patches centred at the considered points. Indeed, these homolo 
gous points are necessary to feed a photogrammetric bundle ad 
justment to estimate accurately the relative pose of all the images 
within the panorama. The main advantage of this method is that 
it finds corresponding points in any situation, even if the surface 
is uniform or regular. The second advantage is that it chooses the 
number of corresponding points per images as well as the repar 
tition of these points. 
Our mathematical calibration model 
Saint-Mandé, France, September 1-3, 2010 
For our calibration process, we need to have all the images ac 
quired from the same point of view. To minimize manually the 
parallax due to the mounting of the camera on the acquisition de 
vice, one takes a pair of rotated images with near and far objects. 
If the two objects remain in the same order in both images, there 
is no or only a small parallax. If the order is changed, there is 
some parallax. Fig. 2 shows two images taken with our system. 
Most calibration techniques try to minimize the residuals in the 
image space of known points, or geometrical characteristics ex 
tracted from images. In our case, we only want to minimize the 
angle between the corresponding photogrammetric rays of ho 
mologous points directly in panoramic space (see eq. 1). This 
explains why our process does not require ground points. 
Each image is fixed in the panoramic space by a rotation noted 
Ri, p . Our perspective model contains the Principal Point of Au- 
tocollimation (the intersection of the focal plane with the optical 
axis) of coordinates (cppaJppa) and the focal length denoted 
/- 
1.3.1 Ray in 3D space To transform a point (c, l) in image 
coordinates to a ray (x',y',z') in panoramic space, we use a 
function g which depends on Rj, p , f and (cppaJppa) (see 
eq. 1). 
yjx 2 + y 2 + Z 2 
(1) 
where: 
= C — CppA 
= IppA — I 
= ~f 
Figure 2: Rotation with a decentered camera around pan-tilt axes 
We can see that the order of the wire in the foreground and the 
TV antenna changes between the two images. Thus, the position 
of the camera on the device is shifted manually until this effect 
disappears. 
1.2 Description of our matching process 
1.3.2 Distortion modeling We consider an additive radial dis 
tortion model which amplitude is modelled by a polynomial of 
the form p(r) = a.r 3 + b.r 5 + c.r ‘ where r is the radius centred 
on the Principal Point of Symmetry (cpps,Ipps) which is dif 
ferent from the PPA. 
Eq. 2 shows how to compute a corrected measure from a real 
measure, where (Cb,lb) is the measure directly taken in image 
space, and (c c , l c ) is the corrected measurement: 
To compute homologous points, you can use different strategies. 
For example by extracting and matching interest points such as 
SIFT point (Lowe, 2004). We have used also a matching of ho 
mologous points in two neighbouring images based on anony 
mous features (Craciun et al., 2009), following a two steps ap 
proach. In a similar way to the process describe in (Coorg and 
Teller, 2000), the first step of our pose estimation method consists 
in finding for each pair of overlapping images the rotation which 
optimises the Normalised Cross Correlation similarity score com 
puted on the overlap of the first image with a rigid transform of 
the second image to put it in the geometry of the first. The opti 
misation is performed in a greedy way within a multi-resolution 
framework. 
T = yf (Cb — cpps) 2 + (h — Ipps) 2 
dr = ar 2 + br 4 + cr G 
Cb + (Cb — cpps)dr 
l c — h 4- (h — lpps)dr 
(2) 
In this process, we must ensure that the model is not correlated 
with intrinsic parameters. For example, a model of purely linear 
radial distortion shows that an increase of the focal length has 
the same effect as reducing the coefficient of distortion. The sys 
tem thus cannot converge or converges to a mathematical minima 
which is physically incorrect. 
System to solve 
To compute the pose of images in a panorama and calibrate a 
camera, one must find a set of parameters: rotation of each image 
in the panoramic system (i?i lP , • ■ • , Rn, p ) and intrinsic parame 
ters (/ and the PPA image coordinates (cppaJppa))• This can 
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