Camera calibration can be conducted through bundle adjustment with additional parameters

1. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
Camera calibration can be conducted through bundle adjustment with additional
parameters
1. Why camera calibration is necessary for UAV photogrammetry and how to achieve
high accuracy and high reliability results?
The purpose of camera calibration is to mathematically describe the internal
geometry of the imaging system, particularly after a light ray passes through the
camera’s perspective center. Those internal geometries are determined by a self-
calibrating bundle adjustment method with some additional parameters. It would
automatically measure the image coordinates of the coded targets on our camera
calibration field. We can conclude that the purposes of Camera Calibration are:
a) Modelling the internal geometry within a camera
b) Correct the systematic displacement to fulfill the property of collinearity
 Principal point coordinates (x0, y0)
This is specified by coordinates of the principal point given with respect to the
x and y coordinates of the fiducial marks
 Focal length (f)
This is the focal length that produces an overall mean distribution of lens
distortion.
 Radial lens distortion (K1, K2, K3)
This is the symmetric component of distortion that occurs along radial lines
from the principal point. Although the amount may be negligible, this type of
distortion is theoretically always present even if the lens system is perfectly
manufactured to design specifications. Radial lens distortion shown as 3rd,
5th, 7th, 9th, and 11th-order term and their symbols are K1, K2, K3, K4, and
K5 respectively. The value of K4 and K5 are both zero. It means, only the K1
K2 and K3 that affect the distortion and K4 and K5 are not useful.
Figure 1. Effect of radial distortion (a) barrel; (b) pincushion distortion
 Decentering lens distortion (P1, P2)
This is the lens distortion that remains after compensation for symmetric
radial lens distortion. Decentering distortion can be further broken down into
asymmetric radial and tangential lens distortion components. These
distortions are caused by imperfections in the manufacture and alignment of
the lens system

2. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
Figure 2. Tangential distortion
In order to achieve a high accuracy and high reliability results, we can perform a
significance test and stability test. The significance and stability test were conducted
to check the reliability of their additional parameters. So that we would be able to
know, which additional parameters are necessary to be included in the process of
camera calibration. Thus, our results will achieve a higher accuracy and moreover the
results would be reliable.
2. What are its mathematic equations, observations, unknowns, etc.?
Those internal geometries are determined by a self-calibrating bundle adjustment
method with some additional parameters. The process uses collinearity equations
that have been augmented with additional terms to account for adjustment of the
calibrated focal length, principal-point offsets, and symmetric radial and decentering
lens distortion. The mathematics equation would be
𝑥 𝑎 = 𝑥0 − 𝑥̅ 𝑎(𝑘1 𝑟𝑎
2
+ 𝑘2 𝑟𝑎
4
+ 𝑘3 𝑟𝑎
6) − (1 + 𝑝3
2
𝑟𝑎
2)[𝑝1(3𝑥̅ 𝑎
2
+ 𝑦̅𝑎
2) + 2𝑝2 𝑥̅ 𝑎 𝑦̅𝑎] − 𝑓
𝑟
𝑞
𝑦𝑎 = 𝑦0 − 𝑦̅𝑎(𝑘1 𝑟𝑎
2
+ 𝑘2 𝑟𝑎
4
+ 𝑘3 𝑟𝑎
6) − (1 + 𝑝3
2
𝑟𝑎
2)[2𝑝1 𝑥̅ 𝑎 𝑦̅𝑎 + 𝑝2(𝑥̅ 𝑎
2
+ 3𝑦̅𝑎
2)] − 𝑓
𝑠
𝑞
Where:
𝑥 𝑎, 𝑦𝑎 = 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑝ℎ𝑜𝑡𝑜 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑟𝑒𝑙𝑎𝑡𝑒𝑑 𝑡𝑜 𝑓𝑖𝑑𝑢𝑐𝑖𝑎𝑙𝑠
𝑥0 , 𝑦0 = 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑝𝑜𝑖𝑛𝑡
𝑥̅ 𝑎 = 𝑥 𝑎 − 𝑥0
𝑦̅𝑎 = 𝑦𝑎 − 𝑦0
𝑟𝑎
2
= 𝑥̅ 𝑎
2
+ 𝑦̅𝑎
2
𝑓 = 𝑐𝑎𝑙𝑖𝑏𝑟𝑎𝑡𝑒𝑑 𝑓𝑜𝑐𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ
𝑘1, 𝑘2, 𝑘3 = 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑟𝑎𝑑𝑖𝑎𝑙 𝑙𝑒𝑛𝑠 𝑑𝑖𝑠𝑡𝑜𝑟𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠
𝑝1, 𝑝2, 𝑝3 = 𝑑𝑒𝑐𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔 𝑑𝑖𝑠𝑡𝑜𝑟𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠
𝑟, 𝑠, 𝑞 = 𝑐𝑜𝑙𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑡𝑒𝑟𝑚𝑠
The unknown parameters are EOPs, IOPs, Terrain points, Image coordinates,
Additional parameters. These equations are of course nonlinear, and therefore
Taylor's series is used to linearize them, and an iterative solution is made.

3. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
3. Which parameters (unknowns) have high correlation and how to decouple their
correlation?
With the inclusion of the extra unknowns. it follows that additional independent
equations will be needed to obtain a solution. Also, the numerical stability of
analytical self-calibration is of serious concern. Maybe including the additional
parameters does not guarantee their solution. It is necessary to have special
constraints and geometric configurations to ensure their solution. For example, with
nominally vertical photography if the object points are at roughly the same elevation,
then x0, y0, and f are strongly correlated with XL, YL, and ZL, respectively. Given this
correlation, the solution may not produce satisfactory results. This problem can be
overcome or at least alleviated if there is significant elevation variation in the object
field, or by using highly convergent (non-vertical) photography, by making
observations of the camera position or by using redundant photographic coverage at
different κ angles (Wolf and Dewitt, 2000). In order to recover the lens distortion
parameters, we need to have many redundant object points whose images are well
distributed across the format of the photos. Thus, the coupling between IO and EO
can be reduced.