CIPA 2003 XIX th International Symposium, 30 September - 04 October, 2003, Antalya, Turkey 
295 
id the 
e user 
nodel, 
actual 
only, 
higher 
ce the 
dicate 
;nt. 
fluent 
model 
atures 
it. An 
to get 
o this 
been 
results 
simple 
aled a 
espect 
lained 
they 
intent, 
inique 
ontext 
inique 
M 
lirther 
;ed on 
irlying 
;r and 
other 
vledge 
:-point 
¡ective 
vel of 
image 
in be 
mages 
imuni- 
server 
g new 
scene 
:rmine 
i view 
i-l, is 
~orma- 
which 
B. the 
erence 
of the 
imand 
pixel 
the Z 
4) Then, a pixel by pixel difference between view B and 
the predicted view is calculated by the server. A so 
called error-image is obtained, which is compressed 
with LZ77 algorithm and sent to the client. 
Client-side operations 
1) The client receives the projective transformation 
parameters and the error-image, which is decompressed, 
retrieving the original image. 
2) The projective transformation is applied to the view A, 
generating a so called predicted view. 
3) Then, the error-image is added pixel by pixel to the 
predicted view, in order to create the requested view B, 
which is then displayed by the client in the user’s GUI. 
Figure 3: The VRML Split-browser structure 
The decription presented above shows that the computational 
load on the client is greatly reduced, respect with the classical 
approach (transmission of the whole VRML model). Indeed, 
the client is required only to apply the projective 
transformation to the current view and to decompress the 
error-image as fast as possible. Nowadays, requested 
computational capabilities to perform such tasks are 
commonly available on most home desktop PCs. 
In the following subsection, some mathematics will be 
provided about the developed frame compression algorithm. 
4.1 The projective transformation 
A projective transformation can be considered as a subset of 
the more general group of coordinate transformations, which 
maps a given input 2D image point x = [x b x 2 ] into a new 
image point y = [y,, y 2 ]. Adopting a matrix notation, this 
mapping fucntion can be defined as follows: 
Ax + b 
c'x + d 
(1) 
Typically, each projective transformation is associated a 
matrix Pe 91 3x3 , called projective matrix, allowing a more 
compact notation for eq. (1): 
A 
M 
b' 
a \i 
0,2 
w 
p = 
Л 
Л 
Л 
= 
a 2\ 
o 22 
b 2 
c' 
M 
d 
_ C \ 
C 2 
d 
(2) 
In the following, the procedure adopted to compute the 
projective transformation parameters will be described using 
a simple polygon as target object in the 3D space. This 
assumption doesn’t limit the effectiveness of our procedure, 
since it is well known that each 3D object can be described in 
terms of a number of interconnected polygons (typically 
triangles). Therefore, in principle, it would be sufficient to 
apply the algorithm to each composing polygon. 
According to figure 4, the aim of our procedure deals with 
the computation of the position of point T 2 , by application of 
a projective transformation to point T| (identified by P 3 ). 
Figure 4: Geometric model for the projective transformation 
If projective transformations P! and P 2) . mapping T on T| and 
T 2 respectively, are known, following relationship can be 
established: 
T 2 =P 2 (T) = P 2 (P,,„ V (T 1 )) (3) 
In practice, the projective transformation which maps the 
view-plane a] (part of polygon p displayed on the user’s GUI 
at time n-1) on a 2 (view of polygon p at time n) can be 
defined in terms of a matrix product: 
P 3 = P 2 P, ' (4) 
where Pj denotes the projective matrix associated to the i-th 
projective transformation. As shown by eq (4), in order to 
determine P 3 , we need to compute P| and P 2 before. To this 
aim, we consider firstly the parametric equations of a and P 
planes (see figure 5): 
where 
<2-»x" = A-t"+d /? —>■ X' = A • t'+C (5) 
A = 
cit 
Û-,, Cl-, 
L“21 
; b = [6, b 2 ]' ; c' = [c, c 2 ] \ d e R 
where A, BeiR 3x2 ; c, d e9f 3 ; t, t”e9? 2 and x’, x”e s Jl 3 . Then 
we add two constrains: a) vectors generating the planes are 
orthogonal (eq. 6); b) vectors c and d, defining the distance of