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.

Abstract. The roto-translation group [Formula: see text] has been of active interest in image analysis due to methods that lift the image data to multiorientation representations defined on this Lie group. This has led to impactful applications of crossing-preserving flows for image denoising, 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 [Formula: see text]. We make several theoretical contributions (generalizable to matrix Lie groups) such as the nonoptimality 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 [Formula: see text]. We observe that our framework of lifting images to [Formula: see text] 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 [Formula: see text].

Breast cancer is the most common invasive cancer with the highest cancer occurrence in females. Handheld ultrasound is one of the most efficient ways to identify and diagnose the breast cancer. The area and the shape information of a lesion is very helpful for clinicians to make diagnostic decisions. In this study we propose a new deep-learning scheme, semi-pixel-wise cycle generative adversarial net (SPCGAN) for segmenting the lesion in 2D ultrasound. The method takes the advantage of a fully convolutional neural network (FCN) and a generative adversarial net to segment a lesion by using prior knowledge. We compared the proposed method to a fully connected neural network and the level set segmentation method on a test dataset consisting of 32 malignant lesions and 109 benign lesions. Our proposed method achieved a Dice similarity coefficient (DSC) of 0.92 while FCN and the level set achieved 0.90 and 0.79 respectively. Particularly, for malignant lesions, our method increases the DSC (0.90) of the fully connected neural network to 0.93 significantly (p 0.001). The results show that our SPCGAN can obtain robust segmentation results. The framework of SPCGAN is particularly effective when sufficient training samples are not available compared to FCN. Our proposed method may be used to relieve the radiologists' burden for annotation.

We propose a template matching method for the detection of 2D image objects that are characterized by orientation patterns. Our method is based on data representations via orientation scores, which are functions on the space of positions and orientations, and which are obtained via a wavelet-type transform. This new representation allows us to detect orientation patterns in an intuitive and direct way, namely via cross-correlations. Additionally, we propose a generalized linear regression framework for the construction of suitable templates using smoothing splines. Here, it is important to recognize a curved geometry on the position-orientation domain, which we identify with the Lie group SE(2): the roto-translation group. Templates are then optimized in a B-spline basis, and smoothness is defined with respect to the curved geometry. We achieve state-of-the-art results on three different applications: detection of the optic nerve head in the retina (99.83 percent success rate on 1,737 images), of the fovea in the retina (99.32 percent success rate on 1,616 images), and of the pupil in regular camera images (95.86 percent on 1,521 images). The high performance is due to inclusion of both intensity and orientation features with effective geometric priors in the template matching. Moreover, our method is fast due to a cross-correlation based matching approach.

Retinal image analysis is a challenging problem due to the precise quantification required and the huge numbers of images produced in screening programs. This paper describes a series of innovative brain-inspired algorithms for automated retinal image analysis, recently developed for the RetinaCheck project, a large-scale screening program for diabetic retinopathy and other retinal diseases in Northeast China. The paper discusses the theory of orientation scores, inspired by cortical multi-orientation pinwheel structures, and presents applications for automated quality assessment, optic nerve head detection, crossing-preserving enhancement and segmentation of retinal vasculature, arterio-venous ratio, fractal dimension, and vessel tortuosity and bifurcations. Many of these algorithms outperform state-of-the-art techniques. The methods are currently validated in collaborating hospitals, with a rich accompanying base of metadata, to phenotype and validate the quantitative algorithms for optimal classification power.

This paper presents a robust and fully automatic filter-based approach for retinal vessel segmentation. We propose new filters based on 3D rotating frames in so-called orientation scores, which are functions on the Lie-group domain of positions and orientations [Formula: see text]. By means of a wavelet-type transform, a 2D image is lifted to a 3D orientation score, where elongated structures are disentangled into their corresponding orientation planes. In the lifted domain [Formula: see text], vessels are enhanced by means of multi-scale second-order Gaussian derivatives perpendicular to the line structures. More precisely, we use a left-invariant rotating derivative (LID) frame, and a locally adaptive derivative (LAD) frame. The LAD is adaptive to the local line structures and is found by eigensystem analysis of the left-invariant Hessian matrix (computed with the LID). After multi-scale filtering via the LID or LAD in the orientation score domain, the results are projected back to the 2D image plane giving us the enhanced vessels. Then a binary segmentation is obtained through thresholding. The proposed methods are validated on six retinal image datasets with different image types, on which competitive segmentation performances are achieved. In particular, the proposed algorithm of applying the LAD filter on orientation scores (LAD-OS) outperforms most of the state-of-the-art methods. The LAD-OS is capable of dealing with typically difficult cases like crossings, central arterial reflex, closely parallel and tiny vessels. The high computational speed of the proposed methods allows processing of large datasets in a screening setting.

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton--Jacobi--Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case $\mathcal{C}=1$. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.

HARDI (High Angular Resolution Diffusion Imaging) is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. In this article we study left-invariant diffusion on the group of 3D rigid body movements (i.e. 3D Euclidean motion group) SE(3) and its application to crossing-preserving smoothing of HARDI images. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space of positions and orientations in 3D embedded in SE(3) and can be solved by ℝ3 ⋊ S 2-convolution with the corresponding Green’s functions. We provide analytic approximation formulas and explicit sharp Gaussian estimates for these Green’s functions. In our design and analysis for appropriate (nonlinear) convection-diffusions on HARDI data we explain the underlying differential geometry on SE(3). We write our left-invariant diffusions in covariant derivatives on SE(3) using the Cartan connection. This Cartan connection has constant curvature and constant torsion, and so have the exponential curves which are the auto-parallels along which our left-invariant diffusion takes place. We provide experiments of our crossing-preserving Euclidean-invariant diffusions on artificial HARDI data containing crossing-fibers.

Many image processing problems require the enhancement of crossing elongated structures. These problems cannot easily be solved by commonly used coherence-enhancing diffusion methods. Therefore, we propose a method for coherence-enhancing diffusion on the invertible orientation score of a 2D image. In an orientation score, the local orientation is represented by an additional third dimension, ensuring that crossing elongated structures are separated from each other. We consider orientation scores as functions on the Euclidean motion group, and use the group structure to apply left-invariant diffusion equations on orientation scores. We describe how we can calculate regularized left-invariant derivatives, and use the Hessian to estimate three descriptive local features: curvature, deviation from horizontality, and orientation confidence. These local features are used to adapt a nonlinear coherence-enhancing, crossing-preserving, diffusion equation on the orientation score. We propose two explicit finite-difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis. Experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherence-enhancing diffusion. The use of curvature leads to improved enhancement of curves with high curvature. Furthermore, the use of deviation from horizontality makes it feasible to reduce the number of sampled orientations while still preserving crossings.

The reconstruction problem is usually formulated as a variational problem in which one searches for that image that minimizes a so called prior (image model) while insisting on certain image features to be preserved. When the prior can be described by a norm induced by some inner product on a Hilbert space, the exact solution to the variational problem can be found by orthogonal projection. In previous work we considered the image as compactly supported in $\mathbb{L}_{2}(\mathbb{R}^{2})$ and we used Sobolev norms on the unbounded domain including a smoothing parameter γ>0 to tune the smoothness of the reconstructed image. Due to the assumption of compact support of the original image, components of the reconstructed image near the image boundary are too much penalized. Therefore, in this work we minimize Sobolev norms only on the actual image domain, yielding much better reconstructions (especially for γ≫0). As an example we apply our method to the reconstruction of singular points that are present in the scale space representation of an image.

In this paper we describe a method for estimating curvature of elongated structures in images. The curvature estimation is performed on an invertible orientation score, which is a 3D entity obtained from a 2D image by convolution with a rotating kernel. By considering the group structure we can define left-invariant derivatives, which are essential to construct operations on the orientation score that amount to rotationally invariant operations on the corresponding image. The problem of estimating curvature of an oriented structure is stated as a minimization problem, which can be solved by eigenvector analysis of a matrix constructed from the non-symmetric Hessian matrix. The experiments show the method performs well for a wide range of curvatures and noise levels. The method clearly outperforms a related curvature estimation method by Van Ginkel et al. that tends to give estimates that are too small. We show how we can incorporate the curvature estimate in our method for coherence-enhancing diffusion in orientation scores. This method has superior performance in enhancing crossing contours, which is demonstrated on medical images.

It is well-known that scale space theory and Tikhonov regularization are close-knit. In previous studies, qualitative analogies and formal relations have already been found, but none of these established both an exact as well as operational connection. We establish such a connection for the case of a Gaussian scale space representation and a first order Tikhonov scheme. The free parameter of the latter turns out to be related to a particular attenuation of scales in a procedure whereby one "collapses" the scale space image along the scale axis via the Laplace transform. The result provides a physical interpretation of first order Tikhonov regularization and its associated control parameter.