Deformable medical image registration is a fundamental task in medical image analysis. While deep learning-based methods have demonstrated superior accuracy and efficiency, they often overlook the critical role of regularization in ensuring robustness and anatomical plausibility. We propose DARE (Deformable Adaptive Regularization Estimator), a novel registration framework that dynamically adjusts elastic regularization based on the gradient norm of the deformation field. The method integrates strain and shear energy terms, adaptively modulated to balance deformation stability and flexibility, and includes a folding-prevention mechanism that penalizes regions with negative deformation Jacobian to reduce folding artifacts and encourage topologically plausible mappings. Extensive experiments on the IXI, OASIS, and MUI-P datasets demonstrate that DARE consistently improves registration accuracy while preserving anatomical plausibility. Compared with classical regularizers and state-of-the-art learning-based methods, DARE achieves higher Dice scores and reduces folding artifacts, while maintaining low strain energy and realistic volume changes in key brain structures. These results confirm that the proposed adaptive regularization mechanism delivers robust, accurate, and physiologically consistent deformation fields across clinical scenarios related to brain registration. • We introduce DARE, a novel Deformable Adaptive Regularization Estimator for learning-based medical image registration. It is an unsupervised method designed to enhance both anatomical plausibility and registration accuracy. • Our approach incorporates adaptively modulated strain and shear energy terms to dynamically balance stability and flexibility, addressing the limitations of traditional uniform regularization in learning-based methods. • The proposed framework is highly adaptive and context-aware, making it particularly effective for clinical applications that require precision and structural consistency. It demonstrates strong potential for improving the reliability and effectiveness of medical image registration.

Late gadolinium enhancement (LGE) cardiac magnetic resonance (CMR) imaging is considered the in vivo reference standard for assessing infarct size (IS) and microvascular obstruction (MVO) in ST-elevation myocardial infarction (STEMI) patients. However, the exact quantification of those markers of myocardial infarct severity remains challenging and very time-consuming. As LGE distribution patterns can be quite complex and hard to delineate from the blood pool or epicardial fat, automatic segmentation of LGE CMR images is challenging. In this work, we propose a cascaded framework of two-dimensional and three-dimensional convolutional neural networks (CNNs) which enables to calculate the extent of myocardial infarction in a fully automated way. By artificially generating segmentation errors which are characteristic for 2D CNNs during training of the cascaded framework we are enforcing the detection and correction of 2D segmentation errors and hence improve the segmentation accuracy of the entire method. The proposed method was trained and evaluated on two publicly available datasets. We perform comparative experiments where we show that our framework outperforms state-of-the-art reference methods in segmentation of myocardial infarction. Furthermore, in extensive ablation studies we show the advantages that come with the proposed error correcting cascaded method. The code of this project is publicly available at https://github.com/matthi99/EcorC.git.

Abstract. Inverse problems are fundamental in fields like medical imaging, geophysics, and computerized tomography, aiming to recover unknown quantities from observed data. However, these problems often lack stability due to noise and ill-conditioning, leading to inaccurate reconstructions. To mitigate these issues, regularization methods are employed, introducing constraints to stabilize the inversion process and achieve a meaningful solution. Recent research has shown that the application of regularizing filters to diagonal frame decompositions (DFDs) yields regularization methods. These filters dampen certain frame coefficients to prevent noise amplification. This paper introduces a nonlinear filtered DFD method, inspired by a learning strategy. In experiments, we learned optimal nonlinear filters from training data pairs without imposing constraints. Although the learned filters were found to be strictly increasing, they did not satisfy the nonexpansiveness condition required to link them with convex regularizers and prove stability and convergence in the sense of regularization methods in previous works. Motivated by this, the paper relaxes the nonexpansiveness condition, resulting in weakly convex regularization. Despite the relaxed assumptions, we managed to derive stability, convergence, and convergence rates with respect to the absolute symmetric Bregman distance for the learned nonlinear regularizing filters. We illustrate our findings by applying the method to the inversion of the Radon transform using 500 image-sinogram training pairs from real CT scans. Our results show that the learned optimal nonlinear filters in the filtered DFD not only yield accurate reconstructions but also closely align with our theoretical assumptions.

Volumetric medical image registration plays a vital role in computer-aided diagnosis and has recently benefited from Transformer-based learning techniques. Despite their effective global modeling capabilities, Transformers require a large number of parameters and significant computational resources, which limits their use in low-resource environments. Furthermore, they depend heavily on large datasets for optimal performance. Conversely, CNN-based methods excel in capturing detailed local information but face difficulties with global modeling and long-distance interactions. This study aims to develop a method that offers an extended receptive field, uses fewer parameters, and achieves notable results with a limited training dataset. We introduce the SIRU-Net (Smoothing Inception Residual U-Net) architecture, which combines an inception module to extend the receptive field, a residual module to enhance feature quality, and Gaussian convolution to improve information quality in the deeper stages of the network. Evaluations on inter-patient and atlas-patient datasets with limited data show that our approach outperforms transformer-based methods while utilizing only 5 % of their parameters.

Non-uniqueness and instability are characteristic features of image reconstruction methods. As a result, it is necessary to develop regularization methods that can be used to compute reliable approximate solutions. A regularization method provides a family of stable reconstructions that converge to a specific solution of the noise-free problem as the noise level tends to zero. The standard regularization technique is defined by a variational image reconstruction that 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 (PnP) image reconstruction 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 have been studied so far. In this work, we extend the idea of PnP by considering families of PnP iterations, each 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 as mathematically justified for robust image reconstruction as variational methods.

State-of-the-art methods for quantifying wear in cylinder liners of large internal combustion engines for stationary power generation require disassembly and cutting of the examined liner. This is followed by laboratory-based high-resolution microscopic surface depth measurement that quantitatively evaluates wear based on bearing load curves (also known as Abbott-Firestone curves). Such reference methods are destructive, time-consuming and costly. The goal of the research presented here is to develop nondestructive yet reliable methods for quantifying the surface topography. A novel machine learning framework is proposed that allows prediction of the bearing load curves representing the depth profiles from reflection RGB images of the liner surface. These images can be collected with a simple handheld microscope. A joint deep learning approach involving two neural network modules optimizes the prediction quality of surface roughness parameters as well. The network stack is trained using a custom-built database containing 422 perfectly aligned depth profile and reflection image pairs of liner surfaces of large gas engines. The observed success of the method suggests its great potential for on-site wear assessment of engines during service.

Abstract Real-time estimation of actual object depth is an essential module for various autonomous system tasks such as 3D reconstruction, scene understanding and condition assessment. During the last decade of machine learning, extensive deployment of deep learning methods to computer vision tasks has yielded approaches that succeed in achieving realistic depth synthesis out of a simple RGB modality. Most of these models are based on paired RGB-depth data and/or the availability of video sequences and stereo images. However, the lack of RGB-depth pairs, video sequences, or stereo images makes depth estimation a challenging task that needs to be explored in more detail. This study builds on recent advances in the field of generative neural networks in order to establish fully unsupervised single-shot depth estimation. Two generators for RGB-to-depth and depth-to-RGB transfer are implemented and simultaneously optimized using the Wasserstein-1 distance, a novel perceptual reconstruction term, and hand-crafted image filters. We comprehensively evaluate the models using a custom-generated industrial surface depth data set as well as the Texas 3D Face Recognition Database, the CelebAMask-HQ database of human portraits and the SURREAL dataset that records body depth. For each evaluation dataset, the proposed method shows a significant increase in depth accuracy compared to state-of-the-art single-image transfer methods.

Segmentation of specific brain tissue from MRI volumes is of great significance for brain disease diagnosis, progression assessment, and monitoring of neurological conditions. Manual segmentation is time-consuming, laborious, and subjective, which significantly amplifies the need for automated processes. Over the last decades, the active development in the field of deep learning, especially convolutional neural networks (CNNs), and the associated performance improvements have increased the demand for the application of CNN-based methods to provide consistent measurements and quantitative analyses. In this paper, we present an efficient deep learning approach for the segmentation of brain tissue. More specifically, we address the problem of segmentation of the posterior limb of the internal capsule (PLIC) in preterm neonates. To this end, we propose a CNN-based pipeline comprised of slice-selection modules and a multi-view segmentation model, which exploits the 3D information contained in the MRI volumes to improve segmentation performance. One special feature of the proposed method is its ability to identify one desired slice out of the whole image volume, which is relevant for pediatricians in terms of prognosis. To increase computational efficiency, we apply a strategy that automatically reduces the information contained in the MRI volumes to its relevant parts. Finally, we conduct an expert rating alongside standard evaluation metrics, such as dice score, to evaluate the performance of the proposed framework. We demonstrate the benefit of the multi-view technique by comparing it with its single-view counterparts, which reveals that the proposed method strikes a good balance between exploiting the available image information and reducing the required computing power compared to 3D segmentation networks. Standard evaluation metrics as, well as expert-based assessment, confirm the good performance of the proposed framework, with the latter being more relevant in terms of clinical applicability. We demonstrate that the proposed deep learning pipeline can compete with the experts in terms of accuracy. To prove the generalisability of the proposed method, we additionally assess our deep learning pipeline to data from the Developing Human Connectome Project (dHCP).

Photoacoustic tomography (PAT) is a non-invasive imaging modality that requires recovering the initial data of the wave equation from certain measurements of the solution outside the object. In the standard PAT measurement setup, the used data consist of time-dependent signals measured on an observation surface. In contrast, the measured data from the recently invented full-field detection technique provide the solution of the wave equation on a spatial domain at a single instant in time. While reconstruction using classical PAT data has been extensively studied, not much is known for the full field PAT problem. In this paper, we build mathematical foundations of the latter problem for variable sound speed and settle its uniqueness and stability. Moreover, we introduce an exact inversion method using time-reversal and study its convergence. Our results demonstrate the suitability of both the full field approach and the proposed time-reversal technique for high resolution photoacoustic imaging.

Performing a large number of spatial measurements enables high-resolution photoacoustic imaging without specific prior information. However, the acquisition of spatial measurements is time-consuming, costly, and technically challenging. By exploiting nonlinear prior information, compressed sensing techniques in combination with sophisticated reconstruction algorithms allow reducing the number of measurements while maintaining high spatial resolution. To this end, in this work we propose a multiscale factorization for the wave equation that decomposes the measured data into a low-frequency factor and sparse high-frequency factors. By extending the acoustic reciprocity principle, we transfer sparsity in the measurement domain into spatial sparsity of the initial pressure, which allows the use of sparse reconstruction techniques. Numerical results are presented that demonstrate the feasibility of the proposed framework.

In this article we study the problem of recovering the initial data of the two-dimensional wave equation from Neumann measurements on a convex domain $\Omega\subset\mathbb{R}^2$ with smooth boundary. We derive an explicit inversion formula of a so-called back-projection type and deduce exact inversion formulas for circular and elliptical domains. In addition, for circular domains, we show that the initial data can also be recovered from any linear combination of its solution and its normal derivative on the boundary. Numerical results of our implementation of the derived inversion formulas are presented demonstrating their accuracy and stability.

Block coordinate descent (BCD) methods approach optimization problems by performing gradient steps along alternating subgroups of coordinates. This is in contrast to full gradient descent, where a gradient step updates all coordinates simultaneously. BCD has been demonstrated to accelerate the gradient method in many practical large-scale applications. Despite its success no convergence analysis for inverse problems is known so far. In this paper, we investigate the BCD method for solving linear inverse problems. As the main theoretical result, we show that for operators having a particular tensor product form, the BCD method combined with an appropriate stopping criterion yields a convergent regularization method. To illustrate the theory, we perform numerical experiments comparing the BCD and the full gradient descent method for a system of integral equations. We also present numerical tests for a nonlinear inverse problem not covered by our theory, namely one-step inversion in multispectral X-ray tomography.

We introduce a new iterative regularization method for solving inverse problems that can be written as systems of linear or nonlinear equations in Hilbert spaces. The proposed averaged Kaczmarz (AVEK) method can be seen as a hybrid method between the Landweber and the Kaczmarz methods. As the Kaczmarz method, the proposed method only requires evaluation of one direct and one adjoint subproblem per iterative update. On the other hand, similar to the Landweber iteration, it uses an average over previous auxiliary iterates which increases stability. We present a convergence analysis of the AVEK iteration. Further, detailed numerical studies are presented for a tomographic image reconstruction problem, namely the limited data problem in photoacoustic tomography. Thereby, the AVEK is compared with other iterative regularization methods including standard Landweber and Kaczmarz iterations, as well as recently proposed accelerated versions based on error minimizing relaxation strategies.

In this article, we revisit iterative methods for solving the inverse source problem of photoacoustic tomography in free space. Recently, there have been interesting developments on explicit formulations of the adjoint operator, demonstrating that iterative methods are an attractive choice for photoacoustic image reconstruction. In this work, we propose several modifications of current formulations of the adjoint operator which help speed up the convergence and yield improved error estimates. We establish a stability analysis and show that, with our choices of the adjoint operator, the iterative methods can achieve a linear rate of convergence, in the $L^2$-norm as well as $H^1$-norm. In addition, we analyze the normal operator from the microlocal analysis point of view. This gives insight into the convergence speed of the iterative methods and choosing proper weights for the mapping spaces. Finally, we present numerical results using various iterative reconstruction methods for full as well as limited view data. Our results demonstrate that Nesterov's fast gradient and the CG methods converge faster than Landweber's and iterative time reversal methods in the visible as well as the invisible case.

Single photon emission computed tomography (SPECT) is a well-established clinical tool for functional imaging. A limitation of current SPECT systems is the use of mechanical collimation, where only a small fraction of the emitted photons are actually used for image reconstruction. This results in a large noise level and finally in a limited spatial resolution. In order to decrease the noise level and to increase the imaging resolution, Compton cameras have been proposed as an alternative to mechanical collimators. Image reconstruction in SPECT with Compton cameras yields the problem of recovering a marker distribution from integrals over conical surfaces. Due to this and other applications, such conical Radon transforms recently got significant attention. In the current paper we consider the case where the cones of integration have vertices on a circular cylinder and axis pointing to the symmetry axis of the cylinder. Our setup does not use all emitted photons but a much larger fraction than systems based on mechanical collimation. Further, it may be simpler to be fabricated than a Compton camera system collecting full five-dimensional data. As main theoretical results in this paper we derive analytic reconstruction methods for the considered transform. We also investigate the V-line transform with vertices on a circle and symmetry axis orthogonal to the circle, which arises in the special case where the absorber distribution is located in a horizontal plane.

Recovering a function from circular or spherical mean values is the basis of many modern imaging technologies, such as photo and thermoacoustic computed tomography or ultrasound reflection tomography. Recently, much progress has been made concerning the problem of recovering a function from its circular mean values (its circular Radon transform). In particular, theoretically exact inversion formulas of the back-projection type have been discovered using continuously sampled data. In practical applications, however, only a discrete number of circular mean values can be collected. In this paper, we address this issue in the context of the Shannon sampling theory. We derive sharp sampling conditions for the number of angular and radial samples, such that any essentially b0-bandlimited function can be recovered from a finite number of such circular mean values.

Determining a function from its circular means with restricted centers is crucial for many modern medical imaging and remote sensing applications. Examples include photo- and thermoacoustic tomography, ultrasound imaging, and synthetic aperture radar. In this paper we derive an explicit inversion formula for the case where the centers of the circles of integration are restricted to the boundary of parabolic domains. A similar result is obtained for recovering the initial data of the wave equation from data on the boundary of a parabolic domain. These results show that the universal back-projection formulas, previously known to be exact for half-spaces, discs, and ellipsoids, are also exact for parabolic domains. We further present numerical simulations supporting our theoretical results. There, we also include two previously proposed strategies (dynamic aperture length correction and weight factor correction) accounting for the truncation of the parabola in practical applications.

In this paper we study the inversion of a generalized Radon transform that maps a function in three dimensional space to a family of cylindrical integrals. We derive local backprojection-type inversion formulas for this cylindrical Radon transform. Our inversion formulas can be implemented in a straightforward manner with $\mathcal{O}(\mathtt{N}^{4/3})$ floating point operations, where $\mathtt{N}$ is the number of spatial reconstruction points. Numerical results are presented which demonstrate the accuracy and validity of the derived algorithms.

Fourier reconstruction algorithms significantly outperform conventional backprojection algorithms in terms of computation time. In photoacoustic imaging, these methods require interpolation in the Fourier space domain, which creates artifacts in reconstructed images. We propose a novel reconstruction algorithm that applies the one-dimensional nonuniform fast Fourier transform to photoacoustic imaging. It is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of Fourier reconstruction.