arXiv subjects

Remco Duits

Publications and source records attributed to Remco Duits.

At least 19 recordsLinked to original sources

arXiv2026

Evaluation of AI-based Visual Crack Detection in Steel Bridges Using Probability of Detection

Bridge structures are regularly inspected for structural damage such as cracks and corrosion in order to ensure public safety and reduce maintenance costs. Much research has been done on automating this process using computer vision methods, which are often evaluated and compared using metrics such as intersection over union, mean average precision, etc. However, predicting the actual effectiveness of an inspection method within the field of structural engineering from these metrics remains challenging. To enable the systematic use of these increasingly popular methods in engineering practice, evaluating the performance of these methods in a way that is compatible with standard engineering approaches is therefore an urgent necessity. We present a new statistical evaluation framework to allow the comparison of computer vision methods with conventional visual inspection for crack detection in steel bridges. The framework is based on probability of detection curves and can account for the influence of image resolution. We apply this evaluation method to the real-world ``Cracks in Steel Bridges'' dataset, which contains annotated images of cracks in bridge structures. The quantification of the probability of detection and its uncertainty enables a practical assessment of the effect of automated methods for damage detection in structural reliability analyses. In turn, this enables the wide-spread use of automated (AI-based) damage detection in safety critical applications. This evaluation method provides evidence that the proposed computer vision approach approach is robust for the crack detection task and can have a high added value as an addition to conventional visual inspection methods.

cs.CV↗

arXiv2026

Sub-Riemannian Snakes on the Projective Line Bundle with Applications to Segmentation of SEM Images

Geodesic tracking on the projective line bundle $\R^2 \times P^1 $ has many uses, including the segmentation of objects in images. However, global tracking requires expensive distance map computations. We provide a practical solution to this problem by introducing a snake model on $\R^2 \times P^1$, where we only compute the distance map where needed. Our method introduces a geometric criterion for switching between fast spatial snakes and computing minimizing geodesics of a new projective line bundle model. The new pseudo-distance underlying our geometric model is both symmetric and cusp-free, in contrast to previous geodesic sub-Riemannian models on $\R^2 \times P^1$. Our pseudo-distance satisfies the triangle inequality on a large set that we characterize, and includes a connected-component-informed cost function, which is highly advantageous in applications. Experiments on Scanning Electron Microscopy (SEM) images demonstrate our method's robust, automatic segmentation of overlapping electronic structures.

math.DG↗

arXiv2026

Generalized Reduction to the Isotropy for Flexible Equivariant Neural Fields

Many geometric learning problems require invariants on heterogeneous product spaces, i.e., products of distinct spaces carrying different group actions, where standard techniques do not directly apply. We show that, when a group $G$ acts transitively on a space $M$, any $G$-invariant function on a product space $X \times M$ can be reduced to an invariant of the isotropy subgroup $H$ of $M$ acting on $X$ alone. Our approach establishes an explicit orbit equivalence $(X \times M)/G \cong X/H$, yielding a principled reduction that preserves expressivity. We apply this characterization to Equivariant Neural Fields, extending them to arbitrary group actions and homogeneous conditioning spaces, and thereby removing the major structural constraints imposed by existing methods.

cs.LG↗

arXiv2025

Diffusion-Shock PDEs for Deep Learning on Position-Orientation Space

We extend Regularised Diffusion-Shock (RDS) filtering from Euclidean space $\mathbb{R}_2$ [1] to position-orientation space $\mathbb{M}_2 \cong \mathbb{R}^2 \times S^1$. This has numerous advantages, e.g. making it possible to enhance and inpaint crossing structures, since they become disentangled when lifted to $\mathbb{M}_2$. We create a version of the algorithm using gauge frames to mitigate issues caused by lifting to a finite number of orientations. This leads us to study generalisations of diffusion, since the gauge frame diffusion is not generated by the Laplace-Beltrami operator. RDS filtering compares favourably to existing techniques such as Total Roto-Translational Variation (TR-TV) flow, NLM, and BM3D when denoising images with crossing structures, particularly if they are segmented. Furthermore, we see that $\mathbb{M}_2$ RDS inpainting is indeed able to restore crossing structures, unlike $\mathbb{R}^2$ RDS inpainting. In addition to the contributions of our SSVM submission "Diffusion-Shock Filtering on the Space of Positions and Orientations", in this extended work we provide new theorical results and automate RDS filtering by integrating it into a geometric deep learning framework. Regarding our theoretical contributions, we prove that our generalised diffusions are still well-posed, smoothing, and analytic. We developed an RDS filtering PDE layer for the PDE-CNN and PDE-G-CNN deep learning frameworks, using a novel gating mechanism. We show that these new RDS PDE layers can be beneficial in various impainting and denoising tasks.

math.DG↗

arXiv2025

Equivariant Eikonal Neural Networks: Grid-Free, Scalable Travel-Time Prediction on Homogeneous Spaces

We introduce Equivariant Neural Eikonal Solvers, a novel framework that integrates Equivariant Neural Fields (ENFs) with Neural Eikonal Solvers. Our approach employs a single neural field where a unified shared backbone is conditioned on signal-specific latent variables - represented as point clouds in a Lie group - to model diverse Eikonal solutions. The ENF integration ensures equivariant mapping from these latent representations to the solution field, delivering three key benefits: enhanced representation efficiency through weight-sharing, robust geometric grounding, and solution steerability. This steerability allows transformations applied to the latent point cloud to induce predictable, geometrically meaningful modifications in the resulting Eikonal solution. By coupling these steerable representations with Physics-Informed Neural Networks (PINNs), our framework accurately models Eikonal travel-time solutions while generalizing to arbitrary Riemannian manifolds with regular group actions. This includes homogeneous spaces such as Euclidean, position-orientation, spherical, and hyperbolic manifolds. We validate our approach through applications in seismic travel-time modeling of 2D, 3D, and spherical benchmark datasets. Experimental results demonstrate superior performance, scalability, adaptability, and user controllability compared to existing Neural Operator-based Eikonal solver methods.

cs.LG↗

arXiv2025

Analysis and Computation of Geodesic Distances on Reductive Homogeneous Spaces

Many geometric machine learning and image analysis applications, require a left-invariant metric on the 5D homogeneous space of 3D positions and orientations SE(3)/SO(2). This is done in Equivariant Neural Networks (G-CNNs), or in PDE-Based Group Convolutional Neural Networks (PDE-G-CNNs), where the Riemannian metric enters in multilayer perceptrons, message passing, and max-pooling over Riemannian balls. In PDE-G-CNNs it is proposed to take the minimum left-invariant Riemannian distance over the fiber in SE(3)/SO(2), whereas in G-CNNs and in many geometric image processing methods an efficient SO(2)-conjugation invariant section is advocated. The conjecture rises whether that computationally much more efficient section indeed always selects distance minimizers over the fibers. We show that this conjecture does NOT hold in general, and in the logarithmic norm approximation setting used in practice we analyze the small (and sometimes vanishing) differences. We first prove that the minimal distance section is reached by minimal horizontal geodesics with constant momentum and zero acceleration along the fibers, and we generalize this result to (reductive) homogeneous spaces with legal metrics and commutative structure groups.

math.DG↗

arXiv2025

Universal Collection of Euclidean Invariants between Pairs of Position-Orientations

Euclidean E(3) equivariant neural networks that employ scalar fields on position-orientation space M(3) have been effectively applied to tasks such as predicting molecular dynamics and properties. To perform equivariant convolutional-like operations in these architectures one needs Euclidean invariant kernels on M(3) x M(3). In practice, a handcrafted collection of invariants is selected, and this collection is then fed into multilayer perceptrons to parametrize the kernels. We rigorously describe an optimal collection of 4 smooth scalar invariants on the whole of M(3) x M(3). With optimal we mean that the collection is independent and universal, meaning that all invariants are pertinent, and any invariant kernel is a function of them. We evaluate two collections of invariants, one universal and one not, using the PONITA neural network architecture. Our experiments show that using a collection of invariants that is universal positively impacts the accuracy of PONITA significantly.

math.DG↗

arXiv2025

Crossing-Preserving Geodesic Tracking on Spherical Images

In image analysis one often encounters spherical images, for instance in retinal imaging. The behavior of the vessels in the retina is an indicator of several diseases. To automate disease diagnosis using retinal images, it is necessary to develop an algorithm that automatically identifies and tracks vessels. To deal with crossings due to projected blood vessels in the image it is common to lift retinal images to the space of planar positions and orientations $\mathbb{M} := \mathbb{R}^2 \times S^1$. This implicitly assumes that the flat image accurately represents the geometry of the retina. As the eyeball is a sphere (and not a plane), we propose to compute the cusp-free, crossing-preserving geodesics in the space of spherical positions and orientations $\mathbb{W}$ on wide-field images. We clarify how to relate both manifolds and compare the calculated geodesics. The results show clear advantages of crossing-preserving tracking in $\mathbb{W}$ over non-crossing-preserving tracking in $\mathbb{W}$ and are comparable to tracking results in $\mathbb{M}$.

math.DG↗

arXiv2025

Orientation Scores should be a Piece of Cake

We axiomatically derive a family of wavelets for an orientation score, lifting from position space $\mathbb{R}^2$ to position and orientation space $\mathbb{R}^2\times S^1$, with fast reconstruction property, that minimise position-orientation uncertainty. We subsequently show that these minimum uncertainty states are well-approximated by cake wavelets: for standard parameters, the uncertainty gap of cake wavelets is less than 1.1, and in the limit, we prove the uncertainty gap tends to the minimum of 1. Next, we complete a previous theoretical argument that one does not have to train the lifting layer in (PDE-)G-CNNs, but can instead use cake wavelets. Finally, we show experimentally that in this way we can reduce the network complexity and improve the interpretability of (PDE-)G-CNNs, with only a slight impact on the model's performance.

math.DG↗

arXiv2025

Diffusion-Shock Filtering on the Space of Positions and Orientations

We extend Regularised Diffusion-Shock (RDS) filtering from Euclidean space $\mathbb{R}^2$ to the space of positions and orientations $\mathbb{M}_2 := \mathbb{R}^2 \times S^1$. This has numerous advantages, e.g. making it possible to enhance and inpaint crossing structures, since they become disentangled when lifted to $\mathbb{M}_2$. We create a version of the algorithm using gauge frames to mitigate issues caused by lifting to a finite number of orientations. This leads us to study generalisations of diffusion, since the gauge frame diffusion is not generated by the Laplace-Beltrami operator. RDS filtering compares favourably to existing techniques such as Total Roto-Translational Variation (TR-TV) flow, NLM, and BM3D when denoising images with crossing structures, particularly if they are segmented. Additionally, we see that $\mathbb{M}_2$ RDS inpainting is indeed able to restore crossing structures, unlike $\mathbb{R}^2$ RDS inpainting.

math.DG↗

arXiv2024

Connected Components on Lie Groups and Applications to Multi-Orientation Image Analysis

We develop and analyze a new algorithm to find the connected components of a compact set $I$ from a Lie group $G$ endowed with a left-invariant Riemannian distance. For a given $\delta>0$, the algorithm finds the largest cover of $I$ such that all sets in the cover are separated by at least distance $\delta$. We call the sets in the cover the $\delta$-connected components of I (closely related to $\check{\text{C}}$ech complexes of radius $\delta/2$). The grouping relies on an iterative procedure involving morphological dilations with Hamilton-Jacobi-Bellman kernels on $G$ and notions of $\delta$-thickened sets. We prove that the algorithm converges in finitely many iteration steps. We find the optimal value for $\delta$ using persistence diagrams. We also propose specific affinity matrices that allow for grouping of $\delta$-connected components based on their local proximity and alignment. Among the many different applications of the algorithm, in this article, we focus on illustrating that the method can efficiently identify (possibly overlapping) branches in complex vascular trees on retinal images. This is done by applying an orientation score transform to the images that allows us to view them as functions from $\mathbb{L}_2(G)$ where $G=SE(2)$, the Lie group of roto-translations. By applying our algorithm in this Lie group, we illustrate that we obtain $\delta$-connected components that differentiate between crossing structures and that group well-aligned, nearby structures. This contrasts standard connected component algorithms in $\mathbb{R}^2$.

math.DG↗

arXiv2024

Segmentation tool for images of cracks

Safety-critical infrastructures, such as bridges, are periodically inspected to check for existing damage, such as fatigue cracks and corrosion, and to guarantee the safe use of the infrastructure. Visual inspection is the most frequent type of general inspection, despite the fact that its detection capability is rather limited, especially for fatigue cracks. Machine learning algorithms can be used for augmenting the capability of classical visual inspection of bridge structures, however, the implementation of such an algorithm requires a massive annotated training dataset, which is time-consuming to produce. This paper proposes a semi-automatic crack segmentation tool that eases the manual segmentation of cracks on images needed to create a training dataset for a machine learning algorithm. Also, it can be used to measure the geometry of the crack. This tool makes use of an image processing algorithm, which was initially developed for the analysis of vascular systems on retinal images. The algorithm relies on a multi-orientation wavelet transform, which is applied to the image to construct the so-called "orientation scores", i.e. a modified version of the image. Afterwards, the filtered orientation scores are used to formulate an optimal path problem that identifies the crack. The globally optimal path between manually selected crack endpoints is computed, using a state-of-the-art geometric tracking method. The pixel-wise segmentation is done afterwards using the obtained crack path. The proposed method outperforms fully automatic methods and shows potential to be an adequate alternative to the manual data annotation.

cs.CV↗

arXiv2024

Deep Learning for Segmentation of Cracks in High-Resolution Images of Steel Bridges

Automating the current bridge visual inspection practices using drones and image processing techniques is a prominent way to make these inspections more effective, robust, and less expensive. In this paper, we investigate the development of a novel deep-learning method for the detection of fatigue cracks in high-resolution images of steel bridges. First, we present a novel and challenging dataset comprising of images of cracks in steel bridges. Secondly, we integrate the ConvNext neural network with a previous state-of-the-art encoder-decoder network for crack segmentation. We study and report, the effects of the use of background patches on the network performance when applied to high-resolution images of cracks in steel bridges. Finally, we introduce a loss function that allows the use of more background patches for the training process, which yields a significant reduction in false positive rates.

cs.CV↗

arXiv2024

PDE-CNNs: Axiomatic Derivations and Applications

PDE-based Group Convolutional Neural Networks (PDE-G-CNNs) use solvers of evolution PDEs as substitutes for the conventional components in G-CNNs. PDE-G-CNNs can offer several benefits simultaneously: fewer parameters, inherent equivariance, better accuracy, and data efficiency. In this article we focus on Euclidean equivariant PDE-G-CNNs where the feature maps are two-dimensional throughout. We call this variant of the framework a PDE-CNN. From a machine learning perspective, we list several practically desirable axioms and derive from these which PDEs should be used in a PDE-CNN, this being our main contribution. Our approach to geometric learning via PDEs is inspired by the axioms of scale-space theory, which we generalize by introducing semifield-valued signals. Our theory reveals new PDEs that can be used in PDE-CNNs and we experimentally examine what impact these have on the accuracy of PDE-CNNs. We also confirm for small networks that PDE-CNNs offer fewer parameters, increased accuracy, and better data efficiency when compared to CNNs.

cs.LG↗

arXiv2024

Optimal Transport on the Lie Group of Roto-translations

The roto-translation group SE2 has been of active interest in image analysis due to methods that lift the image data to multi-orientation representations defined on this Lie group. This has led to impactful applications of crossing-preserving flows for image de-noising, geodesic tracking, and roto-translation equivariant deep learning. In this paper, we develop a computational framework for optimal transportation over Lie groups, with a special focus on SE2. We make several theoretical contributions (generalizable to matrix Lie groups) such as the non-optimality of group actions as transport maps, invariance and equivariance of optimal transport, and the quality of the entropic-regularized optimal transport plan using geodesic distance approximations. We develop a Sinkhorn like algorithm that can be efficiently implemented using fast and accurate distance approximations of the Lie group and GPU-friendly group convolutions. We report valuable advancements in the experiments on 1) image barycentric interpolation, 2) interpolation of planar orientation fields, and 3) Wasserstein gradient flows on SE2. We observe that our framework of lifting images to SE2 and optimal transport with left-invariant anisotropic metrics leads to equivariant transport along dominant contours and salient line structures in the image. This yields sharper and more meaningful interpolations compared to their counterparts on R^2

cs.CV↗

arXiv2022

Analysis of (sub-)Riemannian PDE-G-CNNs

Group equivariant convolutional neural networks (G-CNNs) have been successfully applied in geometric deep learning. Typically, G-CNNs have the advantage over CNNs that they do not waste network capacity on training symmetries that should have been hard-coded in the network. The recently introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalises G-CNNs. PDE-G-CNNs have the core advantages that they simultaneously 1) reduce network complexity, 2) increase classification performance, and 3) provide geometric interpretability. Their implementations primarily consist of linear and morphological convolutions with kernels. In this paper we show that the previously suggested approximative morphological kernels do not always accurately approximate the exact kernels accurately. More specifically, depending on the spatial anisotropy of the Riemannian metric, we argue that one must resort to sub-Riemannian approximations. We solve this problem by providing a new approximative kernel that works regardless of the anisotropy. We provide new theorems with better error estimates of the approximative kernels, and prove that they all carry the same reflectional symmetries as the exact ones. We test the effectiveness of multiple approximative kernels within the PDE-G-CNN framework on two datasets, and observe an improvement with the new approximative kernels. We report that the PDE-G-CNNs again allow for a considerable reduction of network complexity while having comparable or better performance than G-CNNs and CNNs on the two datasets. Moreover, PDE-G-CNNs have the advantage of better geometric interpretability over G-CNNs, as the morphological kernels are related to association fields from neurogeometry.

cs.LG↗

arXiv2022

Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

We introduce a data-driven version of the plus Cartan connection on the homogeneous space $\mathbb{M}_2$ of 2D positions and orientations. We formulate a theorem that describes all shortest and straight curves (parallel velocity and parallel momentum, respectively) with respect to this new data-driven connection and corresponding Riemannian manifold. Then we use these shortest curves for geodesic tracking of complex vasculature in multi-orientation image representations defined on $\mathbb{M}_{2}$. The data-driven Cartan connection characterizes the Hamiltonian flow of all geodesics. It also allows for improved adaptation to curvature and misalignment of the (lifted) vessel structure that we track via globally optimal geodesics. We compute these geodesics numerically via steepest descent on distance maps on $\mathbb{M}_2$ that we compute by a new modified anisotropic fast-marching method. Our experiments range from tracking single blood vessels with fixed endpoints to tracking complete vascular trees in retinal images. Single vessel tracking is performed in a single run in the multi-orientation image representation, where we project the resulting geodesics back onto the underlying image. The complete vascular tree tracking requires only two runs and avoids prior segmentation, placement of extra anchor points, and dynamic switching between geodesic models. Altogether we provide a geodesic tracking method using a single, flexible, transparent, data-driven geodesic model providing globally optimal curves which correctly follow highly complex vascular structures in retinal images. All experiments in this article can be reproduced via documented Mathematica notebooks available at GitHub (https://github.com/NickyvdBerg/DataDrivenTracking).

math.DG↗

arXiv2020

Roto-Translation Equivariant Convolutional Networks: Application to Histopathology Image Analysis

Rotation-invariance is a desired property of machine-learning models for medical image analysis and in particular for computational pathology applications. We propose a framework to encode the geometric structure of the special Euclidean motion group SE(2) in convolutional networks to yield translation and rotation equivariance via the introduction of SE(2)-group convolution layers. This structure enables models to learn feature representations with a discretized orientation dimension that guarantees that their outputs are invariant under a discrete set of rotations. Conventional approaches for rotation invariance rely mostly on data augmentation, but this does not guarantee the robustness of the output when the input is rotated. At that, trained conventional CNNs may require test-time rotation augmentation to reach their full capability. This study is focused on histopathology image analysis applications for which it is desirable that the arbitrary global orientation information of the imaged tissues is not captured by the machine learning models. The proposed framework is evaluated on three different histopathology image analysis tasks (mitosis detection, nuclei segmentation and tumor classification). We present a comparative analysis for each problem and show that consistent increase of performances can be achieved when using the proposed framework.

cs.CV↗