to a conventional approach, where the forward operator is approximated using the pseudospectral $k$-space method. The results show that the learned forward operator can be used to approximate the photoacoustic wave propagation with good accuracy, and that it can be utilised as a computationally efficient forward operator in solving the inverse problem of photoacoustic tomography.

physics.comp-ph↗

arXiv2026

Transformer Causality Regularization for Dynamic Inverse Problems

We study the concept of including the causality principle as regularizer into the solution of linear time-dependent inverse problems. This is achieved by combining transformer-based predictions with classical variational regularization, resulting in what we call transformer causality regularization (TCR). The causality principle states that an object at time $t'$ depends only on its previous states at $t < t'$ and is independent of future states at $t > t'$. Since the transformer architecture represents sequence-to-sequence functions and can be equipped with a causal attention mask, transformers are the natural choice for a learned causality function that predicts the state of an object at time $t'$ given the previous states at $t < t'$. We combine this with the inductive bias of convolutional neural networks (CNNs) for imaging tasks to treat the spatial variable. The output of the spatial-temporal transformer is then used as a prior for variational regularization, such that classical results on regularization and convergence for solution methods directly transfer to our case. Using the example of dynamic computerized tomography, we compare TCR to a static and dynamic version of the earlier introduced unrolled adversarial regularizer for simulated and measured data. The results show that using TCR within a variational framework improves reconstruction results and data-consistency.

math.NA↗

arXiv2025

Learned iterative networks: An operator learning perspective

Learned image reconstruction has become a pillar in computational imaging and inverse problems. Among the most successful approaches are learned iterative networks, which are formulated by unrolling classical iterative optimisation algorithms for solving variational problems. While the underlying algorithm is usually formulated in the functional analytic setting, learned approaches are often viewed as purely discrete. In this survey we present a unified operator view for learned iterative networks. Specifically, we formulate a learned reconstruction operator, defining how to compute, and separately the learning problem, which defines what to compute. In this setting we present common approaches and show that many approaches are closely related in their core. We review linear as well as non-linear inverse problems in this framework and present a short numerical study to conclude.

eess.IV↗

arXiv2025

Towards robust quantitative photoacoustic tomography via learned iterative methods

Photoacoustic tomography (PAT) is a medical imaging modality that can provide high-resolution tissue images based on the optical absorption. Classical reconstruction methods for quantifying the absorption coefficients rely on sufficient prior information to overcome noisy and imperfect measurements. As these methods utilize computationally expensive forward models, the computation becomes slow, limiting their potential for time-critical applications. As an alternative approach, deep learning-based reconstruction methods have been established for faster and more accurate reconstructions. However, most of these methods rely on having a large amount of training data, which is not the case in practice. In this work, we adopt the model-based learned iterative approach for the use in Quantitative PAT (QPAT), in which additional information from the model is iteratively provided to the updating networks, allowing better generalizability with scarce training data. We compare the performance of different learned updates based on gradient descent, Gauss-Newton, and Quasi-Newton methods. The learning tasks are formulated as greedy, requiring iterate-wise optimality, as well as end-to-end, where all networks are trained jointly. The implemented methods are tested with ideal simulated data as well as against a digital twin dataset that emulates scarce training data and high modeling error.

eess.IV↗

arXiv2025

Fast algorithms enabling optimization and deep learning for photoacoustic tomography in a circular detection geometry

The inverse source problem arising in photoacoustic tomography and in several other coupled-physics modalities is frequently solved by iterative algorithms. Such algorithms are based on the minimization of a certain cost functional. In addition, novel deep learning techniques are currently being investigated to further improve such optimization approaches. All such methods require multiple applications of the operator defining the forward problem, and of its adjoint. In this paper, we present new asymptotically fast algorithms for numerical evaluation of the forward and adjoint operators, applicable in the circular acquisition geometry. For an $(n \times n)$ image, our algorithms compute these operators in $\mathcal{O}(n^2 \log n)$ floating point operations. We demonstrate the performance of our algorithms in numerical simulations, where they are used as an integral part of several iterative image reconstruction techniques: classic variational methods, such as non-negative least squares and total variation regularized least squares, as well as deep learning methods, such as learned primal dual. A Python implementation of our algorithms and computational examples is available to the general public.

eess.IV↗

arXiv2025

Learning the detector in optical tomography

We propose a method to reconstruct the optical absorption of a highly-scattering medium probed by diffuse light. The method consists of learning the optical detection system and then using this result to reconstruct the absorption. Our results are illustrated by numerical simulations.

physics.optics↗

arXiv2025

Deep Learning Based Reconstruction Methods for Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is a powerful imaging modality widely used in medical diagnostics, industrial monitoring, and environmental studies. The EIT inverse problem is about inferring the internal conductivity distribution of the concerned object from the voltage measurements taken on its boundary. This problem is severely ill-posed, and requires advanced computational approaches for accurate and reliable image reconstruction. Recent innovations in both model-based reconstruction and deep learning have driven significant progress in the field. In this review, we explore learned reconstruction methods that employ deep neural networks for solving the EIT inverse problem. The discussion focuses on the complete electrode model, one popular mathematical model for real-world applications of EIT. We compare a wide variety of learned approaches, including fully-learned, post-processing and learned iterative methods, with several conventional model-based reconstruction techniques, e.g., sparsity regularization, regularized Gauss-Newton iteration and level set method. The evaluation is based on three datasets: a simulated dataset of ellipses, an out-of-distribution simulated dataset, and the KIT4 dataset, including real-world measurements. Our results demonstrate that learned methods outperform model-based methods for in-distribution data but face challenges in generalization, where hybrid methods exhibit a good balance of accuracy and adaptability.

eess.IV↗

arXiv2025

Regularization for time-dependent inverse problems: Geometry of Lebesgue-Bochner spaces and algorithms

We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover a function from given observations where the function or the data depend on time. Lebesgue-Bochner spaces allow to easily incorporate the different nature of time and space. In this manuscript, we present two different regularization methods in Lebesgue Bochner spaces: 1. classical Tikhonov regularization in Banach spaces 2. temporal variational regularization by penalizing the time-derivative In the first case, we additionally investigate geometrical properties of Lebesgue Bochner spaces. In particular, we compute the duality mapping and show that these spaces are smooth of power type. With this we can implement Tikhononv regularization in Lebesgue-Bochner spaces using different regularities for time and space. We test both methods using the example of dynamic computerized tomography.

math.NA↗

arXiv2025

Digital twins enable full-reference quality assessment of photoacoustic image reconstructions

Quantitative comparison of the quality of photoacoustic image reconstruction algorithms remains a major challenge. No-reference image quality measures are often inadequate, but full-reference measures require access to an ideal reference image. While the ground truth is known in simulations, it is unknown in vivo, or in phantom studies, as the reference depends on both the phantom properties and the imaging system. We tackle this problem by using numerical digital twins of tissue-mimicking phantoms and the imaging system to perform a quantitative calibration to reduce the simulation gap. The contributions of this paper are two-fold: First, we use this digital-twin framework to compare multiple state-of-the-art reconstruction algorithms. Second, among these is a Fourier transform-based reconstruction algorithm for circular detection geometries, which we test on experimental data for the first time. Our results demonstrate the usefulness of digital phantom twins by enabling assessment of the accuracy of the numerical forward model and enabling comparison of image reconstruction schemes with full-reference image quality assessment. We show that the Fourier transform-based algorithm yields results comparable to those of iterative time reversal, but at a lower computational cost. All data and code are publicly available on Zenodo: https://doi.org/10.5281/zenodo.15388429.

physics.med-ph↗

arXiv2025

Learned enclosure method for experimental EIT data

Electrical impedance tomography (EIT) is a non-invasive imaging method with diverse applications, including medical imaging and non-destructive testing. The inverse problem of reconstructing internal electrical conductivity from boundary measurements is nonlinear and highly ill-posed, making it difficult to solve accurately. In recent years, there has been growing interest in combining analytical methods with machine learning to solve inverse problems. In this paper, we propose a method for estimating the convex hull of inclusions from boundary measurements by combining the enclosure method proposed by Ikehata with neural networks. We demonstrate its performance using experimental data. Compared to the classical enclosure method with least squares fitting, the learned convex hull achieves superior performance on both simulated and experimental data.

eess.IV↗

arXiv2024

Self-Supervised Denoiser Framework

Reconstructing images using Computed Tomography (CT) in an industrial context leads to specific challenges that differ from those encountered in other areas, such as clinical CT. Indeed, non-destructive testing with industrial CT will often involve scanning multiple similar objects while maintaining high throughput, requiring short scanning times, which is not a relevant concern in clinical CT. Under-sampling the tomographic data (sinograms) is a natural way to reduce the scanning time at the cost of image quality since the latter depends on the number of measurements. In such a scenario, post-processing techniques are required to compensate for the image artifacts induced by the sinogram sparsity. We introduce the Self-supervised Denoiser Framework (SDF), a self-supervised training method that leverages pre-training on highly sampled sinogram data to enhance the quality of images reconstructed from undersampled sinogram data. The main contribution of SDF is that it proposes to train an image denoiser in the sinogram space by setting the learning task as the prediction of one sinogram subset from another. As such, it does not require ground-truth image data, leverages the abundant data modality in CT, the sinogram, and can drastically enhance the quality of images reconstructed from a fraction of the measurements. We demonstrate that SDF produces better image quality, in terms of peak signal-to-noise ratio, than other analytical and self-supervised frameworks in both 2D fan-beam or 3D cone-beam CT settings. Moreover, we show that the enhancement provided by SDF carries over when fine-tuning the image denoiser on a few examples, making it a suitable pre-training technique in a context where there is little high-quality image data. Our results are established on experimental datasets, making SDF a strong candidate for being the building block of foundational image-enhancement models in CT.

eess.IV↗

arXiv2024

Inverse Problems with Learned Forward Operators

Solving inverse problems requires the knowledge of the forward operator, but accurate models can be computationally expensive and hence cheaper variants that do not compromise the reconstruction quality are desired. This chapter reviews reconstruction methods in inverse problems with learned forward operators that follow two different paradigms. The first one is completely agnostic to the forward operator and learns its restriction to the subspace spanned by the training data. The framework of regularisation by projection is then used to find a reconstruction. The second one uses a simplified model of the physics of the measurement process and only relies on the training data to learn a model correction. We present the theory of these two approaches and compare them numerically. A common theme emerges: both methods require, or at least benefit from, training data not only for the forward operator, but also for its adjoint.

math.NA↗

arXiv2023

Sparsity promoting reconstructions via hierarchical prior models in diffuse optical tomography

Diffuse optical tomography (DOT) is a severely ill-posed nonlinear inverse problem that seeks to estimate optical parameters from boundary measurements. In the Bayesian framework, the ill-posedness is diminished by incorporating {\em a priori} information of the optical parameters via the prior distribution. In case the target is sparse or sharp-edged, the common choice as the prior model are non-differentiable total variation and $\ell^1$ priors. Alternatively, one can hierarchically extend the variances of a Gaussian prior to obtain differentiable sparsity promoting priors. By doing this, the variances are treated as unknowns allowing the estimation to locate the discontinuities. In this work, we formulate hierarchical prior models for the nonlinear DOT inverse problem using exponential, standard gamma and inverse-gamma hyperpriors. Depending on the hyperprior and the hyperparameters, the hierarchical models promote different levels of sparsity and smoothness. To compute the MAP estimates, the previously proposed alternating algorithm is adapted to work with the nonlinear model. We then propose an approach based on the cumulative distribution function of the hyperpriors to select the hyperparameters. We evaluate the performance of the hyperpriors with numerical simulations and show that the hierarchical models can improve the localization, contrast and edge sharpness of the reconstructions.

math.NA↗

arXiv2023

Model-based reconstructions for quantitative imaging in photoacoustic tomography

The reconstruction task in photoacoustic tomography can vary a lot depending on measured targets, geometry, and especially the quantity we want to recover. Specifically, as the signal is generated due to the coupling of light and sound by the photoacoustic effect, we have the possibility to recover acoustic as well as optical tissue parameters. This is referred to as quantitative imaging, i.e, correct recovery of physical parameters and not just a qualitative image. In this chapter, we aim to give an overview on established reconstruction techniques in photoacoustic tomography. We start with modelling of the optical and acoustic phenomena, necessary for a reliable recovery of quantitative values. Furthermore, we give an overview of approaches for the tomographic reconstruction problem with an emphasis on the recovery of quantitative values, from direct and fast analytic approaches to computationally involved optimisation based techniques and recent data-driven approaches.

physics.med-ph↗