arXiv subjects

Felix Rydell

Publications and source records attributed to Felix Rydell.

17 recordsLinked to original sources

arXiv2026

Neuromorphic Monocular Depth Estimation with Uncertainty Modeling

Event cameras offer distinct advantages over conventional frame-based sensors, including microsecond-level temporal resolution, high dynamic range, and low bandwidth. In this paper, we predict per-pixel depth distributions from monocular event streams using deep neural networks. We estimate uncertainty using Gaussian, log-normal, and evidential learning frameworks. We compare six event representations: spatio-temporal voxel grids with 1, 5, 10, and 20 temporal bins, the Compact Spatio-Temporal Representation (CSTR), and Time-Ordered Recent Event (TORE) volumes. Our U-Net-based models are trained on synthetic data and then fine-tuned on real sequences. We evaluate performance using absolute relative error, root mean squared error, and the area under the sparsification error. Quantitative results show that the representations perform similarly, while 10 bin log-normal and 5 bin evidential learning perform best across metrics. Our experiments demonstrate that uncertainty estimation can be successfully integrated into event-based monocular depth estimation, and be used to indicate pixels with reliable depth.

cs.CV↗

arXiv2025

Infinite Euclidean Distance Discriminants

We study infinite Euclidean distance discriminants of algebraic varieties, defined as the loci of data points whose fibers under the second projection from the Euclidean distance correspondence are positive-dimensional. In particular, these discriminants contain all data points with infinitely many critical points for the nearest-point problem. We present computer code that computes the infinite Euclidean distance discriminant, and use it to present numerous varieties with nonempty such discriminants. Moreover, we prove that for any data point, the fiber under the second projection is contained in a finite union of hyperspheres centered at that point. For curves, we include a complete characterization; their infinite Euclidean distance discriminants turn out to be affine linear spaces. Finally, we introduce and characterize skew-tube surfaces in three-dimensional space. By construction, these have a one-dimensional infinite Euclidean distance discriminant. We further demonstrate that many skew-tubes have significantly lower Euclidean distance degrees than generic surfaces of the same degree.

math.AG↗

arXiv2025

A Framework for Reducing the Complexity of Geometric Vision Problems and its Application to Two-View Triangulation with Approximation Bounds

In this paper, we present a new framework for reducing the computational complexity of geometric vision problems through targeted reweighting of the cost functions used to minimize reprojection errors. Triangulation - the task of estimating a 3D point from noisy 2D projections across multiple images - is a fundamental problem in multiview geometry and Structure-from-Motion (SfM) pipelines. We apply our framework to the two-view case and demonstrate that optimal triangulation, which requires solving a univariate polynomial of degree six, can be simplified through cost function reweighting reducing the polynomial degree to two. This reweighting yields a closed-form solution while preserving strong geometric accuracy. We derive optimal weighting strategies, establish theoretical bounds on the approximation error, and provide experimental results on real data demonstrating the effectiveness of the proposed approach compared to standard methods. Although this work focuses on two-view triangulation, the framework generalizes to other geometric vision problems.

cs.CV↗

arXiv2024

Three's company in six dimensions: irreducible, isospectral, non-isometric flat tori

In 1964, John Milnor, using a construction of two lattices by Witt, produced the first example of two flat tori that are not globally isometric and whose Laplacians for exterior forms have the same sequence of eigenvalues. The aforementioned flat tori are sixteen-dimensional. One is reducible while the second is irreducible. In the ensuing years, pairs of non-isometric flat tori that share a common Laplace spectrum have been shown to exist in dimensions four and higher. In dimensions three and lower, Alexander Schiemann proved in 1994 that any flat tori that are isospectral are in fact isometric, so four is the lowest dimension in which such pairs exist. Using a four-dimensional such pair, one can easily construct an eight-dimensional such triplet. However, triplets of mutually non-isometric flat tori that share a common Laplace spectrum in dimensions 4, 5, 6, and 7 have eluded researchers - until now. We present here the first example.

math.SP↗

arXiv2024

Revisiting Sampson Approximations for Geometric Estimation Problems

Many problems in computer vision can be formulated as geometric estimation problems, i.e. given a collection of measurements (e.g. point correspondences) we wish to fit a model (e.g. an essential matrix) that agrees with our observations. This necessitates some measure of how much an observation ``agrees" with a given model. A natural choice is to consider the smallest perturbation that makes the observation exactly satisfy the constraints. However, for many problems, this metric is expensive or otherwise intractable to compute. The so-called Sampson error approximates this geometric error through a linearization scheme. For epipolar geometry, the Sampson error is a popular choice and in practice known to yield very tight approximations of the corresponding geometric residual (the reprojection error). In this paper we revisit the Sampson approximation and provide new theoretical insights as to why and when this approximation works, as well as provide explicit bounds on the tightness under some mild assumptions. Our theoretical results are validated in several experiments on real data and in the context of different geometric estimation tasks.

cs.CV↗

arXiv2024

Line Multiview Ideals

We study the following problem in computer vision from the perspective of algebraic geometry: Using $m$ pinhole cameras we take $m$ pictures of a line in $\mathbb P^3$. This produces $m$ lines in $\mathbb P^2$ and the question is which $m$-tuples of lines can arise that way. We are interested in polynomial equations and therefore study the complex Zariski closure of all such tuples of lines. The resulting algebraic variety is a subvariety of $(\mathbb P^2)^m$ and is called line multiview variety. In this article, we study its ideal. We show that for generic cameras the ideal is generated by $3\times 3$-minors of a specific matrix. We also compute Gröbner bases and discuss to what extent our results carry over to the non-generic case.

math.AG↗

arXiv2024

Projections of Higher Dimensional Subspaces and Generalized Multiview Varieties

We present a generalization of multiview varieties as closures of images obtained by projecting subspaces of a given dimension onto several views, from the photographic and geometric points of view. Motivated by applications in Computer Vision for triangulation of world features, we investigate when the associated projection map is generically injective; an essential requirement for successful triangulation. We give a complete characterization of this property by determining two formulae for the dimensions of these varieties. Similarly, we describe for which center arrangements calibration of camera parameters is possible. We explore when the multiview variety is naturally isomorphic to its associated blowup. In the case of generic centers, we give a precise formula for when this occurs.

math.AG↗

arXiv2024

Projections of Curves and Conic Multiview Varieties

We present an algebraic study of the projection of plane curves and twisted cubics in space onto multiple images of pinhole cameras. The Zariski closure of the image of the projection of conics is a conic multiview varieties. Extending previous work for point and line multiview varieties, we use back-projected cones to describe these varieties. For two views, we provide the defining ideal of the multiview variety. For any number of views, we state when the simplest possible set-theoretic description is achieved based on the geometry of the camera centers. Finally, we investigate the complexity of the associated triangulation problem and conjecture the Euclidean distance degree for the conic multiview variety for two cameras.

math.AG↗

arXiv2024

Caustics by Refraction of Circles and Lines

This short note revisits the classical result that the complete caustic by refraction of a circle is the evolute of Cartesian ovals. We provide additional details to the statement and geometric proof of this fact, as presented in G. Salmon's 1879 book `Higher Plane Curves'. We observe that as the circle tends to a line, the Cartesian ovals collapse into an ellipse or a branch of a hyperbola. Further, we derive a general formula for caustics by refraction of circles using a computer algebra system, providing a modern computational perspective on this classical problem.

math.AG↗

arXiv2024

Geometric Interpretations of Compatibility for Fundamental Matrices

In recent work, algebraic computational software was used to provide the exact algebraic conditions under which a sixtuple $\{F^{ij}\}$ of fundamental matrices, corresponding to $4$ images, will be compatible, i.e. there will exist cameras $\{P_i\}_{i=1}^4$ such that each pair $P_i,P_j$ has fundamental matrix $F^{ij}$; it has been further demonstrated that quadruplewise compatibility is sufficient for the problem of $n>4$ images. We expand on these prior results by proving equivalent geometric conditions for compatibility. We find that when the camera centers are in general position, compatibility can be characterized via the intersections of epipolar lines in each image. When the camera centers are coplanar, compatibility occurs when the prior condition holds and additionally any one camera center can be reconstructed via the other three.

math.AG↗

arXiv2024

Metric Multiview Geometry -- a Catalogue in Low Dimensions

We systematically compile an exhaustive catalogue of multiview varieties and anchored multiview varieties arising from projections of points and lines in 1, 2, and 3-dimensional projective space. We say that two such varieties are ED-equivalent if there is a linear isomorphism between that that preserve ED-critical points. This gives rise to fourteen equivalence classes, and we determine various properties - dimension, set-theoretic equations, and multidegrees - for all varieties featured in our catalogue. In the case of points, we also present a complementary study of resectioning varieties and their singular loci. Finally, we propose conjectures for the Euclidean distance degrees of all varieties appearing in our comprehensive compilation.

math.AG↗

arXiv2023

Theoretical and Numerical Analysis of 3D Reconstruction Using Point and Line Incidences

We study the joint image of lines incident to points, meaning the set of image tuples obtained from fixed cameras observing a varying 3D point-line incidence. We prove a formula for the number of complex critical points of the triangulation problem that aims to compute a 3D point-line incidence from noisy images. Our formula works for an arbitrary number of images and measures the intrinsic difficulty of this triangulation. Additionally, we conduct numerical experiments using homotopy continuation methods, comparing different approaches of triangulation of such incidences. In our setup, exploiting the incidence relations gives both a faster point reconstruction and in three views more accurate.

math.AG↗

arXiv2023

Compatibility of Fundamental Matrices for Complete Viewing Graphs

This paper studies the problem of recovering cameras from a set of fundamental matrices. A set of fundamental matrices is said to be compatible if a set of cameras exists for which they are the fundamental matrices. We focus on the complete graph, where fundamental matrices for each pair of cameras are given. Previous work has established necessary and sufficient conditions for compatibility as rank and eigenvalue conditions on the n-view fundamental matrix obtained by concatenating the individual fundamental matrices. In this work, we show that the eigenvalue condition is redundant in the generic and collinear cases. We provide explicit homogeneous polynomials that describe necessary and sufficient conditions for compatibility in terms of the fundamental matrices and their epipoles. In this direction, we find that quadruple-wise compatibility is enough to ensure global compatibility for any number of cameras. We demonstrate that for four cameras, compatibility is generically described by triple-wise conditions and one additional equation involving all fundamental matrices.

math.AG↗

arXiv2022

Line Multiview Varieties

We present an algebraic study of line correspondences for pinhole cameras, in contrast to the thoroughly studied point correspondences. We define the line multiview variety as the Zariski closure of the image of the map projecting lines in 3-space to tuples of image lines in 2-space. We prove that in the case of generic camera matrices the line multiview variety is a determinantal variety and we provide a complete set-theoretic description for any camera arrangement. We investigate basic properties of this variety such as dimension, smoothness, and multidegree. Finally, we give experimental results for the Euclidean distance degree and robustness under noise for the triangulation of lines.

math.AG↗

arXiv2022

The isospectral problem for flat tori from three perspectives

Flat tori are among the only types of Riemannian manifolds for which the Laplace eigenvalues can be explicitly computed. In 1964, Milnor used a construction of Witt to find an example of isospectral non-isometric Riemannian manifolds, a striking and concise result that occupied one page in the Proceedings of the National Academy of Science of the USA. Milnor's example is a pair of 16-dimensional flat tori, whose set of Laplace eigenvalues are identical, in spite of the fact that these tori are not isometric. A natural question is: what is the \em lowest \em dimension in which such isospectral non-isometric pairs exist? This isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language. We explore this question from all three perspectives and describe its resolution by Schiemann in the 1990s. Moreover, we share a number of open problems.

math.SP↗

arXiv2021

Adjoints and Canonical Forms of Polypols

Polypols are natural generalizations of polytopes, with boundaries given by nonlinear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an algebro-geometric perspective. We relate planar polypols to positive geometries introduced originally in particle physics, and identify the adjoint curve of a planar polypol with the numerator of the canonical differential form associated with the positive geometry. We settle several cases of a conjecture by Wachspress claiming that the adjoint curve of a regular planar polypol does not intersect its interior. In particular, we provide a complete characterization of the real topology of the adjoint curve for arbitrary convex polygons. Finally, we determine all types of planar polypols such that the rational map sending a polypol to its adjoint is finite, and explore connections of our topic with algebraic statistics.

math.AG↗

arXiv2021

Maximum Likelihood Estimation for Nets of Conics

We study the problem of maximum likelihood estimation for $3$-dimensional linear spaces of $3\times 3$ symmetric matrices from the point of view of algebraic statistics where we view these nets of conics as linear concentration or linear covariance models of Gaussian distributions on $\mathbb{R}^3$. In particular, we study the reciprocal surfaces of nets of conics which are rational surfaces in $\mathbb{P}^5$. We show that the reciprocal surfaces are projections from the Veronese surface and determine their intersection with the polar nets. This geometry explains the maximum likelihood degrees of these linear models. We compute the reciprocal maximum likelihood degrees. This work is based on Wall's classification of nets of conics from 1977.

math.AG↗