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Fundamental Theories
in Photogrammetry
Mathematical model of
Correspondence between Imagepoint and Object point
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Coordinate Reference Frames
Image space coordinate system
Object space coordinate system
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Figure 1. Object and image space
coordinate systems
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Sensor Model (Interior Orientation)
Focal length or principal distance (f)
The location of principal point in theimage plane
The description of lens distortion
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00 , yx
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Image coordinate system
T
00 ),(2D yyxx
T00 ),,(3D fyyxx
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Figure 2. Elements defining the image
coordinate system
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Lens distortion
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Radial component of lens distortion
Tangential component
Figure 3. Lens distortion
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The elements of exterior orientation are described byposition of perspective center and thepose of the ray axis
Platform Model (Exterior Orientation)
T
LLL ZYXL ),,(
=
L
L
L
ZZ
YY
XX
kM
f
yy
xx
0
0
Where k: a scale factor
M: rotation matrix
Point correspondence between
image space and object space
(4)
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),,(
Figure 4. Object and image
coordinate systems
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M: Rotation Matrix
The standard approach to constructing M is by
using three sequential rotation:about the X-axis
about the once-rotated Y-axis
about the twice-rotated Z-axis
=
cossin-0sincos0
001
M
=
cos0sin
010
sin-0cos
M
=
100
0cossin-
0sincos
M
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M: Rotation Matrix - continued
MMMM =
+
+
=
coscoscossinsin
sinsincoscossinsinsinsinscossincoscossincos-sinsincossinsinsincoscoscos
coM
=
333231
232221
131211
mmm
mmm
mmm
M
(5)
(6)
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Derivation of Collinearity equations
=
L
L
L
ZZ
YY
XX
mmm
mmm
mmm
k
f
yy
xx
333231
232221
131211
0
0
=
L
L
L
ZZ
YY
XX
kM
f
yy
xx
0
0
[ ]
[ ]
[ ])()()(
)()()(
)()()(
333231
2322210
1312110
LLL
LLL
LLL
ZZmYYmXXmkf
ZZmYYmXXmkyy
ZZmYYmXXmkxx
++=
++=
++=
[ ][ ]
[ ]
[ ])()()(
)()()(
)()()()()()(
333231
2322210
333231
1312110
LLL
LLL
LLL
LLL
ZZmYYmXXm
ZZmYYmXXmfyy
ZZmYYmXXmZZmYYmXXmfxx
++
++=
++
++=
(7-1)
(7-2)
(7-3)
Further processes : dividing (7-1) by (7-3) and dividing (7-2) by (7-3)
(8-1)
(8-2)
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Derivation of Collinearity equations-continued
Likewise, one can come to the following relationship:
[ ][ ]
[ ][ ])()()(
)()()()(
)()()(
)()()()(
33023013
32022012
33023013
31021011
fmyymxxm
fmyymxxmZZYY
fmyymxxm
fmyymxxmZZXX
LL
LL
++
++=
++
++
= (9-1)
(9-2)
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The Interpretation of Collinearity Equations
Mathematically
Whether the balance between unknowns and
number of equations (measurements) would leadto the solution?
Geometrically
?
?
?
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The Interpretation of Collinearity Equations-continued
Resection: Derive exterior orientation via photo
measurements and the known object points. Intersection: Determine the position
(coordinates) of object point by intersecting, atleast, two rays originated from differentperspective centers.
Back-Projection: Predict the photo coordinatesfor a known object point and exterior orientation.
Note: To get better understanding of the configurations,one should be able to distinguish observations from(unknown) parameters in each case.
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From Collinearity equations toCoplanarity equations
=
=
12
12
12
LL
LL
LL
Z
Y
X
ZZ
YY
XX
b
b
b
b
2
0
0
2
2
2
2
2
1
0
0
1
1
1
1
1
=
=
=
=
f
yy
xx
M
w
v
u
a
f
yy
xx
M
w
v
u
a
T
T
coplanarareand, 21 aab 0)( 21 =
aab
Therefore, the determinant of thevector components ends up zero
0
222
111 ==
wvu
wvu
bbb
F
ZYX
(10)
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From Collinearity equations toCoplanarity equations continued
Note that the object point coordinateshave been eliminated from the equations.
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What if Exterior Orientation is not known?
How to solve exterior orientation parameters?;namely how to determine the position and thepose of the perspective center w.r.t. the
assumed object coordinate system.
?
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Solutions of Exterior Orientation
Single Photo (Single Photo Resection)
Stereo model
Block Triangulation
Always to think about the balance between
number of parameters and the number of
equations; most likely one would deal with
redundant observations for increasing accuracy and
reliability purposes.
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Computational concerns Collinearity equations are nonlinear
systems, linearization and iterativecomputation are needed for theconvergent solution.
Due to the linearization, the approximationof parameters need to be provided prior to
the computation. What are the alternatives?
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