Linda-Sophie Schneider

Mareike Thies, Fabian Wagner, Noah Maul et al.

Incorporating computed tomography (CT) reconstruction operators into differentiable pipelines has proven beneficial in many applications. Such approaches usually focus on the projection data and keep the acquisition geometry fixed. However, precise knowledge of the acquisition geometry is essential for high quality reconstruction results. In this paper, the differentiable formulation of fan-beam CT reconstruction is extended to the acquisition geometry. This allows to propagate gradient information from a loss function on the reconstructed image into the geometry parameters. As a proof-of-concept experiment, this idea is applied to rigid motion compensation. The cost function is parameterized by a trained neural network which regresses an image quality metric from the motion affected reconstruction alone. Using the proposed method, we are the first to optimize such an autofocus-inspired algorithm based on analytical gradients. The algorithm achieves a reduction in MSE by 35.5 % and an improvement in SSIM by 12.6 % over the motion affected reconstruction. Next to motion compensation, we see further use cases of our differentiable method for scanner calibration or hybrid techniques employing deep models.

Mareike Thies, Fabian Wagner, Noah Maul et al.

In computed tomography (CT), the projection geometry used for data acquisition needs to be known precisely to obtain a clear reconstructed image. Rigid patient motion is a cause for misalignment between measured data and employed geometry. Commonly, such motion is compensated by solving an optimization problem that, e.g., maximizes the quality of the reconstructed image with respect to the projection geometry. So far, gradient-free optimization algorithms have been utilized to find the solution for this problem. Here, we show that gradient-based optimization algorithms are a possible alternative and compare the performance to their gradient-free counterparts on a benchmark motion compensation problem. Gradient-based algorithms converge substantially faster while being comparable to gradient-free algorithms in terms of capture range and robustness to the number of free parameters. Hence, gradient-based optimization is a viable alternative for the given type of problems.

Linda-Sophie Schneider, Mareike Thies, Christopher Syben et al.

We present a method for selecting valuable projections in computed tomography (CT) scans to enhance image reconstruction and diagnosis. The approach integrates two important factors, projection-based detectability and data completeness, into a single feed-forward neural network. The network evaluates the value of projections, processes them through a differentiable ranking function and makes the final selection using a straight-through estimator. Data completeness is ensured through the label provided during training. The approach eliminates the need for heuristically enforcing data completeness, which may exclude valuable projections. The method is evaluated on simulated data in a non-destructive testing scenario, where the aim is to maximize the reconstruction quality within a specified region of interest. We achieve comparable results to previous methods, laying the foundation for using reconstruction-based loss functions to learn the selection of projections.

Linda-Sophie Schneider, Yipeng Sun, Chengze Ye et al.

Deep learning has brought significant advancements to X-ray Computed Tomography (CT) reconstruction, offering solutions to challenges arising from modern imaging technologies. These developments benefit from methods that combine classical reconstruction techniques with data-driven approaches. Differentiable operators play a key role in this integration by enabling end-to-end optimization and the incorporation of physical modeling within neural networks. In this work, we present an updated version of PYRO-NN, a Python-based library for differentiable CT reconstruction. The updated framework extends compatibility to PyTorch and introduces native CUDA kernel support for efficient projection and back-projection operations across parallel, fan, and cone-beam geometries. Additionally, it includes tools for simulating imaging artifacts, modeling arbitrary acquisition trajectories, and creating flexible, end-to-end trainable pipelines through a high-level Python API. Code is available at: https://github.com/csyben/PYRO-NN

Linda-Sophie Schneider, Andreas Maier

This work introduces a continuous soft near-orthogonality score and a resolution-aware saturated coverage objective for projection selection in region-of-interest focused cone-beam CT, grounded in Tuy's completeness theory. Replacing the binary hit-or-miss model of classical Tuy completeness with a graded, differentiable formulation preserves a direct link to achievable feature sizes while enabling both efficient approximate and exact optimisation. We establish that the underlying discrete decision problems are NP-complete via polynomial-time reductions from Set Cover, motivating a submodular greedy algorithm with proven $(1-1/\mathrm{e})$ approximation guarantees and a mixed-integer linear program (MILP) that provides certified optimality bounds. The MILP serves as a quality certificate for the greedy solution rather than a competing optimiser. The primary empirical finding confirms this relationship: across a systematic benchmark spanning six target regions, multiple projection budgets, and four controlled occlusion conditions, the pooled median greedy-to-MILP objective ratio was 0.998, with a substantial fraction of cases certified globally optimal. A binary formulation is included as a diagnostic baseline; it strengthens hard directional completeness but is weaker on the continuous coverage scale. We additionally introduce Effective Spatial Resolution (ESR), a physically interpretable trajectory-level diagnostic that maps directional sampling gaps to achievable feature sizes. ESR correlates reliably with matched reconstruction quality across projection budgets and occlusion levels, providing a practical bridge between the selection stage and the image domain without requiring reconstruction.

Yipeng Sun, Yixing Huang, Linda-Sophie Schneider et al.

Computed Tomography (CT) image reconstruction is crucial for accurate diagnosis and deep learning approaches have demonstrated significant potential in improving reconstruction quality. However, the choice of loss function profoundly affects the reconstructed images. Traditional mean squared error loss often produces blurry images lacking fine details, while alternatives designed to improve may introduce structural artifacts or other undesirable effects. To address these limitations, we propose Eagle-Loss, a novel loss function designed to enhance the visual quality of CT image reconstructions. Eagle-Loss applies spectral analysis of localized features within gradient changes to enhance sharpness and well-defined edges. We evaluated Eagle-Loss on two public datasets across low-dose CT reconstruction and CT field-of-view extension tasks. Our results show that Eagle-Loss consistently improves the visual quality of reconstructed images, surpassing state-of-the-art methods across various network architectures. Code and data are available at \url{https://github.com/sypsyp97/Eagle_Loss}.

Yipeng Sun, Linda-Sophie Schneider, Fuxin Fan et al.

In this study, we introduce a Fourier series-based trainable filter for computed tomography (CT) reconstruction within the filtered backprojection (FBP) framework. This method overcomes the limitation in noise reduction by optimizing Fourier series coefficients to construct the filter, maintaining computational efficiency with minimal increment for the trainable parameters compared to other deep learning frameworks. Additionally, we propose Gaussian edge-enhanced (GEE) loss function that prioritizes the $L_1$ norm of high-frequency magnitudes, effectively countering the blurring problems prevalent in mean squared error (MSE) approaches. The model's foundation in the FBP algorithm ensures excellent interpretability, as it relies on a data-driven filter with all other parameters derived through rigorous mathematical procedures. Designed as a plug-and-play solution, our Fourier series-based filter can be easily integrated into existing CT reconstruction models, making it an adaptable tool for a wide range of practical applications. Code and data are available at https://github.com/sypsyp97/Trainable-Fourier-Series.

Yipeng Sun, Linda-Sophie Schneider, Mingxuan Gu et al.

Effective denoising is crucial in low-dose CT to enhance subtle structures and low-contrast lesions while preventing diagnostic errors. Supervised methods struggle with limited paired datasets, and self-supervised approaches often require multiple noisy images and rely on deep networks like U-Net, offering little insight into the denoising mechanism. To address these challenges, we propose an interpretable self-supervised single-image denoising framework -- Filter2Noise (F2N). Our approach introduces an Attention-Guided Bilateral Filter that adapted to each noisy input through a lightweight module that predicts spatially varying filter parameters, which can be visualized and adjusted post-training for user-controlled denoising in specific regions of interest. To enable single-image training, we introduce a novel downsampling shuffle strategy with a new self-supervised loss function that extends the concept of Noise2Noise to a single image and addresses spatially correlated noise. On the Mayo Clinic 2016 low-dose CT dataset, F2N outperforms the leading self-supervised single-image method (ZS-N2N) by 4.59 dB PSNR while improving transparency, user control, and parametric efficiency. These features provide key advantages for medical applications that require precise and interpretable noise reduction. Our code is demonstrated at https://github.com/sypsyp97/Filter2Noise.git .

Thomas Gorges, Janne van der Loop, Lukas Hüttner et al.

This paper presents CircleID, a large-scale ICDAR 2026 competition on writer identification and pen classification from scanned hand-drawn circles. The primary objective is to investigate how biometric writer characteristics and physical pen features naturally entangle within minimal, static traces. CircleID comprises two distinct tasks: (1) open-set writer identification, requiring models to recognize known writers while explicitly rejecting unknown ones, and (2) cross-writer pen classification, evaluated across both seen and unseen writers. Participants were provided with a new, controlled dataset of 46,155 tightly cropped circle images, digitized at 400 DPI and annotated for writer identity and pen type. The dataset comprises samples from 50 known and 16 unknown writers using eight different pens. Hosted on Kaggle as two separate tracks with public and private leaderboards, the competition provided participants with a ResNet baseline. In total, 389 teams (436 participants) made 3,185 submissions for the pen classification task, and 113 teams (141 participants) made 1,737 submissions for the writer identification track. The best-performing private leaderboard submissions achieved a Top-1 accuracy of 64.801% for writer identification and 92.726% for pen classification. This paper details the dataset, evaluates the winning methodologies, and analyzes the impact of out-of-distribution writers on model generalization and feature disentanglement. In this large-scale competition, CircleID establishes a new baseline for minimal-trace analysis.

Mareike Thies, Fabian Wagner, Noah Maul et al.

Cone-beam computed tomography (CBCT) systems, with their flexibility, present a promising avenue for direct point-of-care medical imaging, particularly in critical scenarios such as acute stroke assessment. However, the integration of CBCT into clinical workflows faces challenges, primarily linked to long scan duration resulting in patient motion during scanning and leading to image quality degradation in the reconstructed volumes. This paper introduces a novel approach to CBCT motion estimation using a gradient-based optimization algorithm, which leverages generalized derivatives of the backprojection operator for cone-beam CT geometries. Building on that, a fully differentiable target function is formulated which grades the quality of the current motion estimate in reconstruction space. We drastically accelerate motion estimation yielding a 19-fold speed-up compared to existing methods. Additionally, we investigate the architecture of networks used for quality metric regression and propose predicting voxel-wise quality maps, favoring autoencoder-like architectures over contracting ones. This modification improves gradient flow, leading to more accurate motion estimation. The presented method is evaluated through realistic experiments on head anatomy. It achieves a reduction in reprojection error from an initial average of 3mm to 0.61mm after motion compensation and consistently demonstrates superior performance compared to existing approaches. The analytic Jacobian for the backprojection operation, which is at the core of the proposed method, is made publicly available. In summary, this paper contributes to the advancement of CBCT integration into clinical workflows by proposing a robust motion estimation approach that enhances efficiency and accuracy, addressing critical challenges in time-sensitive scenarios.

Chengze Ye, Linda-Sophie Schneider, Yipeng Sun et al.

This study presents a novel approach for reconstructing cone beam computed tomography (CBCT) for specific orbits using known operator learning. Unlike traditional methods, this technique employs a filtered backprojection type (FBP-type) algorithm, which integrates a unique, adaptive filtering process. This process involves a series of operations, including weightings, differentiations, the 2D Radon transform, and backprojection. The filter is designed for a specific orbit geometry and is obtained using a data-driven approach based on deep learning. The approach efficiently learns and optimizes the orbit-related component of the filter. The method has demonstrated its ability through experimentation by successfully learning parameters from circular orbit projection data. Subsequently, the optimized parameters are used to reconstruct images, resulting in outcomes that closely resemble the analytical solution. This demonstrates the potential of the method to learn appropriate parameters from any specific orbit projection data and achieve reconstruction. The algorithm has demonstrated improvement, particularly in enhancing reconstruction speed and reducing memory usage for handling specific orbit reconstruction.

Chengze Ye, Linda-Sophie Schneider, Yipeng Sun et al.

This paper introduces a novel method for reconstructing cone beam computed tomography (CBCT) images for arbitrary orbits using a differentiable shift-variant filtered backprojection (FBP) neural network. Traditional CBCT reconstruction methods for arbitrary orbits, like iterative reconstruction algorithms, are computationally expensive and memory-intensive. The proposed method addresses these challenges by employing a shift-variant FBP algorithm optimized for arbitrary trajectories through a deep learning approach that adapts to a specific orbit geometry. This approach overcomes the limitations of existing techniques by integrating known operators into the learning model, minimizing the number of parameters, and improving the interpretability of the model. The proposed method is a significant advancement in interventional medical imaging, particularly for robotic C-arm CT systems, enabling faster and more accurate CBCT reconstructions with customized orbits. Especially this method can also be used for the analytical reconstruction of non-continuous orbits like circular plus arc. The experimental results demonstrate that the proposed method significantly accelerates the reconstruction process compared to conventional iterative algorithms. It achieves comparable or superior image quality, as evidenced by metrics such as the mean squared error (MSE), the peak signal-to-noise ratio (PSNR), and the structural similarity index measure (SSIM). The validation experiments show that the method can handle data from different trajectories, demonstrating its flexibility and robustness across different scan geometries. Our method demonstrates a significant improvement, particularly for the sinusoidal trajectory, achieving a 38.6% reduction in MSE, a 7.7% increase in PSNR, and a 5.0% improvement in SSIM. Furthermore, the computation time for reconstruction was reduced by more than 97%.

Yuzhong Zhou, Linda-Sophie Schneider, Fuxin Fan et al.

The rise of deep learning has introduced a transformative era in the field of image processing, particularly in the context of computed tomography. Deep learning has made a significant contribution to the field of industrial Computed Tomography. However, many defect detection algorithms are applied directly to the reconstructed domain, often disregarding the raw sensor data. This paper shifts the focus to the use of sinograms. Within this framework, we present a comprehensive three-step deep learning algorithm, designed to identify and analyze defects within objects without resorting to image reconstruction. These three steps are defect segmentation, mask isolation, and defect analysis. We use a U-Net-based architecture for defect segmentation. Our method achieves the Intersection over Union of 92.02% on our simulated data, with an average position error of 1.3 pixels for defect detection on a 512-pixel-wide detector.

Yuedong Yuan, Linda-Sophie Schneider, Andreas Maier

Recent advances in computed tomography (CT) imaging, especially with dual-robot systems, have introduced new challenges for scan trajectory optimization. This paper presents a novel approach using Gated Recurrent Units (GRUs) to optimize CT scan trajectories. Our approach exploits the flexibility of robotic CT systems to select projections that enhance image quality by improving resolution and contrast while reducing scan time. We focus on cone-beam CT and employ several projection-based metrics, including absorption, pixel intensities, contrast-to-noise ratio, and data completeness. The GRU network aims to minimize data redundancy and maximize completeness with a limited number of projections. We validate our method using simulated data of a test specimen, focusing on a specific voxel of interest. The results show that the GRU-optimized scan trajectories can outperform traditional circular CT trajectories in terms of image quality metrics. For the used specimen, SSIM improves from 0.38 to 0.49 and CNR increases from 6.97 to 9.08. This finding suggests that the application of GRU in CT scan trajectory optimization can lead to more efficient, cost-effective, and high-quality imaging solutions.

Linda-Sophie Schneider, Gabriel Herl, Andreas Maier

X-ray computed tomography (CT) plays a key role in digitizing three-dimensional structures for a wide range of medical and industrial applications. Traditional CT systems often rely on standard circular and helical scan trajectories, which may not be optimal for challenging scenarios involving large objects, complex structures, or resource constraints. In response to these challenges, we are exploring the potential of twin robotic CT systems, which offer the flexibility to acquire projections from arbitrary views around the object of interest. Ensuring complete and mathematically sound reconstructions becomes critical in such systems. In this work, we present an integer programming-based CT trajectory optimization method. Utilizing discrete data completeness conditions, we formulate an optimization problem to select an optimized set of projections. This approach enforces data completeness and considers absorption-based metrics for reliability evaluation. We compare our method with an equidistant circular CT trajectory and a greedy approach. While greedy already performs well in some cases, we provide a way to improve greedy-based projection selection using an integer optimization approach. Our approach improves CT trajectories and quantifies the optimality of the solution in terms of an optimality gap.

Yipeng Sun, Linda-Sophie Schneider, Chengze Ye et al.

Cone-Beam Computed Tomography (CBCT) is essential in medical imaging, and the Feldkamp-Davis-Kress (FDK) algorithm is a popular choice for reconstruction due to its efficiency. However, FDK is susceptible to noise and artifacts. While recent deep learning methods offer improved image quality, they often increase computational complexity and lack the interpretability of traditional methods. In this paper, we introduce an enhanced FDK-based neural network that maintains the classical algorithm's interpretability by selectively integrating trainable elements into the cosine weighting and filtering stages. Recognizing the challenge of a large parameter space inherent in 3D CBCT data, we leverage wavelet transformations to create sparse representations of the cosine weights and filters. This strategic sparsification reduces the parameter count by $93.75\%$ without compromising performance, accelerates convergence, and importantly, maintains the inference computational cost equivalent to the classical FDK algorithm. Our method not only ensures volumetric consistency and boosts robustness to noise, but is also designed for straightforward integration into existing CT reconstruction pipelines. This presents a pragmatic enhancement that can benefit clinical applications, particularly in environments with computational limitations.

Chengze Ye, Linda-Sophie Schneider, Yipeng Sun et al.

The differentiable shift-variant filtered backprojection (FBP) model enables the reconstruction of cone-beam computed tomography (CBCT) data for any non-circular trajectories. This method employs deep learning technique to estimate the redundancy weights required for reconstruction, given knowledge of the specific trajectory at optimization time. However, computing the redundancy weight for each projection remains computationally intensive. This paper presents a novel approach to compress and optimize the differentiable shift-variant FBP model based on Principal Component Analysis (PCA). We apply PCA to the redundancy weights learned from sinusoidal trajectory projection data, revealing significant parameter redundancy in the original model. By integrating PCA directly into the differentiable shift-variant FBP reconstruction pipeline, we develop a method that decomposes the redundancy weight layer parameters into a trainable eigenvector matrix, compressed weights, and a mean vector. This innovative technique achieves a remarkable 97.25% reduction in trainable parameters without compromising reconstruction accuracy. As a result, our algorithm significantly decreases the complexity of the differentiable shift-variant FBP model and greatly improves training speed. These improvements make the model substantially more practical for real-world applications.

Linda-Sophie Schneider, Patrick Krauss, Nadine Schiering et al.

Mathematical models are vital to the field of metrology, playing a key role in the derivation of measurement results and the calculation of uncertainties from measurement data, informed by an understanding of the measurement process. These models generally represent the correlation between the quantity being measured and all other pertinent quantities. Such relationships are used to construct measurement systems that can interpret measurement data to generate conclusions and predictions about the measurement system itself. Classic models are typically analytical, built on fundamental physical principles. However, the rise of digital technology, expansive sensor networks, and high-performance computing hardware have led to a growing shift towards data-driven methodologies. This trend is especially prominent when dealing with large, intricate networked sensor systems in situations where there is limited expert understanding of the frequently changing real-world contexts. Here, we demonstrate the variety of opportunities that data-driven modeling presents, and how they have been already implemented in various real-world applications.