STRATIFIED SELF CALIBRATION FROM SCREW-TRANSFORM MANIFOLDS
RUSSELL MANNING / CHARLES DYER / UNIVERSITY OF WISCONSIN
ABSTRACT
(4) STRATIFIED SELF CALIBRATION FROM
SCREW-TRANSFORM MANIFOLDS
This paper introduces a new, stratified approach for the metric self
calibration of a camera with fixed internal parameters. The method works by
intersecting modulus-constraint manifolds, which are a specific type of
screw-transform manifold. Through the addition of a single scalar parameter,
a 2-dimensional modulus-constraint manifold can become a 3-dimensional
Kruppa-constraint manifold allowing for direct self calibration from disjoint
pairs of views. In this way, we demonstrate that screw-transform manifolds
represent a single, unified approach to performing both stratified and direct
self calibration. This paper also shows how to generate the screw-transform
manifold arising from turntable (i.e., pairwise-planar) motion and discusses
some important considerations for creating a working algorithm from these
ideas.
"Sketch out" several screw-transform manifolds
and find their mutual intersection point (e.g., with
voting scheme).
Requires at least three manifolds, so need at least
three fundamental matrices.
At most two mutual intersection points. Manifolds
exactly describe modulus constraint.
(1) SCREW TRANSFORMATION
(3) STRATIFIED SELF CALIBRATION
When same camera is moved to two positions,
the physical transformation of the camera can be
decomposed into a "screw transformation."
theta = angle of rotation
projective
affine
gamma = amount of translation parallel to
screw axis (as a multiple of
distance from screw axis)
metric
kappa = determines where vanishing point
of screw axis appears in first view
(a) take several views of a scene with same camera
(b) find common projective basis for all views (i.e.,
perform projective reconstruction and find camera
matrix for each view in this reconstruction)
(c) upgrade projective reconstruction to affine by
finding "plane at infinity"
(d) pairwise "infinity homographies" contain rotation
information, which is sufficient to upgrade affine
reconstruction to metric
(2) SCREW-TRANSFORM MANIFOLD
Found directly from fundamental matrix
between two views taken by same camera.
Every choice of real number kappa and real
number theta leads to a legal affine calibration
for given fundamental matrix.
Affine calibration is converted to a 3-vector:
Manifold "lives" in space of legal affine
calibrations, which is R3.
(5) REMARKS
Generating the screw-transform manifold for turntable motion follows a different algorithm than for
general motion. The following theorem helps distinguish between the two cases:
Fundamental matrix plus 3 real numbers <kappa, theta, gamma> gives
internal calibration matrix K.
Running over all possible <kappa, theta, gamma> triplets yields a screwtransform manifold in K-space. Intersecting several of these manifolds
allows K to be found directly (see Manning and Dyer, CVPR01).
Fundamental matrix plus 2 real numbers <kappa, theta> gives relative
calibration (i.e., infinity homography).
WISCONSIN
Running over all possible pairs <kappa, theta> yields a screw-transform
manifold in a-space. Intersecting several of these manifolds allows for
affine reconstruction, followed by metric reconstruction.
Reconstruction of a real
box covered with a dot
pattern. Dot centers
were automatically
extracted to give highlyaccurate point
correspondences
between views.