2D Coordinate Transformations Nick Battjes, Senior Student

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2D Coordinate TransformationsNick Battjes,
Senior Student
Spatial data without coordinates
Control Points
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Applications

• Integrating of maps and spatial data in local
  coordinate system into a world database system.
• Note how the vector data (USGS Road layer) should
  be transformed to match the raster data (Ikonos 1
  meter res. Image)

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2D Spatial Transformations

• Three different transformation primitives for
  the Similarity transformation
• Translation- origin is moved, axes do not rotate
  i.e.
• Xn X0 DX0 Yn Y0 DY0
• Scaling - both origin and axes are fixed, scale
  change
• Xn sX X0 Yn sY Y0
• Rotation - origin fixed, axes move (rotate about
  origin)
• Xn X0 cos(?) Y0 sin(?) Yn - X0
  sin(?) Y0 cos(?)

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Two-DimensionalGeographic Transformations
Scaling
Rotation
Translation
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Conformal Transformation

• Moves and rotates objects in 2D and 3D space.
  Additionally, you can scale the objects based on
  alignment points when using the 2D option.

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Isogonal Affine Transformationor
Conformal/Similarity Transformation

• Isogonal having equal angles
• Impose additional condition of equal scale (S
  Cx Cy) yielding 4 parameters S, ?, DX0, DY0
• Moves and rotates and scale objects in 2D space.

4 parameters S scale, a rotation, DX0, DY0
shifts in X and Y. Xn,Yn are the transformed
coordinates. Xo, Yo are the original
coordinates. Two given points are required (
X1,Y1 and X2,Y2)
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Conformal Transformation

• Moves and rotates and scale objects in 2D space.
• Often called similarity Transformation since the
  basic shape remain similar after the
  transformation

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Conformal Transformation

• The formulas can be written in different forms
• To compute the parameters given the coordinates
• To compute the new coordinates given the
  parameters

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Conformal/Similarity Transformation

• The formulas can be written in different forms
• 3. To compute the old coordinates given the
  parameters and new coordinates (back
  substitution)

The same operation to obtain Y0
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Conformal Transformation Example

• Moves, rotates and scale objects in 2D space.

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Conformal Transformation Example

• Transform a given point X0121.48, Y022.78
• The point is transformed to Xn310.59,
  Yn373.6773

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Affine Transformation

• Used in photogrammetry for
• Transform comparator coordinates to photo
  coordinates and used for correcting film
  distortion
• Transform model coordinates to survey coordinates
• Property
• Carry parallel lines
• into parallel lines
• Does not have to
• preserve orthogonality

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Affine Transformation
Physical interpretation 6 parameters Cx, Cy,
?, ?, Dx0, Dy0, and in linear form
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2-D Affine Transformation

• The formulas for an affine transformation
• If n control points are measured, this Equation
  is reorganized as follows

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Projective or Polynomial Transformation

• Instead of 4 or 6 parameters we have many
  parameters at least 8 (8 would be the Bi-linear
  or projective transformation)
• With more parameters we need more known points to
  solve the equations
• N-equations and N unknowns.

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Polynomial Transformation

• General form
• Alternatively,

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Projective Transformation

• Frequently used in photogrammetry
• General form

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Orthogonal Affine Transformation

• Impose condition of orthogonality (? 0)
  yielding 5 parameters Cx, Cy, ?, ?x, ?y

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Rigid Body Transformation

• Condition orthogonality and no scale change (Cx
  Cy 1)
• 3 parameters ?, ?x, ?y

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Least Squares Adjustment Review

• Model of adjustment of indirect observation (
  Gauss-Markov model)
• n of observations
• m of parameters
• f is a vector of given observation
• V is the vector of residuals
• B is the matrix of coefficients
• W it the matrix of weights (Variance
  Covariance matrix)
• ?0 is the reference variance
• ? is the vectors of parameters to be
  estimated
• Number of observation is larger than number of
  parameters (redundant observations). The solution
  that minimize the least-squares criterion (vTwv)
  is

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General 2D Conformal Transformation, Example

• Four fiducial marks (1 - 4) and two image points
  (a and b) were measured on a comparator. The
  comparator photo observations and the known
  values from the camera calibration report are
  given in the following spreadsheet.

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General 2D Conformal Transformation, Example
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General 2D Conformal Transformation, Example
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General 2D Conformal Transformation, Example
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General 2D Conformal Transformation, Example
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General 2D Affine Transformation

• Normally shown as
• Unique solution if

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General 2D Affine Transformation, Example

• Four fiducial marks (1 - 4) and two image points
  (a and b) were measured on a comparator. The
  comparator photo observations and the known
  values from the camera calibration report are
  given in the following spreadsheet.

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General Affine Transformation, Example
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General Affine Transformation, Example
Solution forming the B matrix and f vector
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General Affine Transformation, Example
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General Affine Transformation, Example
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General Affine Transformation, Example
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General Affine Transformation, Example