Camera Autocalibration

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 (e.g., checkerboard), but by enforcing simple constraints on the camera parameters, such as constant intrinsic parameters in multiple images, known principal point, known pixel shape, etc.

Autocalibration with the minimum number of cameras with known pixel shape

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.

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Autocalibration of cameras with known pixel shape

3D reconstruction of the King's Courtyard in San Lorenzo de El Escorial, Madrid, Spain.

3D reconstruction of a set of books viewed by multiple cameras with varying intrinsic parameters.

This work provides and evaluates new algorithms for camera autocalibration based on the set of lines intersecting the absolute conic: the absolute quadratic complex (AQC). 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.

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 6x6 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.

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Relationship between geometric objects in Camera Autocalibration: the absolute conic, the dual absolute quadric, the plane at infinity, etc.

The Absolute Line Quadric and camera autocalibration

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. 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.

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

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Camera autocalibration and the calibration pencil

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.

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