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# Full text

Title
International cooperation and technology transfer
Author
Mussio, Luigi

(11)
M'=P T (p-P T Y =P T - \[v o]
0
V-'
0
(5)
Observing that V is a diagonal matrix; for p we get:
'0
i>-'l
p =
0
>*v
• u + w-
=
0
0
w
(6)
Where w 2 is the unknown fourth coordinate of p, that deter
mines the distance between point p and the origin.
If we want to get panoramic images we can rotate the camera
and take photographs all around of an angle of 360°, taking
care that the optic centre is on the rotation axis. As previously
mentioned, this is equivalent to rotate the point p around the
optic centre in the opposite sense, holding the camera itself in
fixed position. In this way the point p appears in a different
photograph as point p
~R
0~
RV~'u
0 T
1
■p =
w
(7)
In the image plane the point u ’ will become :
u'=[V' 0]-/>'=[r 0].
RV-’u
w
= V'RV~‘u
(8)
Therefore u and u' are related through a 2D projective transfor
mation.
Note that V is different from V because the camera focus can
be regulated, this involves changing the focal length between
two photographs (/=/’).
Since for consecutive photographs obtained by camera rotations
around the optic centre this bidimentional relation is
established, it is as if the third dimension, i.e. the depth, would
be missed. Therefore, the images can be overlapped on any
plane to form just one resulting image.
Theoretically it would be possible to build any planar map
given the three independent parameters, defining the camera
rotations, and the employed focal lengths (if they are keeping
fixed only one focal is needed).
These parameters appear in the relationship that relates corre
sponding points of different images, taken with the optic centre
of the camera at fixed position:
u' = MP ■ u
(9)
where MP is the projective matrix.
The simple plane cannot contain the infinity, i.e. a flat angle,
for this reason we used as reference surface the sphere or, as in
the case of the Quick Time VR, the cylindrical surface that is
simpler to manage.
The 3D coordinates of p=(x,y,z) can be projected on the cylin
drical surface according to the following formulas:
w = (S,v) (10)
3 = arc tan
f x
v =
■Jx 2 + z 2
(12)
where Vx 2 + z 2 is the focal length and \$ e (-71,7t).
In this case, however, the transformation requires that for all the
photographs the focal length is held constant, this means that
the camera focus cannot be modified.
If unknown, the focal length is estimated using two consecutive
images.
2. THE IMAGES REFERENCE SURFACE
In order to use a sphere as reference surface we have to divide
the surface in overlapping regions, and to represent each region
with a plane on which the images are mapped (Fig. 2).
Figure 2
The resumption of the map would be difficult, because all the
possible rotations around the projective centre of the camera
have to be performed, requiring a complex tripod to sustain the
camera, hard to manage and of difficult construction. Further
more adopting a sphere, we have to solve for a full coverage of
the surface with image taken at different viewangles, involving
an optimization algorithm for the image overlapping through
the entire sphere. Another issue refers to the memory requi
rements for the data management: still using low resolution
images the memory occupancy can become excessive also for
high level computers (e.g. as Silicon Graphics).
These reasons lead the developers of virtual reality systems to
employ the cylinder as reference surface for image mapping.
3. THE PROJECT
As first step in the realization of a virtual visit using the Quick
Time VR, we have to deal with the visit planning.
This phase involves defining the kind of visit that we want to
realize. We can choose to use:
> real images;
2 w is named projective depth.
2