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 "^
3 |—
fy
is (
Int
inte
ort
ha: