arXiv subjects

Elima Shehu

Publications and source records attributed to Elima Shehu.

8 recordsLinked to original sources

arXiv2026

Triangulation of Points Constrained to a Plane

We study the set of image tuples arising from fixed cameras observing varying planar 3-dimensional point configurations. We derive a formula for the number of complex critical points of the triangulation problem, which seeks to reconstruct such configurations from noisy image data. Valid for an arbitrary number of views, this formula quantifies the intrinsic algebraic complexity of planar triangulation. We validate our theoretical findings through numerical experiments on both synthetic and real data, demonstrating that incorporating the planar incidence constraints leads to faster point reconstruction and improved accuracy compared to unconstrained triangulation.

math.AG↗

arXiv2024

SAGBI and Gr\"obner Bases Detection

We introduce a detection algorithm for SAGBI basis in polynomial rings, analogous to a Gr\"obner basis detection algorithm previously proposed by Gritzmann and Sturmfels. We also present two accompanying software packages named SagbiGbDetection for Macaulay2 and Julia. Both packages allow the user to find one or more term orders for which a set of input polynomials form either Gr\"obner basis for the ideal they generate or a SAGBI basis for the subalgebra. Additionally, we investigate the computational complexity of homogeneous SAGBI detection and apply our implementation to several novel examples.

math.AC↗

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

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 Gr\"obner bases and discuss to what extent our results carry over to the non-generic case.

math.AG↗

arXiv2023

A note on the rate of convergence of integration schemes for closed surfaces

In this paper, we issue an error analysis for integration over discrete surfaces using the surface parametrization presented in [PS22] as well as prove why even-degree polynomials exhibit a higher convergence rate than odd-degree polynomials. Additionally, we provide some numerical examples that illustrate our findings and propose a potential approach that overcomes the problems associated with the original one.

math.NA↗

arXiv2022

Average degree of the essential variety

The essential variety is an algebraic subvariety of dimension $5$ in real projective space $\mathbb R\mathrm P^{8}$ which encodes the relative pose of two calibrated pinhole cameras. The $5$-point algorithm in computer vision computes the real points in the intersection of the essential variety with a linear space of codimension $5$. The degree of the essential variety is $10$, so this intersection consists of 10 complex points in general. We compute the expected number of real intersection points when the linear space is random. We focus on two probability distributions for linear spaces. The first distribution is invariant under the action of the orthogonal group $\mathrm{O}(9)$ acting on linear spaces in $\mathbb R\mathrm P^{8}$. In this case, the expected number of real intersection points is equal to $4$. The second distribution is motivated from computer vision and is defined by choosing 5 point correspondences in the image planes $\mathbb R\mathrm P^2\times \mathbb R\mathrm P^2$ uniformly at random. A Monte Carlo computation suggests that with high probability the expected value lies in the interval $(3.95 - 0.05,\ 3.95 + 0.05)$.

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

Computing algebraic degrees of phylogenetic varieties

A phylogenetic variety is an algebraic variety parameterized by a statistical model of the evolution of biological sequences along a tree. Understanding this variety is an important problem in the area of algebraic statistics with applications in phylogeny reconstruction. In the broader area of algebra statistics, there have been important theoretical advances in computing certain invariants associated to algebraic varieties arising in applications. Beyond the dimension and degree of a variety, one is interested in computing other algebraic degrees, such as the maximum likelihood degree and the Euclidean distance degree. Despite these efforts, the current literature lacks explicit computations of these invariants for the particular case of phylogenetic varieties. In our work, we fill this gap by computing these invariants for phylogenetic varieties arising from the simplest group-based models of nucleotide substitution Cavender-Farris-Neyman model, Jukes-Cantor model, Kimura 2-parameter model, and the Kimura 3-parameter model on small phylogenetic trees with at most 5 leaves.

q-bio.PE↗