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3-D Transformations

3-D Transformations. Brian Romsek Senior Student Surveying Engineering Department. Y Axis. Z Axis. X Axis. Three-Dimensional Conformal Coordinate Transformation. Converting from one three-dimensional system to another, while preserving the true shape .

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3-D Transformations

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  1. 3-D Transformations Brian Romsek Senior Student Surveying Engineering Department

  2. Y Axis Z Axis X Axis 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.

  3. Applications of 3D Conformal Coordinate Transformations • Mobile mapping systems • Relations between different coordinate frames • Sensor frame • Body frame • Mapping frame

  4. Applications of 3D Conformal Coordinate Transformations • Homeland security • E.G., facial pattern recognition • Image processing

  5. Kappa () Z-axis Y-axis (X,Y,Z) Omega () X-axis Phi () 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

  6. Rotation angles Omega In general form: In matrix form: More concisely

  7. Kappa () Omega () Phi () Rotation angles Phi In general form: In matrix form: More concisely Z-axis X-axis

  8. Kappa () Omega () Phi () Rotation angles Kappa In general form: In matrix form: More concisely Z-axis X-axis

  9. Combined Rotation Matrix If we combine all the rotation matrices MG becomes, after multiplication

  10. COMPUTING ROTATION ANGLES • If rotation matrix known, rotation angles can be computed as shown on the right

  11. 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

  12. Y Axis Z Axis X Axis 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:

  13. BURSA-WOLF TRANSFORMATION • Geodesy assumption – rotation angles small • cos  = 1 • sin  =  (radians) • Product of two sines = 0 • Rotation matrix R becomes:

  14. BURSA-WOLF TRANSFORMATION • 3D similarity transformation • Observation Equation:

  15. BURSA-WOLF TRANSFORMATION • Coefficient matrix, B: • Vector of parameters, , and discrepancy vector, f

  16. Three Dimensional Coordinates Transformation General polynomial approach: transformation is not conformal

  17. Three Dimensional Coordinates Transformation Alternative that is conformal in the three planes

  18. Three Dimensional Coordinates Transformation Polynomial projective transformation, 15 parameters

  19. 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) Testing – 4 Methods

  20. 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 Testing – 4 Methods

  21. Expanded Full-Model • Employed method shown in “Photogrammetric Guide” by Abertz & Kreiling • X, Y, Z coordinates translated to relative values based in mean coordinates

  22. 3D Transformations Testing • Data include a set of know control points, transformed from WGS84 system to State Plane Coordinates.

  23. Test Results Reference Variance

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