different interpolation errors (aliasing, lowpass error) can 
occur and the computer operation time is varying. 
2.2. Non-parametric Techniques 
Different plane transformations can be realized by non- 
parametric techniques with sufficient accuracy (Mikhail 
et al., 1975). In contrast with the parametric approach, 
which gives a rigorous solution by the collinearity 
equations, the plane interpolations (non-parametric 
methods) are directly determined between the projective 
plane S'[s'(x',y") andthe image plane S[s(x,y)] using a lot 
of control points Pi: (ui,vi) & (xi,yi) from the reference 
model R = [s(x,y)]. 
  
  
Fig. 2.2.1 Position interpolation with non-parametric 
techniques 
The elevation co-ordinate is influencing the position of 
control points on the projection plane, if these are 
computed from 3D- object co-ordinates P(u,v)=f(X,Y,Z). 
Different projection parameters can be used in each 
application. Distortions caused during the recording can 
indeed only be corrected by a sufficient number of control 
points. 
The transformation of pixel coordinates following the 
indirect methodis givenby S' — S with (x,y') = (x,y), where 
X'=u=u(x,y) andy'=v=v(x,y) represent the transformations. 
2.3. Interpolation and Prediction Methods 
Severalinterpolation and prediction methods are suitable 
for non-parametric transformations depending on the 
plane shape, the distortion and the distribution of control 
points (Hein, 1979; Gópfert, 1987). 
Projective transformation 
800| , |810 801 | y 
Doo| |D1o bo: | 
+ 
5 
V Coo + C10 X + Co1 Y 
X + 
  
  
  
  
  
; Coo = 1 (1) 
  
  
(for planes with projektive distortion, oblique images) 
  
  
  
for irregular and delimited partial planes) 
  
QP Surface Deviation 
  
— 
Linear 1-q | plane affine 
  
— 
Bilinear 1 plane hyperbolic 
  
Quadratic |2 | 2-q | curved global 
  
Biquadratic| 2 | 2 curved irregular 
  
  
  
  
  
  
  
Table 2.3.1 Types of bivariable Polnoms Q-th Order 
For the modelling of differential geometric distortions, 
which result from irregular surfaces, the meshwise linear 
interpolation as a local technique and the multiquadratic 
equations as a global approach appear suitable (Hein, 
1979; Gópfert, 1987). 
Meshwise Linear Transformation 
The triangulation presupposes a steady distribution of 
control points. Places of unsteadiness along the triangle 
sides can come up. Great planes can be transformed. 
SENA A 
Pr v = 
bs t yi T yi] * ly: y) bx: x) 
s em: - x yo . yi] : "m E yi " 7 xi) 
  
(3.2) 
Multiquadratic Equations 
The method shows a superposition of n planes of 2nd 
order. A plausible interpolation betweenthe control points 
is achievedundependently ontheir density anddistribution 
but regarding their quality for a rectification. The shape of 
the plane is not of importance. 
Fi ; Ex (4.1) 
v^ y| 
Bip er "^ 
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