The Absolute Line Quadric and Camera Autocalibration

We introduce a geometrical object providing the same information as the absolute conic: the absolute line quadric (ALQ). After the introduction of the necessary exterior algebra and Grassmannian geometry tools, we analyze the Grassmannian of lines of P^3 from both the projective and Euclidean points of view. The exterior algebra setting allows then to introduce the ALQ as a quadric arising very naturally from the dual absolute quadric. We fully characterize the ALQ and provide clean relationships to solve the inverse problem, i.e., recovering the Euclidean structure of space from the ALQ. Finally we show how the ALQ turns out to be particularly suitable to address the Euclidean autocalibration of a set of cameras with square pixels and otherwise varying intrinsic parameters, providing new linear and non-linear algorithms for this problem. We also provide experimental results showing the good performance of the techniques

Autocalibration with the Minimum Number of Cameras with Known Pixel Shape

In 3D reconstruction, the recovery of the calibration parameters of the cameras is paramount since it provides metric information about the observed scene, e.g., measures of angles and ratios of distances. Autocalibration enables the estimation of the camera parameters without using a calibration device, but by enforcing simple constraints on the camera parameters. In the absence of information about the internal camera parameters such as the focal length and the principal point, the knowledge of the camera pixel shape is usually the only available constraint. Given a projective reconstruction of a rigid scene, we address the problem of the autocalibration of a minimal set of cameras with known pixel shape and otherwise arbitrarily varying intrinsic and extrinsic parameters. We propose an algorithm that only requires 5 cameras (the theoretical minimum), thus halving the number of cameras required by previous algorithms based on the same constraint. To this purpose, we introduce as our basic geometric tool the six-line conic variety (SLCV), consisting in the set of planes intersecting six given lines of 3D space in points of a conic. We show that the set of solutions of the Euclidean upgrading problem for three cameras with known pixel shape can be parameterized in a computationally efficient way. This parameterization is then used to solve autocalibration from five or more cameras, reducing the three-dimensional search space to a two-dimensional one. We provide experiments with real images showing the good performance of the technique.Comment: 19 pages, 14 figures, 7 tables, J. Math. Imaging Vi

Camera Autocalibration using Plücker Coordinates

We present new results on the Absolute Line Quadric (ALQ), the geometric object representing the set of lines that intersect the absolute conic. We include new techniques for the obtainment of the Euclidean structure that lead to an efficient algorithm for the autocalibration of cameras with varying parameters

Line geometry and camera autocalibration

We provide a completely new rigorous matrix formulation of the absolute quadratic complex (AQC), given by the set of lines intersecting the absolute conic. The new results include closed-form expressions for the camera intrinsic parameters in terms of the AQC, an algorithm to obtain the dual absolute quadric from the AQC using straightforward matrix operations, and an equally direct computation of a Euclidean-upgrading homography from the AQC. We also completely characterize the 6×6 matrices acting on lines which are induced by a spatial homography. Several algorithmic possibilities arising from the AQC are systematically explored and analyzed in terms of efficiency and computational cost. Experiments include 3D reconstruction from real images

Linear Camera Autocalibration with Varying Parameters

We provide a new technique for the Euclidean upgrading of a projective calibration for a set of ten or more cameras with known skew angle and aspect ratio and arbitrary varying focal length and principal point. The proposed algorithm, which is purely linear and thus of very low computational cost and not suffering from initialization problems, is based on the geometric object given by the set of lines incident with the absolute conic. We include experiments which show the good performance of the technique

3D Reconstruction with Uncalibrated Cameras Using the Six-Line Conic Variety

We present new algorithms for the recovery of the Euclidean structure from a projective calibration of a set of cameras with square pixels but otherwise arbitrarily varying intrinsic and extrinsic parameters. Our results, based on a novel geometric approach, include a closed-form solution for the case of three cameras and two known vanishing points and an efficient one-dimensional search algorithm for the case of four cameras and one known vanishing point. In addition, an algorithm for a reliable automatic detection of vanishing points on the images is presented. These techniques fit in a 3D reconstruction scheme oriented to urban scenes reconstruction. The satisfactory performance of the techniques is demonstrated with tests on synthetic and real data

Autocalibration of Cameras with Known Pixel Shape

We present new algorithms for the recovery of the Euclidean structure from a projective calibration of a set of cameras of known pixel shape but otherwise arbitrarily varying intrinsic and extrinsic parameters. The algorithms have a geometrical motivation based on the properties of the set of lines intersecting the absolute conic. The theoretical part of the paper contributes with theoretical results that establish the relationship between the geometrical object corresponding to this set of lines and other equivalent objects as the absolute quadric. Finally, the satisfactory performance of the techniques is demonstrated with synthetic and real data

Recursive Camera Autocalibration with the Kalman Filter

Given a projective reconstruction of a 3D scene, we address the problem of recovering the Euclidean structure of the scene in a recursive way. This leads to the application of Kalman filtering to the problem of camera autocalibration and to new algorithms for the autocalibration of cameras with varying parameters. This has benefits in saving memory and computational effort, and obtaining faster updates of the 3D Euclidean structure of the scene under consideration

Hierarchical structure-and-motion recovery from uncalibrated images

This paper addresses the structure-and-motion problem, that requires to find camera motion and 3D struc- ture from point matches. A new pipeline, dubbed Samantha, is presented, that departs from the prevailing sequential paradigm and embraces instead a hierarchical approach. This method has several advantages, like a provably lower computational complexity, which is necessary to achieve true scalability, and better error containment, leading to more stability and less drift. Moreover, a practical autocalibration procedure allows to process images without ancillary information. Experiments with real data assess the accuracy and the computational efficiency of the method.Comment: Accepted for publication in CVI

An Enhanced Structure-from-Motion Paradigm based on the Absolute Dual Quadric and Images of Circular Points

International audienceThis work aims at introducing a new unified Structure-from-Motion (SfM) paradigm in which images of circular point-pairs can be combined with images of natural points. An imaged circular point-pair encodes the 2D Euclidean structure of a world plane and can easily be derived from the image of a planar shape, especially those including circles. A classical SfM method generally runs two steps: first a projective factorization of all matched image points (into projective cameras and points) and second a camera self-calibration that updates the obtained world from projective to Euclidean. This work shows how to introduce images of circular points in these two SfM steps while its key contribution is to provide the theoretical foundations for combining “classical” linear self-calibration constraints with additional ones derived from such images. We show that the two proposed SfM steps clearly contribute to better results than the classical approach. We validate our contributions on synthetic and real images