We address the problem of the Euclidean upgrading of a projective calibration of a minimal set of cameras with known pixel shape and otherwise arbitrarily varying intrinsic and extrinsic parameters.

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 in 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 by means of a one-to-two easily computable mapping and, as a consequence, we propose an algorithm that permits to reduce the number of required cameras to the theoretical minimum of 5 cameras to perform Euclidean upgrading with the pixel shape as the only constraint.

More details

J. I. Ronda, A. Valdés, G. Gallego, 3D Reconstruction with the Minimum Number of Square-pixel Uncalibrated Cameras. In submission.

Pablo Carballeira, José Ignacio Ronda, Antonio Valdés, 3D reconstruction with uncalibrated cameras using the six-line conic variety. In Proceedings of IEEE ICIP 2008: 205-208. DOI: 10.1109/ICIP.2008.4711727

We address the problem of the recovery of Euclidean structure of a projectively distorted n-dimensional space from the knowledge of the, possibly diverse, lengths of a set of segments. This problem is relevant, in particular, for Euclidean reconstruction with uncalibrated cameras, extending previously known results in the affine setting.

The key concept is the Quadric of Segments (QoS), defined in a higher-dimensional space by the set of segments of a fixed length, from which Euclidean structure can be obtained in closed form. We have intended to make a thorough study of the properties of the QoS, including the determination of the minimum number of segments of arbitrary length that determine it and its relationship with the standard geometric objects associated to the Euclidean structure of space. Explicit formulas are given to obtain the dual absolute quadric and the absolute quadratic complex from the QoS.

J. I. Ronda, A. Valdés, Euclidean upgrading from segment lengths, International Journal of Computer Vision, vol. 90, no. 3, pp. 350-368, Dec. 2010. DOI: 10.1007/s11263-010-0369-z

This work provides and evaluates new algorithms for camera autocalibration based on the set of lines intersecting the absolute conic. Besides, in order to make the topic more attractive for the engineering field, a totally new formulation in terms of Plücker matrices and Plücker coordinates is developed. For the sake of completeness, a thorough introduction to Plücker matrices and coordinates is provided.

Examples of application to real images

J. I. Ronda, A. Valdés, G. Gallego, Line geometry and camera autocalibration, Journal of Mathematical Imaging and Vision, vol. 32, no. 2, pp. 193-214, October 2008. DOI: 10.1007/s10851-008-0095-0
Paper preprint

G. Gallego, J. I. Ronda, A. Valdés, N. García, Recursive Camera Autocalibration with the Kalman Filter. In proceedings of IEEE ICIP (5) 2007: 209-212. DOI: 10.1109/ICIP.2007.4379802.
Poster used during presentation at ICIP 2007.

Intersection with the absolute conic ω of the isotropic lines of the k-th camera, with center C_k.

With calibrated cameras you just need two images to get a Euclidean reconstruction. However, with uncalibrated cameras you need three images taken with the same parameters. In this work we show that if the cameras have square pixels, with two cameras you almost have a Euclidean reconstruction, as you can get, in explicit closed-form, a set of reconstructions depending on one parameter. Therefore, if you have a single piece of data from the scene, such as knowing that two lines are parallel or orthogonal, it is straightforward to search the set of solutions for the best-fit case.

As a geometrical by-product, this paper investigates the relationship between the beautiful Poncelet's Porism and the Kruppa equations. It is the analysis of these configurations that results in the explicit parametrization of the solutions of these equations for two arbitrary cameras with the same parameters.

An example of application to real images

J. I. Ronda, A. Valdés, Conic geometry and autocalibration from two images, Journal of Mathematical Imaging and Vision, vol. 28, no. 2, pp. 135-149, June 2007. DOI: 10.1007/s10851-007-0011-z
Paper preprint

  Condition of compatibility of ω with epipolar geometry.

Equivalent to the calibration pencil is the absolute line quadric (ALQ), another geometric object that represents the lines intersecting the absolute conic and can be directly obtained from the absolute quadric. In fact, in this (quite mathematical) paper we show that the ALQ is just the exterior product of the absolute line quadric with itself and employ this fact to recover the absolute quadric from the ALQ.

This paper also provides a practical introduction to exterior algebra and its application in line geometry and related topics.

A. Valdés, J. I. Ronda, G. Gallego, The absolute line quadric and camera autocalibration, International Journal of Computer Vision, vol. 66, no. 3, pp. 283-303, March 2006. DOI: 10.1007/s11263-005-3677-y
Paper preprint

José Ignacio Ronda, Guillermo Gallego, Antonio Valdés, Camera autocalibration using Plucker coordinates. In proceedings of IEEE ICIP (3) 2005: 800-803. DOI: 10.1109/ICIP.2005.1530513

If you have images taken with cameras with square pixels, as most of them are, you have a lot of information in the projective reconstruction that can help you to locate the absolute conic. In fact, for each camera you have two lines that intersect this conic, so that, "all you have to do" is to find a conic that intersects all these lines.

A remarkable fact is that, if you use Plücker coordinates to represent lines, the set of lines that intersect the absolute conic is given by a pencil of quadrics (the "calibration pencil"). This makes autocalibration a problem of linear algebra: given ten or more cameras, an homogeneous linear equation gives you the calibration pencil, and another one extracts the plane at infinity from the calibration pencil.

A. Valdés, J. I. Ronda, Camera autocalibration and the calibration pencil, Journal of Mathematical Imaging and Vision, vol. 23, no. 2, pp. 167-174, Sept. 2005. DOI: 10.1007/s10851-005-6464-z
Paper preprint

Antonio Valdés, José Ignacio Ronda, Guillermo Gallego, Linear camera autocalibration with varying parameters. In proceedings of IEEE ICIP 2004: 3395-3398. DOI: 10.1109/ICIP.2004.1421843.
Poster used during the panel session at ICIP 2004.