Antonio Valdés

The Absolute Line Quadric and Camera Autocalibration

Abstract.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 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.

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.

Camera Autocalibration and the Calibration Pencil

We study the geometric object given by the set of lines incident with the absolute conic. We see that this object is given by a pencil of quadrics of P5, which is characterized. We describe some of its most relevant properties for the camera autocalibration problem. Finally, we illustrate the applicability of the theory proposing a linear algorithm for the metric upgrading of a projective calibration of a set of ten or more cameras with varying parameters and known skew and aspect ratio.

Camera Autocalibration and Horopter Curves

We describe a new algorithm for the obtainment of the affine and Euclidean calibration of a camera under general motion. The algorithm exploits the relationships of the horopter curves associated to each pair of cameras with the plane at infinity and the absolute conic. Using these properties we define cost functions whose minimization by means of general purpose techniques provides the required calibration. The experiments show the good convergence properties, computational efficiency and robust performance of the new techniques.

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.

Camera autocalibration using Plucker 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.

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.

Directional geodesic active contours

We present a non-conformal metric that generalizes the geodesic active contours approach for image segmentation. The new metric is obtained by adding to the Euclidean metric an additional term that penalizes the misalignment of the curve with the image gradient and multiplying the resulting metric by a conformal factor that depends on the edge intensity. In this way, a closer fitting to the edge direction results. The provided experimental results address the computation of the geodesics of the new metric by applying a gradient descent to externally provided curves. The good performance of the proposed techniques is demonstrated in comparison with other active contours methods.

Head-Pose Estimation In-the-Wild Using a Random Forest

Human head-pose estimation has attracted a lot of interest because it is the first step of most face analysis tasks. However, many of the existing approaches address this problem in laboratory conditions. In this paper, we present a real-time algorithm that estimates the head-pose from unrestricted 2D gray-scale images. We propose a classification scheme, based on a Random Forest, where patches extracted randomly from the image cast votes for the corresponding discrete head-pose angle. In the experiments, the algorithm performs similar and better than the state-of-the-art in controlled and in-the-wild databases respectively.

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.