COORDINATE TRANSFORMATIONS - Surveying Engineering 工程测量中坐标转换

Z2 0 sin cosZ1
More concisely C2 MC
Rotation angles Phi
In general form:
X3 X2cosY20Z2(sin) Y3 X20Y2Z20 Z3 X2sinY20Z2cos
In matrix form:
X3 cos 0 sinX2
Y3 0 1
tan m 32 m 33
sin m 31
tan m 21 m 11
Properties of rotation matrix
• The rotation matrix is an orthogonal matrix, which has the property that its inverse is equal
Yn b0 b1x b2 y b3z b4 x2 b5 y2 b6 z 2 b7 xy b8 yz b9 zx b10 xy2 b11x2 y b12 xz 2 L
Zn c0 c1x c2 y c3z c4 x2 c5 y2 c6 z 2 c7 xy c8 yz c9 zx c10 xy2 c11x2 y c12 xz 2 L
• 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
– cos = 1 – sin = (radians) – Product of two sines = 0
• Rotation matrix R becomes:
1 Rκ R
R Rκ 1
Rω
R Rω 1
BURSA-WOLF TRANSFORMATION
• 3D similarity transformation
• Data include a set of know control points, transformed from WGS84 system to State Plane Coordinates.
Test Results
Method Photo Guide Bursa-Wolf Generalized Bursa-Wolf Polynomial 1st Order
M G c o ss i n c o cs o s s is n is ni n s ic n o c s o ss is ni n
s in
s ic n os
c o cs os
COMPUTING ROTATION ANGLES
• If rotation matrix known, rotation angles can be computed as shown on the right
0
0 B 1
0 0
1 0
z1 x2
y1 0
x1 z2
0 0 1 zn yn xn
• Vector of parameters, , and discrepancy vector, f
T
Δ T XT YT ZsR R R
y1
x1
0
y2
0
X
f
Y
Z
Three Dimensional Coordinates Transformation
X'
X1
X1
Y'MGY1MMM Y1
Z'
Z1
Z1
m11 m12 m13
MMMM m21
m22
m23
m31 m32 m33
MG becomes, after multiplication
c o cs oc s o ss i n s is n ic n os s is ni n c o ss ic n o
X' cos sin 0X3 Y'sin cos 0Y3
Z' 0
0 1Z3
More concisely C' MC3
Phi ()
Z-axis
Kappa ()
Omega ()
Combined Rotation Matrix
If we combine all the rotation matrices
– Rubber-sheeting
• Expanded Full- Model
– Photogrammetric approach
– Angles not considered small
– Non-linear: requires a priori estimate of parameters
Expanded Full-Model
Appwk.baidu.comications of 3D Conformal
Coordinate Transformations
•Homeland security •E.G., facial pattern recognition •Image processing
3D Conformal Coordinate
Transformation
C ' sM C T
• Where:
T X T TY
T Z
X
C
'
Y
Z
m11
M
m21
m31
m12 m22 m32
m13
m23
m33
Z Axis
X Axis
BURSA-WOLF TRANSFORMATION
• Geodesy assumption – rotation angles small
Three Dimensional Coordinates Transformation
Alternative that is conformal in the three planes
XnA0A1xA2yA3zA5 x2y2z2 0aA7zx2A6xyL
YnB0A2xA1yA4zA6 x2y2z2 2A7yz02A5xyL
0
Y2
Z3 sin 0 cos Z2
More concisely C3 MC2
Phi ()
Z-axis
Kappa ()
Omega ()
Rotation angles Kappa
In general form:
X' X3cosY3sinZ30
Y' X3sinY3cosZ30
Z' X30Y30Z3
In matrix form:
f(x)X 0 Rκ Rx x TX g(x)YRκ 0 RωysyTY h(x)Z R Rω 0z z TZ
• Observation Equation:
VBΔf
BURSA-WOLF TRANSFORMATION
• Coefficient matrix, B:
1 0 0 x1 0 z1
0 1 0 y1 z1
General polynomial approach: transformation is not conformal
Xn a0 a1x a2 y a3z a4 x2 a5 y 2 a6 z 2 a7 xy a8 yz a9 zx a10 xy2 a11x2 y a12 xz 2 L
ZnC0A3xA4yA1zA7 x2y2z2 2A6yz2A5zx0L
Three Dimensional Coordinates Transformation
Polynomial projective transformation, 15 parameters
X n a1x a2 y a3z a4 d1x d 2 y d 3z 1
RMSE
0.048221 0.130928 0.128114 0.211124
2 0
VTWV nu
RM (X S X ˆE )2 (Y Y ˆ)2 (Z Z ˆ)2/n
• Employed method shown in “Photogrammetric Guide” by Abertz & Kreiling
• X, Y, Z coordinates translated to relative values based in mean coordinates
3D Transformations Testing
X Axis
Z Axis
Applications of 3D Conformal
Coordinate Transformations
•Mobile mapping systems •Relations between different coordinate frames •Sensor frame •Body frame •Mapping frame
to its transpose, or R1RT
• This can be used for inverse relationship
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.
Y n b1x b2 y b3z b4 d1x d 2 y d 3z 1
Z n c1x c2 y c3z c4 d1x d 2 y d 3z 1
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 spaceborne (Kashani, 2019)
Testing – 4 Methods
• Polynomial
– 1st order
– Useful when coordinate systems not uniform in orientation or scale
Kappa ()
Phi ()
(X,Y,Z)
Omega ()
Rotation angles Omega
In general form:
X2 X1Y10Z10 Y2 X10Y1cosZ1sin Z2 X10Y1(sin)Z1cos
In matrix form:
X2 1 0
0 X1
Y2
0
cos
s in Y1
3-D Transformations
Brian Romsek Senior Student Surveying Engineering Department
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.