Pj z xy): Ix- xil « bly - yj « b] (6) 
then the centre of the pixel of (i,j) has the coordinates 
Xi =x,+jb, Yi =y,+ib (7) 
where x, ,y, are the coordinates of the pixel centre with 
indices (0,0). 
The image (4) which has h grey levels is then in the 
digital form the function 
f P,» e, SINT Jjobuy) dxdy) (8) 
mu. 
ij 
For the given point (x,y) the pixel indices are determined 
by the equations 
j=INT(x-x,)/b, i=INT((y, +b/2)/b) (9) 
which we write then as the function 
(1) = e(X,y) (10) 
From the equation (6) it arises that for given x, y, b pixels 
and their indices are explicitly assigned. The digital image 
f, which was made as the result of the scanning process 
of the photograph, corresponds then to the function 
cj =H (Li) (11) 
We will consider now building the terrain image in the 
ground coordinate system. The composition of the 
functions f, and e gives the ordering dependence 
between the image coordinates of the photo and the grey 
levels 
oj =f (ex) (12) 
If we know the transformation of the form (1) from the 
X,Y,Z coordinate system to the x,y coordinate system, 
i.e. the transformation 
(X,y) = g(X, Y,Z) (13) 
then for the given surface , described in the form of DTM 
(digital terrain model), we have the image 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
(X, Y) 2 f, (e(g(X, YH(X, Y)))) (14) 
which we will write as the function 
(X, Y) 2 F(X, Y) (15) 
This is already the image in the ground coordinate 
system obtained from the image f(P,). If would be 
possible to generate all points we will have a cartometric 
image of the terrain in the local ground coordinate 
system at the scale 1:1 i.e. an orthoimage. 
It results that from the equations (14) and (15) the 
following conditions are fulfilled : 
e for the chosen ground area we have DTM, i.e. 
Dr c Dr’ 
e the chosen ground area after the transformation (13) 
is inside the photo area, i.e. D! cD. 
Because all the photo points lie on the opposite side of 
the plane to the projection centre, the condition 
(X,Y,H(X,Y)) eDg is always satisfied. 
The problem of the existence of the image point in the 
photograph for each point of the terrain which belongs to 
D, must be considered 
We notice that from two points lying on the same line 
passing through the project centre, one can see (in the 
photo) point which is closer to the project centre. In this 
way it is not possible to reconstruct the distant point in 
the orthoimage . 
The digital orthoimage ( Fig. 1) is the image composed 
of pixels in the different coordinate system than that 
introduced earlier and assigned by OXY. The 
connections between coordinates X,Y of the point in the 
ground coordinate system and its x,y’ coordinates are 
Xza4*X/A, Y=ag +y/à (16) 
where: a, ‚a, are the coordinates of the O' in the 
system OXY and A is the scale coefficient of the 
creating image. Using the equations (16 ) the image (15) 
can be written as the function 
(xy) 5 F((a4 *x/A),(ao *y/A)) (17) 
which can be finally written as 
(Xy) 2 FAGGy) (18) 
If we make a visualization with the d side of a pixel it 
means that we build the square grid in the terrain with the 
side 
(19) 
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