Reconstruction of 3D Symmetric Curves from Perspective Images without Discrete Features

1
Reconstruction of 3-D Symmetric Curves from
Perspective Images without Discrete Features
Wei Hong, Yi Ma, and Yizhou Yu University of
Illinois at Urbana-Champaign
Epipolar Geometry of Symmetric Curves A pair of
symmetric 3-D curves and satisfy
, Where
is the reflective symmetry.
The image of the symmetric curves are
and . The lines in the image connecting
each pair of corresponding symmetric points will
intersect at a same point ---- the vanishing
point . The angle of the line and
the distance to the vanishing point
establish a polar coordinate so that
and are always two corresponding
symmetric points. The epipolar constraint in
this polar coordinates
Reconstruction of Planar Symmetric Curves from a
Single View The central line of the planar
curves in the image is determined by connecting
the two end points. The triangulation equation
becomes This equation can be written
as The necessary condition for a valid
solution of T for planar symmetric curves
is is the smallest singular value
of .
Simulations To test the robustness, 5 asymmetry
is added onto the 3-D symmetric curves and white
Gaussian noise is added to the projected image
curves with standard deviation corresponding to
approximately one pixel for a 400x320 pixel
image. One view of planar curves is the
angle between the camera optical axis and the
symmetry plane. Two view of general curves
is the difference in
the angles between the two camera axes and the
symmetry plane.
Reconstruction of General Symmetric Curves from
Two Views
Ambiguity in Reconstruction of Symmetric Curves
from a Single View If the actual vanishing point
is given, the 3-D depths for each pair of
corresponding points in image can be uniquely
determined via the following triangulation
equation where is the distance from the
camera center to the symmetry plane. The above
equation always has a unique solution for
and for an arbitrarily chosen
. Lemma 1. Given a pair of curves and
on the image plane and an arbitrary
feasible vanishing point , there exists a
pair of curves in space and
such that and are their
images, respectively. So, there are a
2-parameter family (case c) or 1-parameter family
(case b) of pairs of symmetric curves in 3-D that
give rise to the same pair of image curves.
Experiments on Real Images
Summary of the Reconstruction of Symmetric Curves