Problem: The automatic extraction of the unknown geometric
relations among a set of cameras capturing images of a common scene is
a fundamental problem in computer vision. This work addresses the
problem of estimating the configuration of arbitrarily many cameras
using image points matched across multiple frames, without requiring a
full reconstruction of the viewed scene points.


Motivation:Fast and inexpensive processors, cameras, and
storage are making feasible a variety of applications involving image
sequences and collections, ranging from visually intelligent
automobiles to enhanced sports viewing to home video manipulation. A
prerequisite for many of these applications is knowledge of the
physical positions of the cameras that captured the original images,
whether it be a collection of stationary cameras placed around a
common scene, or a single video camera capturing frames as it moves
through an environment. Accurate estimation of the geometry of
multiple views is the bedrock of such applications.

Previous Work Projective geometric models of multiple views
give rise to algebraic relations among the locations of points in
uncalibrated images that are 2D projections of rigidly moving points
in the scene. These algebraic relations are multilinear in the image
coordinates and their forms depend on the unknown projective
parameters relating the cameras. The projective parameters in turn
depend on the external and internal camera parameters, which may be
directly recovered without having to estimate the 3D structure of the
scene points

There exist 2, 3, and 4view projective multilinear constraints,
expressed via the fundamental matrix, the trifocal tensor, and the
quadrilinear tensor, respectively, that have been wellstudied and
used in many practical situations, while their Nview analogs have
been explored theoretically. Since the
projective relationships among more than 4 cameras in general position
are completely described by the collection of all 2view
relationships, a single Nview model is not necessarily
required. Moreover, the projective multilinearities that
simultaneously describe how N matching points must be related have
exponentially many linear parameters that are highly coupled through
their dependence on the camera properties and positions. Enforcing
these parameter dependencies is a difficult and active area of
research and an important consideration for
modeling the geometry of a large collection of views, such as all the
frames in a video sequence.

Approach:We present an Nfocal projective model of camera
geometry that is minimally parameterized and, by construction,
inherently geometrically consistent. Using the fact that all
multifocal constraints of a collection of cameras in general position
may be reduced to a collection of bifocal constraints, we build an
Nview model from a sparse set bifocal parameters that exploit the
dependencies among the pairwise geometric relationships.


For each new view added to a camera collection with known
configuration, its relation to the entire collection is completely
defined by its bifocal relation to two other views, via the positions
of the epipoles in each view, and the epipolar collineation that maps
epipolar lines from one view to the other. For
an isolated view pair, these quantities have a total of 7 degrees of
freedom, but once the new view's relation to a single existing view is
fixed, its relation to a second view in the collection is constrained
to only 4 degrees of freedom. We explicitly model these dependent
bifocal relations, and show how for the dependent view pair, each
epipole's variation is constrained to a line (leaving 1 DOF each), and
the epipolar collineation is constrained by one line match (leaving 2
DOF). We compute this minimal parameter set from point matches using a
LevenbergMarquardt minimization, initialized from the linear
estimation of the fundamental matrix.

We test the minimally parameterized model on a simulated camera
configuration viewing a block of points at the scene's origin. Two
algorithms are compared: the nonlinear 7parameter optimization of
independent fundamental matrices, and the nonlinear optimization of
dependent 4parameter bifocal models. Since the projective models are
defined only up to an arbitrary projective transformation, we use the
intrinsic parameters of the simulated cameras to recover the Euclidean
positions from the bifocal parameters and display the solutions.
Below are the recovered camera positions overlaid on the ground
truth camera positions, and Table 1 shows the improved accuracy in
camera position gained by enforcing bifocal dependencies.


(a) Ground truth camera positions and viewed
points. 


(b) Independent 7parameter
fundamental matrices. 
(c) Dependent 4parameter bifocal
parameters. 

Camera 
1 
2 
3 
4 
(a) Nonlinear Independent 
2.2068  2.2068  2.2068  2.2068 
(b) Nonlinear Dependent 
0.1593  0.1593  0.1593  0.1593 
Table 1: Translation errors for recovery
of camera positions from bifocal parameters. (a) Independent
7parameter estimates. (b) Dependent 4parameter estimates.


Impact:While camera motion estimation is a mature problem in
the field of computer vision, it is far from being solved. A
geometrically consistent model of arbitrarily many views that can be
automatically recovered from matching points is essential to any
application whose performance depends on first establishing the
relative positions of cameras.

Future Work:A difficult and pervasive issue is the degeneracy
of camera configurations, particularly in video sequences taken with
approximately linear camera motions. If a model has too many
parameters, such a sequence will be overfit, while if a model has too
few parameters, the complexity of some sequences will be too rich to
describe. It would be useful to use information criteria to determine
which subsets of views are degenerate and thus automatically choose
appropriate view pairs to model.

Projective
Minimal Analysis of Camera Geometry. Raquel
A. Romano. Ph.D. Thesis. May 2002. (ps.gz)
