(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 
<y 
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