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2.1 Rational Functions 
Under the model, an image coordinate is determined from a ratio 
of two polynomial functions, in which the image (x,y) and ground 
coordinates (X,Y,Z) have all been normalized (OGC, 1999): 
x = PI(X,Y,Z)/P2(X,Y,Z) = 
m m2 m3 nl n2* n3 
ivJ 7k y ardt 
Sle dase NZ 
i=0 j=0 k=0 : i=0 j=0 k=0 
y=P3(X,Y.Z2)/PAX.Y,Z) = 
ml m2 m3 nl. n2. n3 
EE 7k i t 
$$$ qxve/ESS aacrn 
i=0 j=0 k=0 0 i=0 j=0 k=0 
(1) 
For example RFM with 14, 17 and 20 terms are as follows: 
Rational with 14 terms: 
d, dA +a + az Hay 
  
I+ i X = CY "t €3Z + cs XY 
b, + b,X == b,Y Hr b;Z io b,XY 
  
yc 
l* cX * eY * eZ * eXY 
Rational with 17 terms: 
An TAX EA Fast + A4AY + A5L 
X = 
f CX s e + cl + c XY de csXZ 
  
bb X + hY + h{ b,XY *bsXZ 
y= (3) 
I c,X CS y C32 = c XY + csXZ 
  
Rational with 20 terms: 
4 A tal + a2 + a XY Has AZ tla YZ 
  
Itc, X OYcoXZFX£ CXYcOAZ + est 
B,D, XA bY + BZ + D XY + bs XZ + bY 
y= (4) 
fou d cy EZ CHAT KeNZ vol 
  
The rational function method (RFM) maps three-dimensional 
ground coordinates to image space for all types of sensors, such as 
frame, pushbroom, whiskbroom and SAR systems. The direct 
linear transformation (DLT), self calibration DLT (SDLT), 3D 
affine, 2D projective equations and polynomials are specialized 
forms of the RFM, and we now consider these models. 
2.1.1 Direct Linear Transformation (DLT) 
Eleven linear orientation parameters define the relationship 
between 2D image space and 3D object space: 
at aX + aY + a;Z 
Xu eT res a em ee adig 
jo cuX zx co + Cal 
b, + b,X is b,Y + b3Z 
y = nn nn oe no nt “1 A 
FX Fle Salo AS) 
  
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV. Part B3. Istanbul 2004 
2.1.2 Self Calibration Direct Linear Transformation (SDLT) 
Twelve linear orientation parameters define the relationship 
between 2D image and 3D object space: 
a, + aX + aY + a2 bb, FD XD + DZ 
A a 9 -—————————————— 
L3 c,X + cPY + Ciz 1 CN sk: coY + CiZ (6) 
2.1.3 3D Affine Transformation 
Eight parameters define the relationship between the object and 
image spaces: 
x=a,ta; X+aY+a;Z, y=b,+b,X+b,Y+b;Z (7) 
2.1.4 2D Projective Transformation 
Eight parameters define the relationship between the object and 
image planes: 
a. aud ay B, + 0,X+ bY 
Y= (8) 
] Y C5 C ] V CX TOY 
2.1.5 2D Polynomials 
The model describes the relationship between image and object 
space independent of the sensor geometry: 
AR 5 S d, xy and Y = S. S b xi (9) 
i=0 j=0 j=0 j=0 
In the above functions x,y are the coordinates on the image; X, Y.Z 
are the coordinates on the ground; and A Ed, are 
transformation parameters. 
2.2 Radial Basis Function Methods 
Radial basis functions, such as Hardy's multiquadric functions 
(MQs) and reciprocal multiquadrics (RMQs), thin plate splines 
(TPS) and variations of these methods offer ready alternatives to 
conventional rational function and polynomials methods for 
image to object space transformation where many of GCPs are 
available. The radial basis function methods involve the solution 
of an equation system with the same number of unknown 
parameters as GCPs. Thus, we have a perfect fit for all GCPs. 
2.2.1 Multiquadric Approach (MQ) 
This discussion is limited to the 2D case where radial basis 
functions may be constructed as a linear combination of the 
following equations for x and y: 
M 
N 
Sa h(xy) +S ba xy) =Xandy, i=... 
j=l j=l 
N 
324,0») =0,/=1,...M (10) 
i=l