CIP A 2003 XIX th International Symposium, 30 September - 04 October, 2003, Antalya, Turkey 
a and ß planes from the origin of the absolute reference 
frame, are orthogonal to their corresponding planes (eq. 7). 
B'B = A'A = I 
(6) 
c*B = d‘A = 0 
(7) 
Figure 5: Geometric model (partial view) 
Now let us suppose to know B and c (i.e. the ß plane), the 
position in the 3D space of the eye-point E, the user’s 
viewing direction v, the unit vector up of the eye-point and 
m, the distance from the eye-point to a plane. Given such 
parameters, we can compute the a plane, i.e. parameters A 
and d of eq. 5, according to the constraints reported in eq 6,7. 
These constraints mean that the two vectors generating the a 
plane should be orthonormal and coincident with the columns 
of matrix A, and vector d is orthogonal to them. Assuming 
the view-plane (a) is orthogonal to unit vector v, it can be 
easily demonstrated that matrix A becomes 
A = [up I dx] 
up x 
dx x 
up y 
dx y 
up z 
dx z 
(8) 
where dx = up a v. Similarly, vector d can be easily obtained 
according to following relationship: 
d = (< E, v > -m) • v (9) 
where the symbol <,> denotes the scalar product. 
At this step we have all elements needed to determine the 
projective transformation P, which maps a generic point x’ of 
P plane on point x”, lying on the a plane. Let be 5 the bundle 
of straight lines passing through the eye-point E, 
s—>s = k/ + E; Vk e R 3 , t e R (10) 
With some straightforward algebra, we get the equation of 
the straight line of s, which passes through point x’: 
r = k/ + E ; with k = x'-E (11) 
Thus, in order to compute the value t* assumed by parameter 
t, when the straight line r intersects the a plane on point x”, 
we consider following equality: 
k? *+E = x"= Af"+d (12) 
After some substitutions and taking into account eq. 7, we get 
t* = 
d'(d-E) 
d'Bt’+Cd^-d'E) 
(13) 
Using eq. 10 and 13, the following fundamental relationship 
is obtained, which represents the projective transformation 
we searched for: 
_ F-t'+g 
h‘ t’ + q 
(14) 
Indeed, the 2D vector t’ contains the coordinates of a generic 
point x’ in a p plane fixed reference frame, while the vector 
t”, obtained from t’ by eq. 14 , contains the 2D coordinates of 
point x” on a a plane fixed reference frame. 
5. TEST AND RESULTS 
In order to evaluate the performance of the compression 
algorithm, a server/client data transmission with the 
developed split-browser has been carried out. To this end we 
employed simple geometric shapes (polygons) of different 
colors, as shown in figure 6. Here view A represents the 
scene observed by the user at time n-1, B is the exact view as 
computed by the server, to be displayed at time n, according 
to user input, while the predicted view is the one obtained by 
application of the projective transformation to view A on the 
client. 
Figure 6: Views A (left) - Predicted views (middle) 
Views B (right) 
Following criterions were defined to compare the results each 
other: 
1) Average bit/pixel number of compressed view B. This 
parameter defines the number used on average to code 
each pixel ov compressed view B. As we dealt with 24 
bit RGB images, the resulting value should be lower 
than 24 to pass the test. 
2) Average bit/pixel number of compressed error-image. It 
is similar to the previous parameter, but in this case the 
threshold value for the pass/fail test should be lower 
than the previous one. 
3) Compression ratio. It defines the ratio between the size 
(in bytes) of compressed view B and the sum of the size