computer graphics - 3-dimensional transformations - applied to surveying
TRANSCRIPT
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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.
YAxis
ZAxis
X Axis
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Applications of 3D Conformal
Coordinate Transformations
Mobile mapping systemsRelations between different coordinate frames
Sensor frameBody frameMapping frame
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Applications of 3D Conformal
Coordinate Transformations
Homeland securityE.G., facial pattern recognitionImage processing
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3D Conformal Coordinate
Transformation
Also known as the 7 Parameters transformation since itinvolves: Three rotation angles omega ( ), phi ( ), and kappa ( );
Three translation parameters (TX, TY,TZ) and
a scale factor, S
X-axisY-a
xis
Z-axis
Omega
( )
Kappa ( )
Phi ( )
(X,Y,Z)
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Rotation angles Omega
In general form:
In matrix form:
More concisely
++=
++=
++=
cosZ)sin(Y0XZ
sinZcosY0XY
0Z0YXX
1112
1112
1112
=
1
1
1
2
2
2
Z
Y
X
cossin0
sincos0
001
Z
Y
X
2C M C
=
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Rotation angles Phi
Omega
( )
Kappa
( )
Phi ( )
In general form:
In matrix form:
More concisely X-axis
Z-ax
is
++=
++=++=
cosZ0YsinXZ
0ZY0XY
)sin(Z0YcosXX
2223
2223
2223
=
2
2
2
3
3
3
Z
Y
X
cos0sin
010
sin0cos
Z
Y
X
23 CMC =
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Rotation angles Kappa
Omega
( )
Kappa
( )
Phi ( )
In general form:
In matrix form:
More concisely X-axis
Z-ax
is
( )
333
333
333
Z0Y0X'Z
0ZcosYsinX'Y
0ZsinYcosX'X
++=
++=
++=
=
3
3
3
Z
Y
X
100
0cossin
0sincos
'Z
'Y
'X
3CM'C =
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Combined Rotation Matrix
If we combine all the rotation matrices
MG becomes, after multiplication
=
=
1
1
1
1
1
1
G
Z
Y
X
MMM
Z
Y
X
M
'Z
'Y
'X
++
=coscoscossinsin
sinsincoscossinsinsinsincoscossincos
cossincossinsincossinsinsincoscoscos
M G
11 12 13
21 22 23
31 32 33
m m m
M M M M m m m
m m m
= =
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COMPUTING ROTATION ANGLES
If rotation matrix
known, rotation
angles can be
computed as shown
on the right
11
21
31
33
32
m
mtan
msin
m
mtan
=
=
=
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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
T
RR =1
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Three-Dimensional Conformal
Coordinate Transformation
Finally the 3D Conformal Transformation is derived
by multiplying the system by a scale factors and adding
the translation factors TX, TY, and TZ.
Where:
YAxis
ZAxis
X Axis
'C s M C T = +
=
Z
Y
X
T
T
T
T '
X
C Y
Z
=
11 12 13
21 22 23
31 32 33
m m mM m m m
m m m
=
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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:
=
1
11
RR
RR
RR
R
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BURSA-WOLF TRANSFORMATION
3D similarity transformation
Observation Equation:
+
+
=
=
Z
Y
X
T
T
T
z
y
x
s
z
y
x
RR
RR
RR
Z
Y
X
h(x)
g(x)
f(x)
0
0
0
fBV
=
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BURSA-WOLF TRANSFORMATION
Coefficient matrix, B:
Vector of parameters, , and
discrepancy vector, f
=
0100
0001
0100
0010
0001
222
111
111
111
nnnxyz
yzx
xyz
xzy
yzx
B
[ ]T
RRRsTTTZYX
=
=
Z
Y
X
f
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Three Dimensional Coordinates
Transformation
General polynomial approach: transformation is not
conformal2 2 2
0 1 2 3 4 5 6 7 8
2 2 29 10 11 12
2 2 2
0 1 2 3 4 5 6 7 8
2 2 29 10 11 12
2 2 2
0 1 2 3 4 5 6 7 8
2 2
9 10 11
Xn a a x a y a z a x a y a z a xy a yz
a zx a xy a x y a xz
Yn b b x b y b z b x b y b z b xy b yz
b zx b xy b x y b xz
Zn c c x c y c z c x c y c z c xy c yz
c zx c xy c x
= + + + + + + + +
+ + + + +
= + + + + + + + +
+ + + + +
= + + + + + + + +
+ + +
L
L
2
12y c xz + +L
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Three Dimensional Coordinates
Transformation
Alternative that is conformal in the three planes
( )
( )
( )
2 2 2
0 1 2 3 5 7 6
2 2 2
0 2 1 4 6 7 5
2 2 20 3 4 1 7 6 5
0 2
2 0 2
2 2 0
Xn A A x A y A z A x y z aA zx A xy
Yn B A x A y A z A x y z A yz A xy
Zn C A x A y A z A x y z A yz A zx
= + + + + + + + +
= + + + + + + + +
= + + + + + + + +
L
L
L
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Three Dimensional Coordinates
Transformation
Polynomial
projectivetransformation, 15
parameters
1 2 3 4
1 2 3
1 2 3 4
1 2 3
1 2 3 4
1 2 3
1
1
1
a x a y a z aXn
d x d y d z
b x b y b z bYn
d x d y d z
c x c y c z cZn
d x d y d z
+ + +=
+ + +
+ + +=
+ + +
+ + +=
+ + +
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Testing 4 Methods
Bursa Wolf Linear model
assume smallrotation angles
Best for satelliteto globalsystemtransformations
Bazlov et al:determined PX90 to WGS 84parameters
GeneralizedBursa Wolf Linear model
errors in both
observations andmodel parameters
Usefultransforming
classical to space-borne (Kashani,2006)
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Testing 4 Methods
Polynomial
1st order
Useful when
coordinatesystems not
uniform in
orientation or
scale
Rubber-sheeting
Expanded Full- Model
Photogrammetric
approach
Angles not consideredsmall
Non-linear: requires a
priori estimate of
parameters
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Employed method shown inPhotogrammetric Guide by Abertz &Kreiling
X, Y, Z coordinates translated torelative values based in meancoordinates
Expanded Full-Model
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Data include a set of know control points, transformed from
WGS84 system to State Plane Coordinates.
3D Transformations Testing3D Transformations Testing
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Test Results
Method
Photo Guide
Bursa-Wolf
Refer
ence
Variance
un
2
0 = WVV
T
nZZYYXXRMSE /)()()( 222 ++=