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Mapping without the sun
Zhang, Jixian

Orientation Model
same orientation models
various orientation models
Extent of Complexity
simple, long researching history
complex, many problems, little research
Suitable Scope
Table 1 the differentiation and affiliation between traditional and generalized stereopair
Its practical significance can be described as: (1) Opens the
source: Maximum limit uses the existing image sources, and
expands the information content; (2) Saves money: Saves the
expensive expense; (3) Increases effect: Realizes the fast 3D
information extraction and target reconstruction.
The orientation model of image decides which kind of models
should be used in space intersection. The common orientation
models have the following types: Polynomial Model (PM),
Collinearity Equation Model (CEM), Affine Model (AM),
Direct Linear Transformation (DLT), and Rational Function
Model (RFM). Among them, RFM has already become the
main orientation model for high resolution imagery. Comparing
with other models, the RFM has following advantages: (1)
independence from sensors or cameras; (2) suit for any kind of
coordinate system, and free form the coordinate system; (3)
can imitate rigorous models of any existing sensors or cameras.
According to the above advantages, this paper takes the RFM
as the basis of generalized stereopair space intersection and
makes the principle deduce and analysis.
In the construction of generalized stereopair, the
different-source images have the different sensor models and
imaging techniques. In order to unify all these different sensor
models under the identical kind of model, therefore takes the
RFM as the image orientation model for each single image to
construct generalized stereopair. And all these kind of space
intersection models constructed by this method are all based on
the RFM model.
3.1 Mathematics Model of RFM
mathematics expressing of RFM:
r _ P\(X,Y,H)
P 2 (X,Y,H) (1)
^ P 3 (X,Y,H)
P 4 (X,Y,H)
m ] m 2 tfi 3
P(X,Y,H) = £'£ j Y j a ijk X i Y J H k
i=0 y=0 k=0
— a x + a 2 X + a 3 Y + a 4 H + a s XY + a b XH
+ a 7 YH + a,X 2 + a 9 Y 2 + a l0 H 2 + a n XYH In this
+ a n X + a X2 XY + a x4 XH 2 + a l5 X 2 Y
+ + a xl YH 2 + a xs X 2 H + a X9 Y 2 H + a 20 H 3
Y, C image coordinates after normalization;
X, Y, H ground coordinates after normalization;
a,, b t , c,, d t Rational Polynomial Coefficients of RFM
3.2 Solution Method of RPC
The solution method of RPC can be divided into two kinds: the
first kind is the solution which has nothing to do with the
terrain. This method is under the sensor rigorous physical
model known condition, and establishes the image grid and the
corresponding ground 3D grid, then calculates the RPC by the
least square method according to the grid corresponding
coordinates; The second kind is the solution with terrain. This
method is suitable for the situation of sensor rigorous physical
model is unknown, and needs to select 40 or more GCP, then
calculates the RPC by the least square method according to
these GCP ground 3D coordinates and corresponding image
3.2.1 Solution of Normalized Parameters
Before solves the RPC, in order to strengthen the stability of
normal equation coefficient matrix, and avoids having the rank
defect matrix, it needs to firstly carry on normalization
processing to the GCP ground 3D coordinates and the
corresponding image coordinates, and causes the coordinates
normalized between -1.0 and+1.0. The method of
normalization processing is stated as following formula 2 and
normalization of the image coordinates:
if the size of image is 2m x 2n, an( j t ^ e pj xe j coordinates of
image center is ( m ’ n \ then the corresponding normalization
parameters of image coordinates can be calculated by following
SampOffset = m,LineOffset = n, ^
Samp Scale = m,LineScale = n
normalization of the ground 3D coordinates:
S'[Longitude V Latitude V Height
LongOffset= , LatOffset= , HeightOffet = — (3)
n n n v '
LongScale= maj^|Longtitud^ - LongOffsdf \Longtitud& m - LongOffs^j
LatScale= ma^Latitud^. - LatOffsdf | Latitude - LatOffsdf)
HeightScai = mat^Heigh^ - HeightOffet| |Height - HeightOffe\)
According to above formula, the 10 normalization parameters
of image coordinates can be calculated, then the ground
coordinates (X,Y,H) and corresponding image coordinates
(r )C ) after normalization can be calculate by these normalization
parameters. Finally, the RPC also can be solved by these
coordinates after normalization under the least square method.
3.3 Generalized Stereopair Space Intersection Model Based
on RFM
When the orientation model of left and right image to construct
the generalized stereopair are all RFM, then the corresponding
space intersection mathematical model can be described as the
RFM+RFM model, and its mathematical formula is: