arXiv subjects

Jana Vráblíková

Publications and source records attributed to Jana Vráblíková.

3 recordsLinked to original sources

arXiv2026

Efficient closed-form approaches for pose estimation using Sylvester forms

Solving non-linear least-squares problem for pose estimation (rotation and translation) is often a time consuming yet fundamental problem in several real-time computer vision applications. With an adequate rotation parametrization, the optimization problem can be reduced to the solution of a~system of polynomial equations and solved in closed form. Recent advances in efficient closed form solvers utilizing resultant matrices have shown a promising research direction to decrease the computation time while preserving the estimation accuracy. In this paper, we propose a new class of resultant-based solvers that exploit Sylvester forms to further reduce the complexity of the resolution. We demonstrate that our proposed methods are numerically as accurate as the state-of-the-art solvers, and outperform them in terms of computational time. We show that this approach can be applied for pose estimation in two different types of problems: estimating a pose from 3D to 3D correspondences, and estimating a pose from 3D points to 2D points correspondences.

cs.CV↗

arXiv2025

Lie Group Approach to Envelope Surfaces

In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and Lie algebras, we rigorously capture the inherent symmetry and linearity in the computation of envelopes. In particular, the possible set of characteristic curves (which constitute the envelope surface) can be precomputed as the intersection of a fixed canonical surface and a low-dimensional set of its possible "derivatives." To demonstrate the effectiveness of our approach, we present several examples of surfaces undergoing transformations from various Lie groups. As a remarkable side result, we show that the characteristic curves and the envelopes of cones undergoing rational motions are themselves rational. Furthermore, we provide an explicit rational parameterization of these envelopes and use it to solve the trimming problem.

math.DG↗

arXiv2025

Arc Spline Approximation of Envelopes of Evolving Planar Domains

Computing the envelope of deforming planar domains is a significant and challenging problem with a wide range of potential applications. We approximate the envelope using circular arc splines, curves that balance geometric flexibility and computational simplicity. Our approach combines two concepts to achieve these benefits. First, we represent a planar domain by its medial axis transform (MAT), which is a geometric graph in Minkowski space $\mathbb R^{2,1}$ (possibly with degenerate branches). We observe that circular arcs in the Minkowski space correspond to MATs of arc spline domains. Furthermore, as a planar domain evolves over time, each branch of its MAT evolves and forms a surface in the Minkowski space. This allows us to reformulate the problem of envelope computation as a problem of computing cyclographic images of finite sets of curves on these surfaces. We propose and compare two pairs of methods for approximating the curves and boundaries of their cyclographic images. All of these methods result in an arc spline approximation of the envelope of the evolving domain. Second, we exploit the geometric flexibility of circular arcs in both the plane and Minkowski space to achieve a high approximation rate. The computational simplicity ensures the efficient trimming of redundant branches of the generated envelope using a sweep line algorithm with optimal computational complexity.

cs.CG↗