ntroduced. 
/ersions of 
it the full 
well as an 
reosehens 
nstruktion 
macht die 
fikationen 
rocess of 
lel of the 
isisting of 
quur. 
y values 
ox ^y), 
ion (X,Y) 
(1) 
te values 
the grey 
ibed by a 
Qv 
(X; Y). 
(2) 
Jax 
(3) 
). 
'rojection 
s dX and 
x A y'.y 
dX z LE md. 
Pin Z aZ ed dZ 
Inserting (4) into (3) yields: 
BFP) X-X 
GG, y) e T (GQ*,Y« A ). fle 
,28 Q5 Y) a 
y UE E dt aG'(X,Y)). 
In the Finite Element Method the surface inside a finite 
element is interpolated by the weighted sum of the grid 
values of the element: 
0 
ZX) a XX P)= YY zx That ma Te FE) 
m=0n=0 
(6) 
The sum of the weights a, , is always 1: 
NN ns zl (7) 
m=0n=0 
In Facets Stereo Vision the finite elements are called 
facets. They are used for interpolating the surface Z as 
well as object grey values G(X,Y). Thus, replacing Z by 
G in (6) yields the formula for interpolating object grey 
values from the respective finite elements. 
Differences of the object surface are interpolated by 
differences of the grid points of the finite element 
(replacing Z with G again yields a formula for object 
grey value differences): 
r s 
dZ (xt 10) = SN, LY, y) ay H (8) 
m=0 n=0 
Inserting finite element interpolation for object surface 
and object grey values into (5) forms the basic equation 
of Facets Stereo Vision which describes the relationship 
between image grey values G’(x’,y’) and object surface 
and object grey values. 
GG, y)- T Qc 3a, + SSAC o, + 
m=0n=0 m=0n=0 
role a Pret t) 
rtrd) OX 20-2, 00, 2 -Z 
Ser is quens | 
m=0n=0 
  
9) 
Differences of brightness and contrast between image 
grey values and object grey values are modelled by a 
linear function T: 
TG’ (x',y'))= gö+ gi: 6'b',y')- (10) 
Adding the correction term vg for the observation G' in 
(19) the linearization of the right hand side of (10) leads 
to: 
85° + de * (gi +dg!)- (Gy) vs.) 
- gi « dg, * gi Gr, y) dei G'G' y) g? vs. 
(11) 
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We denote the differences between the grey values of the 
image and their interpolation within the grey value facets 
asl’: 
=g' +g’ Gx, eem GE, Y, .a2) 
m=0 n=0 
Replacing T in (9) by the right hand side of (11) and I’ 
by the right hand side of (12) yields the following 
equation: 
vo =p LAS aur dG (X,Y, )+ 
m=0 n=0 
DNL: pan € XQ 0m, Y iil 
wh) OX 7-2. oY Z2°-Z 
m=0n=0 
He. (X, X, )- de - dgi G'G',y')- I ) 
m=0n=0 
(13) 
  
By forming this equation for any pixel in any image 
which is used as input for FAST Vision a system of 
linear observation equations can be obtained, in which 
the unknowns x are the differences of surface heights and 
surface grey values respectively plus the parameters of 
radiometric transformation: 
v=4-x—-1[. 
From this system of equations a system of normal 
equations can be obtained in the well-known way in 
order to get a least squares solution for the variables x. 
Starting with reasonable Z-values the approximate values 
for the object grey values are calculated from one of the 
images. The shape of the facets is iteratively improved 
until the images calculated from this model surface are 
sufficiently similar to those pictures used as input. By 
taking into account the object surface the method of 
Facet Stereo Vision has an advantage over image space 
based methods of surface reconstruction in case of 
complex surface geometry. 
Up to now, the method of Facet Stereo Vision comprises 
the modelling of object geometry and radiometry by 
three sets of parameters: Object Z-values describe the 
object surface, object grey values describe the object 
radiometry and additional variables for each picture 
model the inter-image differences of brightness and 
contrast. 
3. Modifications of the FAST Vision approach for 
colour images 
Two of the three sets of parameters in the method of 
Facets Stereo Vision have to be modified in order to be 
able to process colour images: Instead of single object 
grey values there are vectors of object colour values - 
usually containing three elements. The scalar variables 
modelling differences of brightness and contrast between 
the pictures also have to be replaced by vectors. Only the 
variables modelling surface geometry remain unchanged. 
All variables in equation (13) containing a capital G have