1. Definitions of the image coordinate system and rotation angles

2. Definitions of the image coordinate system and rotation angles
Exterior orientation parameters define the position and orientation of the camera that captured an image.
The positional elements of exterior orientation include Xo, Yo, and Zo. They define the position of perspective centre (O) with respect to ground space coordinate system (X, Y, and Z).
The angular or rotational elements of exterior orientation describe the relationship between the ground space coordinate system (X, Y, and Z) and the image space coordinate system (x, y, and z).

3. the image space coordinate system (x, y, and z)
the relationship between the ground space coordinate system (X, Y, and Z) and
the image space coordinate system (x, y, and z)

4. Definitions of the image coordinate system and rotation angles
Three rotation angles are commonly used to define angular orientation. They are kappa (κ), omega (ω), and phi (ϕ).
The three rotation angles kappa (κ), omega (ω), and phi (ϕ)

5. Definitions of the image coordinate system and rotation angles
Using the three rotation angles, the relationship between the image space coordinate system (B) and ground space coordinate system (E) can be determined. The relationship between the two systems is defined by 3 × 3 matrix, which is referred to as the orientation or rotation matrix (CEB).
the rotation matrix is derived by applying a sequential primary rotation of kappa about the z-axis followed by a secondary rotation omega about the x-axis, and a tertiary rotation phi about the y-axis. This order for these rotation angles is selected to have a rotation matrix similar to Leica LPS and ORIMA software.

6. Definitions of the image coordinate system and rotation angles
The order of the three rotation angles to create the rotation matrix (CEB)

7. Definitions of the image coordinate system and rotation angles
Consider a point P(x,y) in x-y coordinate system rotated by anα angle relative to the X-Y coordinates system. We require the coordinate X and Y of the Point P in this second system:

9. Definitions of the image coordinate system and rotation angles

11. Definitions of the image coordinate system and rotation angles
By combining the three rotations, a relationship can be defined between the object coordinates system (E), and the image coordinate system (B):

14. Mathematical relation between image and ground coordinates

15. Definitions of the image coordinate system and rotation angles

29. Collinearity Condition Equations
Let:
Coordinates of exposure station be XL, YL, ZL wrt object (ground) coordinate system XYZ
Coordinates of object point A be XA, YA, ZA wrt ground coordinate system XYZ
Coordinates of image point a of object point A be xa, ya, za wrt xy photo coordinate system (of which the principal point o is the origin; correction compensation for it is applied later)
Coordinates of image point a be xa’, ya’, za’ in a rotated image plane x’y’z’ which is parallel to the object coordinate system
Transformation of (xa’, ya’, za’) to (xa, ya, za) is accomplished using rotation equations, which we derive next.
Appendix D
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30. Rotation Equations Omega rotation about x’ axis:
New coordinates (x1,y1,z1) of a point (x’,y’,z’) after rotation of the original coordinate reference frame about the x axis by angle ω are given by:
x1 = x’
y1 = y’ cos ω + z’ sin ω
z1 = -y’sin ω + z’ cos ω
Similarly, we obtain equations for phi rotation about y axis:
x2 = -z1sin Ф + x1 cos Ф
y2 = y1
z2 = z1 cos Ф + x1 sin Ф
And equations for kappa rotation about z axis:
x = x2 cos қ + y2 sin қ
y = -x2 sin қ + y2 cos қ
z = z2
Appendix C
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31. Final Rotation Equations
We substitute the equations at each stage to get the following:
x = m11 x’ + m12 y’ + m13 z’
y = m21 x’ + m22 y’ + m23 z’
z = m31 x’ + m32 y’ + m33 z’
Where m’s are function of rotation angles ω,Ф and қ
In matrix form: X = M X’
where
Are we not rotating x,y,z to the prime x,y and z?
Properties of rotation matrix M:
Sum of squares of the 3 direction cosines (elements of M) in any row or column is unity.
M is orthogonal, i.e. M-1 = MT
Appendix C
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32. Coming back to the collinearity condition…
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33. Collinearity Equations
Using property of similar triangles:
Substitute this into rotation formula:
Now,
factor out za’/(ZA-ZL), divide xa, ya by za
add corrections for offset of principal point (xo,yo)
and equate za=-f, to get:
Appendix D
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Virtual Environment Lab, UTA

34. Review of Collinearity Equations
are nonlinear and
involve 9 unknowns:
omega, phi, kappa inherent in the m’s
Object coordinates (XA, YA, ZA )
Exposure station coordinates (XL, YL, ZL )
Where,
xa, ya are the photo coordinates of image point a
XA, YA, ZA are object space coordinates of object/ground point A
XL, YL, ZL are object space coordinates of exposure station location
f is the camera focal length
xo, yo are the offsets of the principal point coordinates
m’s are functions of rotation angles omega, phi, kappa (as derived earlier)
Ch. 11 & App D
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Virtual Environment Lab, UTA

35. Virtual Environment Lab, UTA
Now that we know about the collinearity condition, lets see where we need to apply it.
First, we need to know what it is that we need to find…
6/26/2019
Virtual Environment Lab, UTA

36. Elements of Exterior Orientation
As already mentioned, the collinearity conditions involve 9 unknowns:
Exposure station attitude (omega, phi, kappa),
Exposure station coordinates (XL, YL, ZL ), and
Object point coordinates (XA, YA, ZA).
Of these, we first need to compute the position and attitude of the exposure station, also known as the elements of exterior orientation.
Thus the 6 elements of exterior orientation are:
spatial position (XL, YL, ZL) of the camera and
angular orientation (omega, phi, kappa) of the camera
All methods to determine elements of exterior orientation of a single tilted photograph, require:
photographic images of at least three control points whose X, Y and Z ground coordinates are known, and
calibrated focal length of the camera.
Chapter 10
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Virtual Environment Lab, UTA

37. Elements of Interior Orientation
As an aside, from earlier discussion:
Elements of Interior Orientation
Elements of interior orientation which can be determined through camera calibration are as follows:
Calibrated focal length (CFL), the focal length that produces an overall mean distribution of lens distortion. Better termed calibrated principal distance since it represents the distance from the rear nodal point of the lens to the principal point of the photograph, which is set as close to optical focal length of the lens as possible.
Principal point location, specified by coordinates of a principal point given wrt x and y coordinates of the fiducial marks.
Fiducial mark coordinates: x and y coordinates of the fiducial marks which provide the 2D positional reference for the principal point as well as images on the photograph.
Symmetric radial lens distortion, the symmetric component of distortion that occurs along radial lines from the principal point. Although negligible, theoretically always present.
Decentering lens distortion, distortion that remains after compensating for symmetric radial lens distortion. Components: asymmetric radial and tangential lens distortion.
Chapter 3
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38. The mathematical model and the collinearity condition
The relationship between the image point measurements and the corresponding ground point coordinates is described by the primary observation equation:

39. The following two non-linear observation equations which referred to as the collinearity equations are resulted from the multiplication of equation
Are elements of 3-D rotation matrix which form the three rotations

42. Exterior Orientation Spatial position (x, y, z) Angular Orientation
Space resection by collinearity is commonly used to determine the six elements.
216

43. Collinearity Condition Equations
The exposure station, an object point, and its photo image all lie on a straight line.
They are perhaps the most useful of all equations to the photogrammetrist.

44. m’s are functions of three rotation angles, --omega, phi, and kappa.

45. Rotation in Terms of Omega, Phi and Kappa
Measurement xyz and rotated x’y’z’ image coordinate systems.

46. Image Coordinate System Rotated

47. The Rotation Formulas
(1)

48. Derive Collinearity Condition Equations
(2)

49. Derive Collinearity Condition Equations
(3-1)
(3-2)
(3-3) -f
(3-1) divided by (3-3)
(3-2) divided by (3-3)

51. Block adjustment with added parameters (Self-Calibration)

52. Block adjustment with added parameters (Self-Calibration)

53. No calibration model Camera Name/GCP/CP
Ground control points RMSE (m) of residuals
Ground check points RMSE (m) of residuals
Image coordinates RMSE (μm) of residuals σo µm X Y Z x y
Vexcel Imaging UltraCamX/9/99
0.052
0.061
0.146
0.075
0.083
0.163
0.79
0.85
1.00
Z/I Imaging DMC/9/95
0.031
0.040
0.178
0.045
0.078
0.184
1.45
1.28
1.90
Z/I Imaging RMK Top 15 film/14/82
0.093
0.130
0.112
0.116
0.154
4.75
4.29
5.90

54. Self-Calibration model
Camera
Name/GCP/CP
Ground control points RMSE (m) of residuals
Ground check points RMSE (m) of residuals
Image coordinates RMSE (μm) of residuals σo µm X Y Z x y
Vexcel Imaging UltraCamX/9/99
0.054
0.059
0.033
0.075
0.082
0.074
0.78
0.82
1.00
Z/I Imaging DMC/9/95
0.029
0.031
0.042
0.056
0.071
1.32
1.14
1.60
Z/I Imaging RMK Top 15 film/14/82
0.114
0.125
0.080
4.86
4.49
5.20

55. Example on the Collinearity Equations

56. Example on the Collinearity Equations

58. Example on the Collinearity Equations

59. Aerial Triangulation
Aerial Triangulation may be defined as the establishment of horizontal and vertical control points by photogrammetric techniques based only on a few ground control points.
Traditionally, it can also be used to calculate the orientation parameters of the photographs.
These orientation parameters are the exposure station coordinates (X0, Y0 and Y0 ) and three rotations(, , and ).
The main advantage of aerial triangulation is that, even within large photographic blocks, it only uses a relatively small framework of surveyed ground points.

60. Data set UltraCam D digital camera Traditional flight plan
60% forward overlap
20% lateral overlap
18 images in full block
Flying height 1500m

61. Data set UltraCam D digital camera Traditional flight plan
60% forward overlap
20% lateral overlap
60 images in full block
Flying height 880m

62. Block Triangulation
For economic reasons and in order to provide an effective way for processing multiple images, block triangulation has been developed.
Without triangulation, expensive ground survey is required, since every stereo model would need two horizontal and three vertical ground control points.
So the block triangulation can be defined as the process of establishment of the mathematical relationship between the block of images and the ground.

63. Block Triangulation
For performing block triangulation, several methods were developed, such as block adjustment. At the present time, most of the block triangulation is performed by using the bundle adjustment.
In addition, the bundle adjustment allows the integration of additional geometric or navigation information, such as GPS and IMU measurements.
The bundle adjustment process is so named because of the many light rays that pass through each lens position.

64. Bundle adjustment
Once the block triangulation has been solved, the following data can be obtained in the bundle adjustment solution:
The X, Y, Z object space coordinates of all object points.
The exterior orientation parameters; the exposure station coordinates and three rotations (, , , X0, Y0 and Z0) of all photographs.
The additional parameters or the interior orientation parameters that describe the camera geometry.
The statistical reports for the accuracy and reliability of the data.

65. The observations data in the bundle adjustment
The measurements (observed quantities) associated with a bundle adjustment are:
Image coordinates: The coordinate system used to define these points is the fiducial coordinates system . This system is defined by the so-called fiducial marks on the photograph, whose coordinates are given in the camera calibration certificate. 
Ground point observations. They split into two groups, the tie points and the control points.  
Direct observations of the exterior orientation parameters (, , , X0, Y0,and Z0) of the photographs.

66. The observations data in the bundle adjustment
The photo coordinates, is the fundamental photogrammetric measurements made with digital photogrammetric workstation. In the bundle adjustment, these observations need to be weighted according to the accuracy and precision with which they were measured.
The ground control points determined through field survey. Although ground control coordinates are indirectly determined quantities, they can be included as observations provided that proper weights are assigned.
The final set of observations, exterior orientation parameters, has recently become important in bundle adjustments with the use of airborne GPS control as well as Inertial Measurement Unit (IMU).

67. The mathematical model and the collinearity condition
The relationship between the image point measurements and the corresponding ground point coordinates is described by the primary observation equation:

68. The following two non-linear observation equations which referred to as the collinearity equations are resulted from the multiplication of equation
Are elements of 3-D rotation matrix which form the three rotations

69. The least squares observation equations are formed from the collinearity equations, and must be linearised using Taylor’s theory.

70. Analysing the quality of results
The results should be evaluated to be sure that they meet the project specifications and requirements, and that the results not contained bad measurements or assumptions.
The statistical analysis techniques are used to find the quality of the adjustment and to see if there are any bad observations.
This operation applies various statistical techniques and provides an indication of the internal and external geometry of the block.
The internal geometry includes the Root Mean Square Error (RMSE) for the ground control points, tie points and image residuals. Further, the RMSE value of the check points (CP) represents the external geometry of the block.
Sigma0 value provides an indication of the validity of the weights used to compute the aerial triangulation, based on the standard deviation of observation data in the bundle adjustment.