
Departament of Geometry and
Topology, Universidad
Complutense de Madrid
Image Processing Group (GTI), Universidad Politécnica de
Madrid 

Research results on
3D reconstruction
with uncalibrated cameras.
Camera autocalibration and other
geometry problems in computer vision




Sample input images: 

Threedimensional Euclidean
reconstruction of the scene viewed by the cameras:



What is camera
autocalibration?
Suppose you want to obtain the 3D structure of a scene (3D
positions of points, relative camera positions) from a set
of images and you have no idea about where the cameras
were located or what their internal parameters (focal
length, etc.) were.
Then you can proceed in the following
way:
 Mark significant points and their correspondances
across the images.
 Obtain from these data a projective
reconstruction of the scene, i.e., a
distorted 3D reconstruction that is related to the
true one by a spacial homography. There are several
wellestablished algorithms for that.
 Find the rectifying homography, i.e., the geometric
transformation that will restore the 3D structure from
the distorted version. This is the camera autocalibration
part, because you get at the same time the camera
positions and their internal parameters.
Different camera autocalibration algorithms cover
different situations (number of images, constant or
varying internal camera parameters, knowledge of some
camera parameters, knowledge about the scene). The first
situation studied was that of three or more cameras with
constant intrinsic parameters, but there are many other
cases of interest. For instance, most cameras have square
pixels, and it makes a lot of sense to employ this
information in the autocalibration process.
Camera autocalibration is an active are of research
because many of the available algorithms are very
noisesensitive, or produce multiple solutions, and also
because there are many particular cases that deserve
special attention.
This page includes some tutorial information on the topic
and introduces the contributions of our group in this
field and other geometry problems in computer vision.


Projective geometry for 3D
reconstruction in 90 minutes
Slides of a talk (PDF, in spanish)
Transparencias de una
charla (PDF, en español)


3D reconstruction with
uncalibrated cameras in 90 minutes
Slides of a talk (PDF, in spanish)
Transparencias de una
charla (PDF, en español) 

3D
Reconstruction with the Minimum Number of Squarepixel
Uncalibrated Cameras
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 sixline 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 onetotwo 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 Squarepixel Uncalibrated Cameras.
In submission.
Pablo Carballeira, José Ignacio
Ronda, Antonio Valdés, 3D reconstruction with
uncalibrated cameras using the sixline conic
variety. In Proceedings of IEEE ICIP 2008:
205208. DOI: 10.1109/ICIP.2008.4711727





Euclidean upgrading from segment
lengths
We address the problem of the
recovery of Euclidean structure of a projectively
distorted ndimensional 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 higherdimensional
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. 350368, Dec. 2010. DOI: 10.1007/s112630100369z





Autocalibration
of cameras with known pixel shape
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. 193214, October 2008. DOI: 10.1007/s1085100800950
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: 209212. 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 kth
camera, with center C_{k}.



Autocalibration
of a camera pair
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 closedform, 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 bestfit case.
As a geometrical byproduct, 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. 135149, June 2007. DOI: 10.1007/s108510070011z
Paper
preprint


Condition of compatibility of
ω with epipolar geometry.



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.
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. 283303, March 2006. DOI: 10.1007/s112630053677y
Paper
preprint
José Ignacio Ronda, Guillermo
Gallego, Antonio Valdés, Camera
autocalibration using Plucker coordinates.
In proceedings of IEEE ICIP (3) 2005: 800803. DOI:
10.1109/ICIP.2005.1530513





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.
A. Valdés, J. I. Ronda, Camera
autocalibration and the calibration pencil,
Journal of Mathematical Imaging and Vision, vol.
23, no. 2, pp. 167174, Sept. 2005. DOI: 10.1007/s108510056464z
Paper
preprint
Antonio Valdés, José Ignacio
Ronda, Guillermo Gallego, Linear camera
autocalibration with varying parameters. In
proceedings of IEEE ICIP 2004: 33953398. DOI: 10.1109/ICIP.2004.1421843.
Poster used during the panel
session at ICIP 2004.






Camera
autocalibration and horopter curves
In this work we provide "yet
another algorithm" for the autocalibration of
three or more cameras with arbitrary constant
intrinsic constant parameters. It has a nice
geometrical motivation, based on the horopter
curves associated to each pair of images and the
special way in which they cut the plane at
infinity.
J. I. Ronda, A. Valdés, F. Jaureguizar, Camera
autocalibration and horopter curves,
International Journal of Computer Vision, vol. 57,
no. 3, pp. 219232, May 2004. DOI:
10.1023/B:VISI.0000013095.05991.55
Paper
preprint





Projective
evolution of plane curves
We show that projectively invariant evolution operators
have unavoidable singularities. In particular, we see that
there exists no nonsingular projective evolution operator
welldefined over straight lines nor conics.
M. Castrillón and A. Valdés, Projective Evolution of
Plane Curves. International Journal of Computer
Vision, 42 (3). pp. 191201, 2001. DOI: 10.1023/A:1011143732632
Paper
preprint


Contact
Please address your questions or comments to José I. Ronda (jir@gti.ssr.upm.es)
(DBLP bibliography)
or Antonio Valdés
(Antonio.Valdes@mat.ucm.es)
(DBLP bibliography)
Last update: Sept. 29th, 2011.
