1,760 research outputs found
Algebra and Geometry of Camera Resectioning
We study algebraic varieties associated with the camera resectioning problem.
We characterize these resectioning varieties' multigraded vanishing ideals
using Gr\"obner basis techniques. As an application, we derive and re-interpret
celebrated results in geometric computer vision related to camera-point
duality. We also clarify some relationships between the classical problems of
optimal resectioning and triangulation, state a conjectural formula for the
Euclidean distance degree of the resectioning variety, and discuss how this
conjecture relates to the recently-resolved multiview conjecture.Comment: 27 page
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.Comment: 27 pages, 5 Figures, 3 table
Self-Calibration of Cameras with Euclidean Image Plane in Case of Two Views and Known Relative Rotation Angle
The internal calibration of a pinhole camera is given by five parameters that
are combined into an upper-triangular calibration matrix. If the
skew parameter is zero and the aspect ratio is equal to one, then the camera is
said to have Euclidean image plane. In this paper, we propose a non-iterative
self-calibration algorithm for a camera with Euclidean image plane in case the
remaining three internal parameters --- the focal length and the principal
point coordinates --- are fixed but unknown. The algorithm requires a set of point correspondences in two views and also the measured relative
rotation angle between the views. We show that the problem generically has six
solutions (including complex ones).
The algorithm has been implemented and tested both on synthetic data and on
publicly available real dataset. The experiments demonstrate that the method is
correct, numerically stable and robust.Comment: 13 pages, 7 eps-figure
An Atlas for the Pinhole Camera
We introduce an atlas of algebro-geometric objects associated with image
formation in pinhole cameras. The nodes of the atlas are algebraic varieties or
their vanishing ideals related to each other by projection or elimination and
restriction or specialization respectively. This atlas offers a unifying
framework for the study of problems in 3D computer vision. We initiate the
study of the atlas by completely characterizing a part of the atlas stemming
from the triangulation problem. We conclude with several open problems and
generalizations of the atlas.Comment: 47 pages with references and appendices, final versio
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