ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
3 3D RECONSTRUCTION OF CYLINDERS 
In this section we use the canonical representation to pro- 
pose different methods for the 3D reconstruction of cylin- 
ders from two, three or more calibrated views. Here we 
suppose that both intrinsic and extrinsic parameters of the 
cameras are known. We also discuss some interesting sce- 
narios, where only one calibrated view of the cylinder and 
some additional information such as the 3D coordinates of 
a point on the cylinder are known. This is often the case 
when markers are attached to cylinders for camera calibra- 
tion, see Fig. 5. We propose a method to bring this addi- 
tional information into the 3D reconstruction framework. 
3.1 Stereo Reconstruction 
The vanishing point is quite noise sensitive. The estimation 
of the orientation of the cylinder 19 from one view is often 
not reliable. In particular when the diameter of the cylinder 
is small compared to its distance from the camera. When 
calibrated stereo views are available, it is more reliable to 
take: 
lo — no x Rn; (11) 
Where no and n, define the viewing plane of the axis of 
the cylinder, i.e the plane which contains the axis of the 
cylinder and the optical center, respectively in the first and 
second camera. The viewing plane of the axis of cylinder 
in the second camera can also be defined by its orientation 
Rn, and its distance from origin n‘T in the first camera 
coordinate system. The depth of the cylinder from the first 
camera ho is then obtained as: 
ni T 
hu E ee 
0 (lo X no)'Rn; 
(12) 
This defines the 3D cylinder, i.e, its canonical coordinates 
as (lo, No = hono, r — ho sin(5.)). 
3.0 3D Reconstruction from Three or more Views 
Here we describe the 3D reconstruction of cylinders from 
three calibrated views. This framework easily extends to 
more than three views. In the rest of this paper, we suppose 
that orientation and position of the third camera relative to 
the first one is always defined by rotation S and translation 
U. We first recover the orientation of the cylinder using all 
observations: 
no 
Rn; lo =0 (13) 
Sn» 
This linear system of equations is of the form AX — 0 
and can be easily solved for X(— 195). The solution is the 
eigenvector associated to the smallest eigenvalue of A*'A. 
This can also be solved using Singular Value Decomposi- 
tion (SVD). 
The depth of the cylinder ho is then estimated as the least 
squares solution to the following system of equations: 
(lo X no)! Rn, s nT (14) 
(lo X no)'Sn; gr niU 
3.2.1 Using a single occluding edge of a cylinder If 
only one occluding edge of the cylinder is visible in an im- 
age, we can still use it for 3D reconstruction from multiple 
views. In fact, in theory we only need to view three oc- 
cluding edges in order to reconstruct the cylinder. Let us 
write the equations for the case, where we see two occlud- 
ing edges in one image and only one in the second one. In 
this case we have the image of the axis of the cylinder no 
in the first view, but only one of the occluding edges of the 
cylinder, nq; in the second view. We still have: 
lo — no X Rn; (15) 
If we also observe a single occluding edge ns; in the third 
image, we will have: 
No 
Rn11 lo =0 (16) 
Sn»i 
Finally, if we see only one occluding edge in each of the 
three images, we have: 
n0; 
Rn; lo zx (17) 
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In order to compute the depth ofthe cylinder and its radius, 
we use the constraints that the tangent plane defined by the 
occluding edge is at distance r to the axis of the cylinder. 
In the first case, where two edges are seen in the first view 
and one in the second one, we have: 
t 
n,, T 
ha = 
9 (lo x no)! Rni; + sin(5) 
  
(18) 
This defines two cylinders with canonical coordinates (19, No — 
hono,r — ho sin(5.)), one for each value of ho in Eq. 18. 
This ambiguity could only be removed if the user has ad- 
ditional information about the position of the cylinder rel- 
ative to the tangent plane. This information is available on 
the image and could be used to obtain a unique solution. 
If there are multiple images with single occluding edges, 
there will be no more ambiguity. This means that only one 
of the two solutions for ho would satisfy the constraints in 
all views. 
3.3 Reconstruction from One View 
In the case we have only one view of the cylinder, as we 
described in the previous sections, the cylinder is defined 
up to one parameter, its depth from the camera or its ra- 
dius. If there are feature points on the cylinder that are 
reconstructed or can be reconstructed from multiple views, 
see Fig. 5, the canonical representation and the associated 
equation of the cylinder allow an elegant and simple solu- 
tion for cylinder reconstruction. For each point M on the 
cylinder, we have: 
IMx1-N| =r (19) 
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