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```									Analytical Photogrammetry

Hande Demirel, PhD,
Assistant Prof.
Istanbul Technical University (ITU),
Faculty of Civil Engineering,
Geodesy and Photogrammetry Department
Division of Photogrammetry
Coordinate Transformations
Division of
PHOTOGRAMMETRY

 A problem frequently encountered in
photogrammetric works is conversion from one
rectangular coordinate system to another.

 This is because photogrammetrists commonly
determine coordinates of unknown points in
convenient arbitrary rectangular coordinate system.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Coordinate Transformations
Division of
PHOTOGRAMMETRY

 The arbibitrary coordinates must be then converted
to a final system such as;
 Photo coordinate system
 Ground coordinate system

 The procedure for converting from one coordinate
system to another is known as coordinate
transformation.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Coordinate Transformations
Division of
PHOTOGRAMMETRY

 The procedure requires that some points have their
coordinates known (or measured) in both the
arbitrary and the final coordinate systems. Such
points are called control points.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Coordinate Transformations
Division of
PHOTOGRAMMETRY

 Two-dimensional conformal coordinate transformation:

 The term two-dimensional means that the coordinate systems lie
on plane surfaces.
 A conformal transformation is one in which true shape is
preserved after transformation.
 To perform:
 At least two points to be known in both the arbitrary and
final coordinate system. If more than two control points are
available, an improved solution may be obtained by applying
the method of least squares.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Two-dimensional conformal
coordinate transformation:
Division of
PHOTOGRAMMETRY

 Three basic steps:
 Scale change
 Rotation
 Translation

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Coordinate Transformations with Redundancy
Division of
PHOTOGRAMMETRY

 Least squares procedure

aX  bY  TE  E  vE
aY  bX  TN  N  vN

 If n points are available whose coordinates are known
in both systems, 2n equations may be formed
containning the four unknown transformation
parameters.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Two-dimensional Affine Coordinate Transformation
Division of
PHOTOGRAMMETRY

 Two-dimensional Affine Coordinate Transformation is
only a slight modification of the two-dimensional
conformal transformation, to include different scale
factors in the x and y directions and to compensate
for nonorthogonality (nonperpendicularity) of the axis
system.

 The affine transformation achieves these additional
features by including two additional unknown
parameters for a total of six.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Three-Dimensional Conformal
Coordinate Transformation
Division of
PHOTOGRAMMETRY

 It is essential in photogrammetry for two basic
problems:
 To transform arbitrary stereomodel coordinates to a
ground or object space system
 To from continuous three dimensional “strip models”
from independent streomodels.
 Seven independent parameters:
 Three rotation angles
 A scale factor
 Three translation parameters.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
More on Coordinate Transformations
Division of
PHOTOGRAMMETRY

 Wolf, P., Dewitt, B., 2000, McGraw- Hill, Elements of
Photogrammetry
 Coordinate Transformations                          pg.518-550

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Projective Equations
Division of
PHOTOGRAMMETRY

 The numerical resection problem involves the
transformation (rotation and translation) of the
ground coordinates to photo coordinates for
comparison purposes in the least squares adjustment.
 Before we begin this process, lets derive the rotation
matrix that will be used to form the collinearity
condition.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Projective Equations
Division of
PHOTOGRAMMETRY

 In photogrammetry, the coordinates of the points
imaged on the photograph are determined through
observations.
 The next procedure is to compare these photo
coordinates with the ground coordinates. On the
photograph, the positive x-axis is taken in the
direction of flight. For any number of reasons, this
will most probably never coincide with the ground X-
axis.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Projective Equations
Division of
PHOTOGRAMMETRY

 The origin of the photographic coordinates is at the principal
point which can be expressed as

X'  x  x o 
Y'   y  y 
           o

 Z'    f 
            
where:
x, y   are the photo coordinates of the imaged point with
reference to the intersection of the fiducial axes
xo, yo are the coordinates from the intersection of the
fiducial axes to the principal point
f      is the focal length

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Projective Equations
Division of
PHOTOGRAMMETRY

 Since the origin of the ground coordinates does not
coincide with the origin of the photographic
coordinate system, a translation is necessary. We can
write this as
X 1  X  X L 
Y   Y  Y 
 1          L

 Z1   Z  Z L 
              
where:
X, Y, Z                      are the ground coordinates of the
point
XL, YL, ZL             are the ground coordinates of the
Spring 2008, Istanbul Technical University, Istanbul-Turkey
Projective Equations
Division of
PHOTOGRAMMETRY

 Thus, in the comparison, both ground coordinates and
photo coordinates are referenced to the same origin
separated only by the flying height.

 Note that the ground nadir coordinates would
correspond to the principal point coordinates in X and
Y if the photograph was truly vertical.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Direction Cosines
Division of
PHOTOGRAMMETRY

 If we look at figure, we can see that point P has
coordinates XP, YP, ZP.

 The length of the vector (distance) can be defined as

Vector OP in 3-D space University, Istanbul-Turkey
Spring 2008, Istanbul Technical
Direction Cosines
Division of
PHOTOGRAMMETRY

                          
1
OP  X  Y  Z     2
P
2
P
2 2
P

 The direction of the vector can be written with
respect to the 3 axes as:
XP
cos  
OP
Y
cos  P
OP
Z
cos   P
OP
Spring 2008, Istanbul Technical University, Istanbul-Turkey
Direction Cosines
Division of
PHOTOGRAMMETRY

 These cosines are called the direction cosines of the vector
from O to P. This concept can be extended to any line in
space. For example, figure 2 shows the line PQ. Here we can
readily see that the vector can be defined as:

XQ  XP 
         
PQ   YQ  YP   QP
Z Z 
 Q     P 

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Line vector PQ in space.
Direction Cosines
Division of
PHOTOGRAMMETRY

 The length of the vector becomes


PQ  X Q  X P   YQ  YP   Z Q  Z P 
2                     2              2

1
2

 and the direction cosines are                                            XQ  XP
cos 
PQ
YQ  YP
cos 
PQ
ZQ  ZP
cos  
Spring 2008, Istanbul Technical University, Istanbul-Turkey                      PQ
Direction Cosines
Division of
PHOTOGRAMMETRY

 If we look at the unit vector as shown in figure, one
can see that the vector from O to P can be defined as

OP  xi  y j  zk

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Unit vectors.
Direction Cosines
Division of
PHOTOGRAMMETRY

 and the point P has coordinates (x, y, z)T.
 Given a second set of coordinates axes (I, J, K), one
can write similar relationships for the same point P.
Each coordinate axes has an angular relationship to
each of the i, j, k coordinate axes.

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Direction Cosines
Division of
PHOTOGRAMMETRY

 For example, figure shows the relationship between
J and i .The angle between the axes is defined as
(xY). Since i has similar angles to the other two
axes, one can write the unit vector in terms of the
direction cosines as:
 i  I  cosxX 
       
i  i  J   cosxY 
                  
 i  K   cosxZ
                  
Rotation between Y and x axes.
Spring 2008, Istanbul Technical University, Istanbul-Turkey
Direction Cosines
Division of
PHOTOGRAMMETRY

 Similarly, we have
cosyX                                   coszX 
j  cosyY
                                      k  coszY 
        
 cosyZ
                                           coszZ
        

 Then, the vector from O to P can be written as

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Direction Cosines
Division of
PHOTOGRAMMETRY

cosxX      cosyX     coszX 
OP  x cosxY   y cosyY  z coszY 
                                 
 cosxZ
              cosyZ
             coszZ
        

cosxX  cosyX coszX   x  X 
 cosxY  cosyY coszY   y   Y 
                            
 cosxZ cosyZ coszZ  z   Z 
                            

Spring 2008, Istanbul Technical University, Istanbul-Turkey
Direction Cosines
Division of
PHOTOGRAMMETRY

 This can be written more generally as

X  Rx
 To solve these unknowns using only three angles, 6 orthogonal
conditions must be applied to the rotation matrix, R. All vectors
must have a length of 1 and any combination of the two must be
orthogonal [Novak, 1993]. Thus, designating R as three column
vectors [R = (r1 r2 r3)], we have

r1T r1  r2t r2  r3T r3  1
r1T r2  r2T r3  r3T r1  0
Spring 2008, Istanbul Technical University, Istanbul-Turkey
Direction Cosines
Division of
PHOTOGRAMMETRY

 Rotation Combinations

Three angles of tilt, swing and azimuth completly define the
angular orientation of a tilted photograph in space.
If the tilt angle is zero....
Spring 2008, Istanbul Technical University, Istanbul-Turkey

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