AFFINE CALIBRATION FROM MOVING OBJECTS

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AFFINE CALIBRATION FROM MOVING OBJECTS
RUSSELL MANNING / CHARLES DYER / UNIVERSITY OF
WISCONSIN
ABSTRACT
EXPERIMENTS WITH REAL IMAGES
This paper introduces a novel linear algorithm
for determining the affine calibration between
two camera views of a dynamic scene. The affine
calibration is computed directly from the
fundamental matrices associated with various
moving objects in the scene, as well as from the
fundamental matrix for the static background if
the cameras are at different locations. A
minimum of two fundamental matrices are required,
but any number of additional fundamental matrices
can be incorporated into the linear system to
improve the stability of the computation.
MATHEMATICAL OVERVIEW
Experiment with one stationary camera, one moving
camera. Could also have been views from a
translating stereo rig or views of a translating
object taken from two stationary cameras.
Calculated and true relative calibrations are
shown below.
(STEP 1) Interpret the views as if both cameras
shared the same optical center.
SAMPLE APPLICATIONS
calculated
A single, moving camera captures two views of a
translating object. The camera can rotate and
change internally (e.g., change focal
length). Alternatively, two (possibly different)
stationary cameras at different positions are
used.
(STEP 2) Fundamental matrix for each moving
object Use the object's direction of motion in
place of the traditional epipole e.
true
(STEP 3) For each object, define the 3x3
matrices to the right labeled p.
Experiment with two cameras and two moving
objects. A fundamental matrix for each object
was calculated, from which the relative
calibration between camera A and camera B was
found. An affine reconstruction is shown below.
(STEP 4) The following equation holds for some
choice of ki's.
(STEP 5) Can find ki's and thus find HAB with a
linear equation. Proof that the linear system
has a unique null eigenvector is given in the
paper.
NOTE When more than two fundamental matrices
are available, a large linear system can be
created that incorporates all the fundamental
matrices at once and still solves for HAB in a
single step.
USES OF RELATIVE CALIBRATION
Affine reconstruction of the dot-covered box
object shown above left.
EXPERIMENTS WITH SYNTHETIC DATA
Some uses are (1) Affine camera calibration and
affine scene reconstruction. (2) Stratified
metric camera calibration and metric scene
reconstruction. (3) Constant-velocity dynamic
view morphing.
Hundreds of experimental trial runs were
performed on synthetic scenes. Each scene
contained several moving objects each moving
object was a spherical cluster of randomly
generated feature points.
Two or more translating objects viewed by
collection of fixed cameras.
Example of dynamic view morphing (CVPR '99).
Table showing error in relative calibration for
various numbers of feature points and various
amounts of noise added to the features after
projection onto the image planes.
Each point in the scatter plots above represents
a single trial with randomly-generated synthetic
data. (a) Calibration error vs. angle (in
degrees) between object motion vectors considered
under the fixed-camera formulation (b)
calibration error vs. average noise added to each
feature point on the image plane (c) calibration
error vs. object area on the image plane (d)
calibration error vs. retinal object motion
WISCONSIN
Table showing error in relative calibration for
various numbers of feature points and moving
objects.