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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