Erich Kobler

Martin Zach, Erich Kobler, Thomas Pock

In the past decades, Computed Tomography (CT) has established itself as one of the most important imaging techniques in medicine. Today, the applicability of CT is only limited by the deposited radiation dose, reduction of which manifests in noisy or incomplete measurements. Thus, the need for robust reconstruction algorithms arises. In this work, we learn a parametric regularizer with a global receptive field by maximizing it's likelihood on reference CT data. Due to this unsupervised learning strategy, our trained regularizer truly represents higher-level domain statistics, which we empirically demonstrate by synthesizing CT images. Moreover, this regularizer can easily be applied to different CT reconstruction problems by embedding it in a variational framework, which increases flexibility and interpretability compared to feed-forward learning-based approaches. In addition, the accompanying probabilistic perspective enables experts to explore the full posterior distribution and may quantify uncertainty of the reconstruction approach. We apply the regularizer to limited-angle and few-view CT reconstruction problems, where it outperforms traditional reconstruction algorithms by a large margin.

Martin Zach, Thomas Pock, Erich Kobler et al.

In this work we tackle the problem of estimating the density $f_X$ of a random variable $X$ by successive smoothing, such that the smoothed random variable $Y$ fulfills $(\partial_t - Δ_1)f_Y(\,\cdot\,, t) = 0$, $f_Y(\,\cdot\,, 0) = f_X$. With a focus on image processing, we propose a product/fields of experts model with Gaussian mixture experts that admits an analytic expression for $f_Y (\,\cdot\,, t)$ under an orthogonality constraint on the filters. This construction naturally allows the model to be trained simultaneously over the entire diffusion horizon using empirical Bayes. We show preliminary results on image denoising where our model leads to competitive results while being tractable, interpretable, and having only a small number of learnable parameters. As a byproduct, our model can be used for reliable noise estimation, allowing blind denoising of images corrupted by heteroscedastic noise.

Erich Kobler, Thomas Pock

In this paper, we propose a unified framework of denoising score-based models in the context of graduated non-convex energy minimization. We show that for sufficiently large noise variance, the associated negative log density -- the energy -- becomes convex. Consequently, denoising score-based models essentially follow a graduated non-convexity heuristic. We apply this framework to learning generalized Fields of Experts image priors that approximate the joint density of noisy images and their associated variances. These priors can be easily incorporated into existing optimization algorithms for solving inverse problems and naturally implement a fast and robust graduated non-convexity mechanism.

Thomas Pinetz, Erich Kobler, Robert Haase et al.

Today Gadolinium-based contrast agents (GBCA) are indispensable in Magnetic Resonance Imaging (MRI) for diagnosing various diseases. However, GBCAs are expensive and may accumulate in patients with potential side effects, thus dose-reduction is recommended. Still, it is unclear to which extent the GBCA dose can be reduced while preserving the diagnostic value -- especially in pathological regions. To address this issue, we collected brain MRI scans at numerous non-standard GBCA dosages and developed a conditional GAN model for synthesizing corresponding images at fractional dose levels. Along with the adversarial loss, we advocate a novel content loss function based on the Wasserstein distance of locally paired patch statistics for the faithful preservation of noise. Our numerical experiments show that conditional GANs are suitable for generating images at different GBCA dose levels and can be used to augment datasets for virtual contrast models. Moreover, our model can be transferred to openly available datasets such as BraTS, where non-standard GBCA dosage images do not exist.

Richard Freinschlag, Timo Bertram, Erich Kobler et al.

Reasoning problems such as Sudoku and ARC-AGI remain challenging for neural networks. The structured problem solving architecture family of Recurrent Reasoning Models (RRMs), including Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), offer a compact alternative to large language models, but currently handle symbol symmetries only implicitly via costly data augmentation. We introduce Symbol-Equivariant Recurrent Reasoning Models (SE-RRMs), which enforce permutation equivariance at the architectural level through symbol-equivariant layers, guaranteeing identical solutions under symbol or color permutations. SE-RRMs outperform prior RRMs on 9x9 Sudoku and generalize from just training on 9x9 to smaller 4x4 and larger 16x16 and 25x25 instances, to which existing RRMs cannot extrapolate. On ARC-AGI-1 and ARC-AGI-2, SE-RRMs achieve competitive performance with substantially less data augmentation and only 2 million parameters, demonstrating that explicitly encoding symmetry improves the robustness and scalability of neural reasoning. Code is available at https://github.com/ml-jku/SE-RRM.

Antonio Ortiz-Gonzalez, Erich Kobler, Lukas Schletter et al.

Magnetic resonance imaging (MRI) is highly susceptible to patient motion due to its relatively long acquisition times and the fact that data are acquired sequentially in k-space. Even small patient movements introduce phase inconsistencies across measurements, leading to severe artifacts such as blurring, ghosting, and geometric distortions that can compromise diagnostic quality. Retrospective motion compensation remains challenging, particularly in accelerated acquisitions, due to the ill-posed nature of the joint reconstruction and motion estimation problem. In this work, we propose a unified Bayesian framework for motion-compensated 3D MRI that jointly estimates the anatomical image, rigid-body motion parameters, and coil sensitivity maps directly from motion-corrupted k-space data. Our approach integrates pretrained 3D complex-valued score-based diffusion models as expressive anatomical image priors within a physics-based forward model. Inference is performed by alternating diffusion posterior image updates with efficient proximal optimization steps for motion and coil sensitivity estimation, enabling fully unsupervised reconstruction without the need for paired motion-free training data. Experiments on simulated and real-motion brain MRI datasets demonstrate that the proposed method achieves improved image quality and motion robustness compared to state-of-the-art classical and learning-based motion correction techniques, particularly in the presence of severe motion and high acceleration.

Jun Ma, Feifei Li, Sumin Kim et al.

Promptable segmentation foundation models have emerged as a transformative approach to addressing the diverse needs in medical images, but most existing models require expensive computing, posing a big barrier to their adoption in clinical practice. In this work, we organized the first international competition dedicated to promptable medical image segmentation, featuring a large-scale dataset spanning nine common imaging modalities from over 20 different institutions. The top teams developed lightweight segmentation foundation models and implemented an efficient inference pipeline that substantially reduced computational requirements while maintaining state-of-the-art segmentation accuracy. Moreover, the post-challenge phase advanced the algorithms through the design of performance booster and reproducibility tasks, resulting in improved algorithms and validated reproducibility of the winning solution. Furthermore, the best-performing algorithms have been incorporated into the open-source software with a user-friendly interface to facilitate clinical adoption. The data and code are publicly available to foster the further development of medical image segmentation foundation models and pave the way for impactful real-world applications.

Mehrsa Pourya, Erich Kobler, Michael Unser et al.

State-of-the-art image reconstruction often relies on complex, highly parameterized deep architectures. We propose an alternative: a data-driven reconstruction method inspired by the classic Tikhonov regularization. Our approach iteratively refines intermediate reconstructions by solving a sequence of quadratic problems. These updates have two key components: (i) learned filters to extract salient image features, and (ii) an attention mechanism that locally adjusts the penalty of filter responses. Our method achieves performance on par with leading plug-and-play and learned regularizer approaches while offering interpretability, robustness, and convergent behavior. In effect, we bridge traditional regularization and deep learning with a principled reconstruction approach.

Thomas Pinetz, Erich Kobler, Robert Haase et al.

Recently, deep learning (DL)-based methods have been proposed for the computational reduction of gadolinium-based contrast agents (GBCAs) to mitigate adverse side effects while preserving diagnostic value. Currently, the two main challenges for these approaches are the accurate prediction of contrast enhancement and the synthesis of realistic images. In this work, we address both challenges by utilizing the contrast signal encoded in the subtraction images of pre-contrast and post-contrast image pairs. To avoid the synthesis of any noise or artifacts and solely focus on contrast signal extraction and enhancement from low-dose subtraction images, we train our DL model using noise-free standard-dose subtraction images as targets. As a result, our model predicts the contrast enhancement signal only; thereby enabling synthesization of images beyond the standard dose. Furthermore, we adapt the embedding idea of recent diffusion-based models to condition our model on physical parameters affecting the contrast enhancement behavior. We demonstrate the effectiveness of our approach on synthetic and real datasets using various scanners, field strengths, and contrast agents.

Johannes Hertrich, Hok Shing Wong, Alexander Denker et al.

In recent years, a variety of learned regularization frameworks for solving inverse problems in imaging have emerged. These offer flexible modeling together with mathematical insights. The proposed methods differ in their architectural design and training strategies, making direct comparison challenging due to non-modular implementations. We address this gap by collecting and unifying the available code into a common framework. This unified view allows us to systematically compare the approaches and highlight their strengths and limitations, providing valuable insights into their future potential. We also provide concise descriptions of each method, complemented by practical guidelines.

Bastian Brandstötter, Erich Kobler

Contrast-enhanced (CE) T1-weighted MRI is central to neuro-oncologic diagnosis but requires gadolinium-based agents, which add cost and scan time, raise environmental concerns, and may pose risks to patients. In this work, we propose a two-stage Posterior-Mean Rectified Flow (PMRF) pipeline for synthesizing volumetric CE brain MRI from non-contrast inputs. First, a patch-based 3D U-Net predicts the voxel-wise posterior mean (minimizing MSE). Then, this initial estimate is refined by a time-conditioned 3D rectified flow to incorporate realistic textures without compromising structural fidelity. We train this model on a multi-institutional collection of paired pre- and post-contrast T1w volumes (BraTS 2023-2025). On a held-out test set of 360 diverse volumes, our best refined outputs achieve an axial FID of $12.46$ and KID of $0.007$ ($\sim 68.7\%$ lower FID than the posterior mean) while maintaining low volumetric MSE of $0.057$ ($\sim 27\%$ higher than the posterior mean). Qualitative comparisons confirm that our method restores lesion margins and vascular details realistically, effectively navigating the perception-distortion trade-off for clinical deployment.

Dominik Narnhofer, Alexander Effland, Erich Kobler et al.

Recent deep learning approaches focus on improving quantitative scores of dedicated benchmarks, and therefore only reduce the observation-related (aleatoric) uncertainty. However, the model-immanent (epistemic) uncertainty is less frequently systematically analyzed. In this work, we introduce a Bayesian variational framework to quantify the epistemic uncertainty. To this end, we solve the linear inverse problem of undersampled MRI reconstruction in a variational setting. The associated energy functional is composed of a data fidelity term and the total deep variation (TDV) as a learned parametric regularizer. To estimate the epistemic uncertainty we draw the parameters of the TDV regularizer from a multivariate Gaussian distribution, whose mean and covariance matrix are learned in a stochastic optimal control problem. In several numerical experiments, we demonstrate that our approach yields competitive results for undersampled MRI reconstruction. Moreover, we can accurately quantify the pixelwise epistemic uncertainty, which can serve radiologists as an additional resource to visualize reconstruction reliability.

Thomas Pinetz, Erich Kobler, Thomas Pock et al.

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.

Elena A. Kaye, Emily A. Aherne, Cihan Duzgol et al.

Purpose: To investigate feasibility of accelerating prostate diffusion-weighted imaging (DWI) by reducing the number of acquired averages and denoising the resulting image using a proposed guided denoising convolutional neural network (DnCNN). Materials and Methods: Raw data from the prostate DWI scans were retrospectively gathered (between July 2018 and July 2019) from six single-vendor MRI scanners. 118 data sets were used for training and validation (age: 64.3 +- 8 years) and 37 - for testing (age: 65.1 +- 7.3 years). High b-value diffusion-weighted (hb-DW) data were reconstructed into noisy images using two averages and reference images using all sixteen averages. A conventional DnCNN was modified into a guided DnCNN, which uses the low b-value DWI image as a guidance input. Quantitative and qualitative reader evaluations were performed on the denoised hb-DW images. A cumulative link mixed regression model was used to compare the readers scores. The agreement between the apparent diffusion coefficient (ADC) maps (denoised vs reference) was analyzed using Bland Altman analysis. Results: Compared to the DnCNN, the guided DnCNN produced denoised hb-DW images with higher peak signal-to-noise ratio and structural similarity index and lower normalized mean square error (p < 0.001). Compared to the reference images, the denoised images received higher image quality scores (p < 0.0001). The ADC values based on the denoised hb-DW images were in good agreement with the reference ADC values. Conclusion: Accelerating prostate DWI by reducing the number of acquired averages and denoising the resulting image using the proposed guided DnCNN is technically feasible.

Erich Kobler, Alexander Effland, Karl Kunisch et al.

Various problems in computer vision and medical imaging can be cast as inverse problems. A frequent method for solving inverse problems is the variational approach, which amounts to minimizing an energy composed of a data fidelity term and a regularizer. Classically, handcrafted regularizers are used, which are commonly outperformed by state-of-the-art deep learning approaches. In this work, we combine the variational formulation of inverse problems with deep learning by introducing the data-driven general-purpose total deep variation regularizer. In its core, a convolutional neural network extracts local features on multiple scales and in successive blocks. This combination allows for a rigorous mathematical analysis including an optimal control formulation of the training problem in a mean-field setting and a stability analysis with respect to the initial values and the parameters of the regularizer. In addition, we experimentally verify the robustness against adversarial attacks and numerically derive upper bounds for the generalization error. Finally, we achieve state-of-the-art results for numerous imaging tasks.

Erich Kobler, Alexander Effland, Karl Kunisch et al.

Diverse inverse problems in imaging can be cast as variational problems composed of a task-specific data fidelity term and a regularization term. In this paper, we propose a novel learnable general-purpose regularizer exploiting recent architectural design patterns from deep learning. We cast the learning problem as a discrete sampled optimal control problem, for which we derive the adjoint state equations and an optimality condition. By exploiting the variational structure of our approach, we perform a sensitivity analysis with respect to the learned parameters obtained from different training datasets. Moreover, we carry out a nonlinear eigenfunction analysis, which reveals interesting properties of the learned regularizer. We show state-of-the-art performance for classical image restoration and medical image reconstruction problems.

Alexander Effland, Erich Kobler, Karl Kunisch et al.

We investigate a well-known phenomenon of variational approaches in image processing, where typically the best image quality is achieved when the gradient flow process is stopped before converging to a stationary point. This paradox originates from a tradeoff between optimization and modelling errors of the underlying variational model and holds true even if deep learning methods are used to learn highly expressive regularizers from data. In this paper, we take advantage of this paradox and introduce an optimal stopping time into the gradient flow process, which in turn is learned from data by means of an optimal control approach. As a result, we obtain highly efficient numerical schemes that achieve competitive results for image denoising and image deblurring. A nonlinear spectral analysis of the gradient of the learned regularizer gives enlightening insights about the different regularization properties.

Kerstin Hammernik, Teresa Klatzer, Erich Kobler et al.

Purpose: To allow fast and high-quality reconstruction of clinical accelerated multi-coil MR data by learning a variational network that combines the mathematical structure of variational models with deep learning. Theory and Methods: Generalized compressed sensing reconstruction formulated as a variational model is embedded in an unrolled gradient descent scheme. All parameters of this formulation, including the prior model defined by filter kernels and activation functions as well as the data term weights, are learned during an offline training procedure. The learned model can then be applied online to previously unseen data. Results: The variational network approach is evaluated on a clinical knee imaging protocol. The variational network reconstructions outperform standard reconstruction algorithms in terms of image quality and residual artifacts for all tested acceleration factors and sampling patterns. Conclusion: Variational network reconstructions preserve the natural appearance of MR images as well as pathologies that were not included in the training data set. Due to its high computational performance, i.e., reconstruction time of 193 ms on a single graphics card, and the omission of parameter tuning once the network is trained, this new approach to image reconstruction can easily be integrated into clinical workflow.