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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
                                     ground nadir point
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