ISPRS Commission III, Vol.34, Part 3A ,Photogrammetric Computer Vision*, Graz, 2002
CANONICAL REPRESENTATION AND THREE VIEW GEOMETRY OF CYLINDERS
Nassir Navab
Siemens Corporate Research, Imaging and Visualization Department
755 College Road East, Princeton, NJ 08540, USA, navab@scr.siemens.com
KEY WORDS: Measurement, Orientation, Reconstruction, Industry, Modelling, Geometry, Vision, Photogrammetry
ABSTRACT
This paper first introduces a canonical representation for cylinders. The canonical representation introduced here is closely
related to the Pliicker line respresentation. It then drives the general equation of a cylinder using this representation. In this
paper, we also show that this representation is an appropriate one for computer vision applications. In particular, it allows
us to easily develop a series of mathematical methods for pose estimation, 3D reconstruction and motion estimation.
Finally, one of the main objectives of this paper is to introduce the main equations dominating the three view geometry of
cylinders. We show the relationship between cylinders’ three-view geometry and that of lines (Spetsakis and Aloimonos,
1990, Weng et al., 1993) and points (Shashua, 1995) defined by the trilinear tensor (Hartley, 1997), and propose a linear
method, which uses the correspondences between six cylinders over three views in order to recover the motion and
structure.
1 INTRODUCTION
Cylinders are the main components in a large number of in-
dustrial sites. Oil platforms, refineries and off-shore instal-
lations are almost made of pipes and pipelines. Chemical
factories, water treatment plants and food processing fac-
tories are also mainly made of cylindrical objects. This is
also the case for power generation plants. Figure 1 shows
a typical image of an off-shore installation.
Figure 1: Typical image of an off-shore installation. Cylin-
ders are the most common shapes found in such environ-
ment.
Majority of these industrial sites does not have a 3D model
of the facility. Those who have a 3D model often need
to update the model on a regular basis. There is an in-
creasing need for creation of the 3D models based on im-
ages. This is the best way for getting an accurate update
of 3D model of the factory. This is called as-built re-
construction. Figure 2 shows an as-built reconstruction
of a power plant. This models has been built based on
thousands of images. Different companies and software
products aim at providing as-built reconstruction software
or services (AsBuilt,FotoG,Geodelta,Leica) and CyliCon
(Navab et al., 1999). The majority of the existing pho-
togrammetric solutions proceed to an off-line image cali-
bration procedure. Often markers are attached on the pipes
and other factory installations. Then a series of pictures are
taken. The markers are automatically detected, the cam-
eras are calibrated through a bundle adujstment procedure,
and the three-dimensional model of the environment in-
cluding the pipes and pipelines are built. In spite of these
Figure 2: An as-built reconstruction of a power plant based
on thousands of images.
activities and the increasing need for calibration and 3D
reconstruction using cylindrical objects, there is neither a
standard representation for cylinders, nor a standard for-
mulation for motion estimation or 3D reconstruction from
cylinders. Some previous work (Chandler and Still, 1994)
used points correspondences and tradional photogramme-
try methods to model such installations. Sayd et al. (Sayd
et al.,1996), Jones et al. (Jones, 1996), Bauer (Bauer, 1997),
and Benning and Schwermann (Benning, 1997) used oc-
cluding edges of cylinders on two or more calibrated views
in order to reconstruct the 3D model of cylinder. Hanek
et al. (Hanek et al., 1999) also used the occluding edge
of cylinders for pose estimation. Veldhuis and Vosselman
(Veldhuis and Vosselman, 1998) and Tangelder et al. (tan-
gelder et al., 2000) propose methods to interactively fit
A - 218
