3-D Transformations

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3-D Transformations

• Brian Romsek
• Senior Student
• Surveying Engineering Department

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Three-Dimensional Conformal Coordinate
Transformation

• Converting from one three-dimensional system to
  another, while preserving the true shape.
• This type of coordinate transformation is
  essential in analytical photogrammetry to
  transform arbitrary stereo model coordinates to a
  ground or object space system.
• It is often used in Geodesy to convert GPS
  coordinates in WGS84 to State Plane Coordinates.

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Applications of 3D Conformal Coordinate
Transformations

• Mobile mapping systems
• Relations between different coordinate frames
• Sensor frame
• Body frame
• Mapping frame

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Applications of 3D Conformal Coordinate
Transformations

• Homeland security
• E.G., facial pattern recognition
• Image processing

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3D Conformal Coordinate Transformation

• Also known as the 7 Parameters transformation
  since it involves
• Three rotation angles omega (?), phi (?), and
  kappa (?)
• Three translation parameters (TX, TY,TZ) and
• a scale factor, S

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Rotation angles Omega
In general form In matrix form More
concisely
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Rotation angles Phi
In general form In matrix form More
concisely
Z-axis
X-axis
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Rotation angles Kappa
In general form In matrix form More
concisely
Z-axis
X-axis
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Combined Rotation Matrix
If we combine all the rotation matrices MG
becomes, after multiplication
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COMPUTING ROTATION ANGLES

• If rotation matrix known, rotation angles can be
  computed as shown on the right

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Properties of rotation matrix

• The rotation matrix is an orthogonal matrix,
  which has the property that its inverse is equal
  to its transpose, or
• This can be used for inverse relationship

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Three-Dimensional Conformal Coordinate
Transformation

• Finally the 3D Conformal Transformation is
  derived by multiplying the system by a scale
  factor s and adding the translation factors TX,
  TY, and TZ.
• Where

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BURSA-WOLF TRANSFORMATION

• Geodesy assumption rotation angles small
• cos ? 1
• sin ? ? (radians)
• Product of two sines 0
• Rotation matrix R becomes

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BURSA-WOLF TRANSFORMATION

• 3D similarity transformation
• Observation Equation

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BURSA-WOLF TRANSFORMATION

• Coefficient matrix, B
• Vector of parameters, ?, and discrepancy vector, f

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Three Dimensional Coordinates Transformation
General polynomial approach transformation is
not conformal
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Three Dimensional Coordinates Transformation
Alternative that is conformal in the three planes
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Three Dimensional Coordinates Transformation

• Polynomial projective transformation, 15
  parameters

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Testing 4 Methods

• Bursa Wolf
• Linear model assume small rotation angles
• Best for satellite to global system
  transformations
• Bazlov et al determined PX 90 to WGS 84
  parameters

• Generalized Bursa Wolf
• Linear model errors in both observations and
  model parameters
• Useful transforming classical to space-borne
  (Kashani, 2006)

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Testing 4 Methods

• Polynomial
• 1st order
• Useful when coordinate systems not uniform in
  orientation or scale
• Rubber-sheeting

• Expanded Full- Model
• Photogrammetric approach
• Angles not considered small
• Non-linear requires a priori estimate of
  parameters

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Expanded Full-Model

• Employed method shown in Photogrammetric Guide
  by Abertz Kreiling
• X, Y, Z coordinates translated to relative values
  based in mean coordinates

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3D Transformations Testing

• Data include a set of know control points,
  transformed from WGS84 system to State Plane
  Coordinates.

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Test Results
Reference Variance