Analytical Photogrammetry

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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
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Coordinate Transformations

• 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.

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Coordinate Transformations

• 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.

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Coordinate Transformations

• 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.

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Coordinate Transformations

• 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.

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Two-dimensional conformal coordinate
transformation

• Three basic steps
• Scale change
• Rotation
• Translation

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Coordinate Transformations with Redundancy

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

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Two-dimensional Affine Coordinate Transformation

• 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.

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Three-Dimensional Conformal Coordinate
Transformation

• 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.

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More on Coordinate Transformations

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

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Projective Equations

• 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.

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Projective Equations

• 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.

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Projective Equations

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

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Projective Equations

• 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
•  
• where
• X, Y, Z are the ground coordinates of the
  point
• XL, YL, ZL are the ground coordinates of the
  ground nadir point

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Projective Equations

• 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.

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Direction Cosines

• 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
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Direction Cosines

• The direction of the vector can be written with
  respect to the 3 axes as

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Direction Cosines

• 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

Line vector PQ in space.
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Direction Cosines

• The length of the vector becomes
• and the direction cosines are

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Direction Cosines

• 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

Unit vectors.
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Direction Cosines

• 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.

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Direction Cosines

• For example, figure shows the relationship
  between
• and .The angle between the axes is
  defined as (xY). Since has similar angles
  to the other two axes, one can write the unit
  vector in terms of the direction cosines as

Rotation between Y and x axes.
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Direction Cosines

• Similarly, we have
• Then, the vector from O to P can be written as

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Direction Cosines
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Direction Cosines

• This can be written more generally as
• 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

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Direction Cosines

• 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....