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2. MATHEMATICAL MODELS 
2.1 Point-Based Projective Equations 
In the case of plane-to-plane central projection, the 8-parameter 
projective equations are used to relate the coordinates of a point 
in one plane to the coordinates of its projection in the second 
plane. Using points, only two independent projective equations 
can be constructed as follows, 
exc fÁvtrs (1) 
ext f yr 
@ + faV+ 8; 
y = 
: ext fytl 
Where X, - point coordinates in the first plane, 
X, J = point coordinates in the second plane, 
(e, Lu 2.) = eight projective parameters. 
The eight parameters are estimated using a minimum of four 
points with known coordinates in both systems, provided that no 
three of which are collinear. Then, for any additional point, its 
unknown coordinates on the first plane can be computed using its 
known coordinates on the second plane, the two projective 
equations and the estimated eight parameters. 
The equations demonstrate the non-linear nature of the projective 
transformation. Therefore, a least squares adjustment program 
was developed in order to uniquely estimate the eight parameters 
(o f tu :2:) and to calculate the coordinates of any additional 
point. 
2.2 Line-Based Projective Equations 
A very important concept in projective geometry is the principle 
of duality. In the projective plane, points and lines are said to be 
dual. Any theorem applying to points also applies to lines and 
vice versa. By this principle, centrally projected straight lines 
may be used to establish the projective equations. The line 
equation using Euclidean coordinates is given in the following 
form, 
gGxT5yrtrlz-0 (2) 
Where a,,b,= line parameters. 
Equations (1) are the classical projective equations based on 
point correspondence. A comparable pair of projective equations 
for two protectively related straight lines is given by, 
ost t si t, 
a => 
ra+s bl 
HE ER t, (3) 
rats b+ 
Where al, a,b are the line parameters in the first and 
second plane, respectively, 
(r,,s,,..,,1,) are the eight projective parameters. 
Le 0 “ 
This equation is derived using equations (1) and (2). Solving the 
3 equations (two of (1) and one of (2)) simultaneously, equation 
(3) is obtained using a simple algebraic manipulation. 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
185 
The relationships between the line projective parameters 
(r, $,24,,5,) and the point projective parameters 
are 2% a 1 : TE 7 Once > 
(e,. f, "esl 25) are presented in subsection 2.3.Since the 
equations are nonlinear, a least squares adjustment program 
was developed to solve for the eight parameters using a 
minimum of 4 straight lines with known line parameters in 
both planes, then the parameters of any additional line can be 
computed, similar to the point case. 
2.3 Combined Point/Line-Based Projective Equations 
Equations (1) and (3) can be used together in the same least 
squares adjustment to effect projectivity between two planes 
based on the combination of points (1) and lines (3). However, 
in this case, two sets of 8-parameters, (o f S er 2) and 
(r,,s,...t,,t,) need to be carried in the adjustment. Since there 
are only 8 independent projective equations in a two-plane 
projectivity, the following 8 constraint equations must be 
carried in the adjustment, 
nell, =f. oe fr =f) 
$, 2 (e,g, —e,)/(e, f, — e, f) 
th 8 (e fo - eS fM f. ef) 
^ SOS foe -ef) (4) 
5, =(e, -€,8i)/(e, f, -ejfi) 
5 7(6, f - 6f e f ef) 
n, 7 (68 - 68;) (6f, - e fi) 
5, 2 (egi 7e ge fe. f) 
These constraint equations are derived from the solution of the 
equations (1) and (3) simultaneously as mentioned earlier. A 
least squares adjustment solution with constraints was 
developed to implement the above equations. 
3. EXPERIMENTS AND ANALYSIS 
Several experiments were carried out to test and study the 
performance of the above derived equations regarding the 
image rectification process that is the process of transforming 
the image data from one grid system into another grid system 
using mathematical models such as (affine, polynomial, 
projective...). 
The point-, line-, and combined point/line-based projective 
equations are used as the mathematical models of the image 
rectification in order to perform the conversion/transformation 
from the image coordinate system into the ground reference 
coordinate system. Rectifying image data involves the 
identification and locating of well distributed ground control 
points/lines, computation and estimation of projective 
parameters, creation of an output rectified image, and, then, 
examination using some check points (points with known 
ground coordinates which are not used in the estimation of the 
projective parameters). 
In the following experiments, several image data sources were 
used. Aerial photographs were used first to check the 
performance of the developed programs, especially for the 
newly derived projective equations for lines and combined 
points/lines solutions, and to compare their results to the well- 
established point-based projective equations. Then, various 
satellite image data of LANDSAT7, SPOT4, IRS-ID, and 
IKONOS were used to investigate the applicability of the