Collinearity Equations Kinds of product that can be derived by the collinearity equation - Space Resection By Collinearity - Space Intersection By Collinearity - Interior Orientation - Relative Orientation - Absolute Orientation - Self-Calibration

1. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
Collinearity Equations
1. The collinearity equations are equations that express the condition in which the exposure
station of a photograph, any object point, and it is photo image all lie along a straight line
in three-dimensional space. The equations expressing this condition are called the
Collinearity Condition Equations. They are perhaps the most useful of all equations to the
photogrammetry. The collinearity condition is illustrated in Figure 1, where 𝐿, 𝑎, and 𝐴 lie
along a straight line. Two equationsexpressthe collinearityconditionforanypointona photo:
one equation for the 𝑥 photo coordinate and another for the 𝑦 photo coordinate.
Figure 1. Collinearity condition
The collinearity equations are:
𝑥 𝑎 = 𝑥0 − 𝑓 [
𝑚11( 𝑋𝐴 − 𝑋 𝐿) + 𝑚12( 𝑌𝐴 − 𝑌𝐿) + 𝑚13( 𝑍𝐴 − 𝑍 𝐿)
𝑚31 ( 𝑋𝐴 − 𝑋 𝐿) + 𝑚32( 𝑌𝐴 − 𝑌𝐿) + 𝑚33( 𝑍𝐴 − 𝑍 𝐿 )
]
𝑦 𝑎 = 𝑦0 − 𝑓 [
𝑚21( 𝑋𝐴 − 𝑋 𝐿) + 𝑚22( 𝑌𝐴 − 𝑌𝐿) + 𝑚23 ( 𝑍𝐴 − 𝑍 𝐿 )
𝑚31 ( 𝑋𝐴 − 𝑋 𝐿) + 𝑚32( 𝑌𝐴 − 𝑌𝐿) + 𝑚33( 𝑍𝐴 − 𝑍 𝐿)
]
where [ 𝑥 𝑎 , 𝑦 𝑎] = image coordinates of point a, [ 𝑥0 , 𝑦0 ] = coordinates of the principal point,
𝑓 = camera focal length, [ 𝑋𝐴 , 𝑌𝐴 , 𝑍𝐴]= object coordinates of point A, and 𝑚 𝑖𝑗 = elements of
the rotation matrix M which is 3x3.
In a simplified development of the equations for the two-dimensional projective
transformation, an 𝑋′
𝑌′
𝑍′
coordinate systemis adopted which is parallel to the 𝑋𝑌𝑍 system
and has it is origin at 𝐿. The 𝑥′
𝑦′
𝑧′
coordinatesof anypoint,suchas 𝑝 of Fig.2, can be expressed
in terms of 𝑋′
𝑌′
𝑍′
coordinates as follows:
𝑥 𝑝
′
= 𝑚11 𝑋 𝑝
′
+ 𝑚12 𝑌𝑝
′
+ 𝑚13 𝑍 𝑝
′
𝑦 𝑝
′
= 𝑚21 𝑋 𝑝
′
+ 𝑚22 𝑌𝑝
′
+ 𝑚23 𝑍 𝑝
′
Eq. 1

2. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
𝑧 𝑝
′
= 𝑚31 𝑋 𝑝
′
+ 𝑚32 𝑌𝑝
′
+ 𝑚33 𝑍 𝑝
′
Figure 2. Geometry of two-dimensional projective transformation
Figure 3. Parallel relationships between 𝑋′
𝑌′
𝑍′
and 𝑋𝑌𝑍 planes in two-dimensional
projective transformation
Consider figure 3, which shows the parallel relationships between the 𝑋′
𝑌′
𝑍′
and 𝑋𝑌𝑍
planes after rotation. From similar triangles of figure 2,
𝑋 𝑝
′
𝑋 𝑝 − 𝑋 𝐿
=
−𝑍 𝑝
′
𝑍 𝐿
From which
𝑋 𝑝
′
=
−𝑍 𝑝
′
𝑍 𝐿
(𝑋 𝑝 − 𝑋 𝐿) (a)
Again from similar triangles of figure 2.

3. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
𝑌𝑝
′
𝑌𝑝 − 𝑌𝐿
=
−𝑍 𝑝
′
𝑍 𝐿
From which
𝑌𝑝
′
=
−𝑍 𝑝
′
𝑍 𝐿
(𝑌𝑝 − 𝑌𝐿) (b)
Also, intuitively the following equation may be written:
𝑍 𝑝
′
=
−𝑍 𝑝
′
𝑍 𝐿
(𝑍 𝐿) (c)
Substituting Eqs. (a), (b), and (c) into Eq. 1 gives
𝑥 𝑝
′
= 𝑚11
−𝑍 𝑝
′
𝑍 𝐿
(𝑋 𝑝 − 𝑋 𝐿) + 𝑚12
−𝑍 𝑝
′
𝑍 𝐿
(𝑌𝑝 − 𝑌𝐿 ) + 𝑚13
−𝑍 𝑝
′
𝑍 𝐿
(𝑍 𝐿)
𝑦 𝑝
′
= 𝑚21
−𝑍 𝑝
′
𝑍 𝐿
(𝑋 𝑝 − 𝑋 𝐿) + 𝑚22
−𝑍 𝑝
′
𝑍 𝐿
(𝑌𝑝 − 𝑌𝐿) + 𝑚23
−𝑍 𝑝
′
𝑍 𝐿
(𝑍 𝐿) Eq. 2
𝑧 𝑝
′
= 𝑚31
−𝑍 𝑝
′
𝑍 𝐿
(𝑋 𝑝 − 𝑋 𝐿) + 𝑚32
−𝑍 𝑝
′
𝑍 𝐿
(𝑌𝑝 − 𝑌𝐿) + 𝑚33
−𝑍 𝑝
′
𝑍 𝐿
(𝑍 𝐿)
Factoring Eq. 2 gives
𝑥 𝑝
′
=
−𝑍 𝑝
′
𝑍 𝐿
[ 𝑚11( 𝑋 𝑃 − 𝑋 𝐿) + 𝑚12( 𝑌𝑃 − 𝑌𝐿) + 𝑚13(− 𝑍 𝐿)] (d)
𝑦 𝑝
′
=
−𝑍 𝑝
′
𝑍 𝐿
[ 𝑚21( 𝑋 𝑃 − 𝑋 𝐿) + 𝑚22 ( 𝑌𝑃 − 𝑌𝐿) + 𝑚23(− 𝑍 𝐿)] (e)
𝑧 𝑝
′
=
−𝑍 𝑝
′
𝑍 𝐿
[ 𝑚31 ( 𝑋 𝑃 − 𝑋 𝐿) + 𝑚32( 𝑌𝑃 − 𝑌𝐿 ) + 𝑚33(− 𝑍 𝐿)] (f)
Dividing Eqs. (d) and (e) by Eqs. (f) yields
𝑥 𝑝
′
𝑧 𝑝
′ =
𝑚11( 𝑋 𝑃− 𝑋 𝐿)+ 𝑚12( 𝑌 𝑃− 𝑌 𝐿 )+ 𝑚13(− 𝑍 𝐿)
𝑚31 ( 𝑋 𝑃− 𝑋 𝐿)+ 𝑚32( 𝑌 𝑃− 𝑌 𝐿)+ 𝑚33(− 𝑍 𝐿)
(g)
𝑦 𝑝
′
𝑧 𝑝
′ =
𝑚21( 𝑋 𝑃− 𝑋 𝐿)+ 𝑚22( 𝑌 𝑃− 𝑌 𝐿)+ 𝑚23(− 𝑍 𝐿)
𝑚31 ( 𝑋 𝑃− 𝑋 𝐿)+ 𝑚32( 𝑌 𝑃− 𝑌 𝐿)+ 𝑚33(− 𝑍 𝐿 )
(h)
Referring to Fig. 2, it can be seen that the x'y' coordinates are offset from the fiducial
coordinates xy by x, and 𝑦0. Furthermore, for a photograph, the z coordinates are equal to
-f. The following equations provide for these relationships:
𝑥 𝑝 = 𝑥 𝑝
′
+ 𝑥0 (i)
𝑦 𝑝 = 𝑦 𝑝
′
+ 𝑦0 (j)
𝑧 𝑝 = −𝑓 (k)
Substituting Eqs. (i), (j), and (k) into Eqs. ( g ) and (h) and rearranging gives
𝑥 𝑝 = 𝑥0 − 𝑓[
𝑚11( 𝑋 𝑃 − 𝑋 𝐿) + 𝑚12( 𝑌𝑃 − 𝑌𝐿) + 𝑚13(− 𝑍 𝐿)
𝑚31 ( 𝑋 𝑃 − 𝑋 𝐿) + 𝑚32 ( 𝑌𝑃 − 𝑌𝐿) + 𝑚33(− 𝑍 𝐿)
]

4. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
𝑦 𝑝 = 𝑦0 − 𝑓 [
𝑚21( 𝑋 𝑃 − 𝑋 𝐿) + 𝑚22 ( 𝑌𝑃 − 𝑌𝐿) + 𝑚23(− 𝑍 𝐿)
𝑚31 ( 𝑋 𝑃 − 𝑋 𝐿) + 𝑚32( 𝑌𝑃 − 𝑌𝐿) + 𝑚33(− 𝑍 𝐿)
]
2. Kinds of product that can be derived by the collinearity equation
- Space Resection By Collinearity
- Space Intersection By Collinearity
- Interior Orientation
- Relative Orientation
- Absolute Orientation
- Self-Calibration
3. Detail explanation about the equations (unknowns & knows) and their production
workflow
- Space Resection By Collinearity
Space resection in photogrammetry is the process of determining the six exterior
orientation parameters of a single tilted photo based on photographic measurements
of object (control) points whose XYZ ground coordinates are known. Than the
linearized forms of the space resection collinearity equations for a point A are
𝑏11 𝑑𝜔 + 𝑏12 𝑑𝜃 + 𝑏13 𝑑𝑍𝐴 − 𝑏14 𝑑𝑋 𝐿 − 𝑏15 𝑑𝑌𝐿 − 𝑏16 𝑑𝑍 𝐿 = 𝐽 + 𝑉𝑥𝑎
𝑏21 𝑑𝜔 + 𝑏22 𝑑𝜃 + 𝑏23 𝑑𝑍𝐴 − 𝑏24 𝑑𝑋 𝐿 − 𝑏25 𝑑𝑌𝐿 − 𝑏26 𝑑𝑍 𝐿 = 𝐾 + 𝑉𝑦𝑎
Figure 2. Geometry of space resection using collinearity condition

5. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
Production workflow
The Rotation
Matrix
Initial Approximations
Ground Control
Coordinates
Photo
Coordinates
The Linearized
Observation Equations
Add corrections
Six Elements of Exterior
Orientation
Iteration
Measurements
A near-vertical aerial photograph taken with focal- length camera contains images of
four ground control points 𝐴 through 𝐷. Refined photo coordinates and ground
control coordinates (in a local vertical system). Calculate the exterior orientation
parameters 𝑤, ∅, 𝑘, 𝑋 𝐿, 𝑌𝐿 and 𝑍 𝐿 for this photograph.
Solution:
a. Determine initial approximations
a) Set 𝜔 = 00 and ∅ = 00.
b) Use the method Pythagorean theorem, based on points A and B, to compute
H.
𝐴𝐵 = √( 𝑋 𝐵 − 𝑋𝐴)2 + ( 𝑌𝐵 − 𝑌𝐴 )2
c) Compute ground coordinates from an assumed vertical photo for every point.
𝑋𝐴 = 𝑥 𝑎 (
𝐻 − ℎ 𝐴
𝑓
)
𝑌𝐴 = 𝑦 𝑎 (
𝐻 − ℎ 𝐴
𝑓
)
d) Compute two-dimensional conformal coordinate transformation parameters
by a least squares solution using all control points, and use the results for
assigning initial approximations.
𝑎, 𝑏, 𝑇𝑋, 𝑇𝑌, 𝑘
Where
𝑋𝐴 = 𝑎𝑥 𝑎 − 𝑏𝑦 𝑎 + 𝑇𝑋
𝑌𝐴 = 𝑎𝑥 𝑎 − 𝑏𝑦 𝑎 + 𝑇𝑌
𝑘 = 𝜃 = 𝑡𝑎𝑛−1
(
𝑎
𝑏
)

6. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
b. Form the rotation matrix
𝑀 = [
𝑚11 𝑚12 𝑚13
𝑚21 𝑚22 𝑚23
𝑚31 𝑚32 𝑚33
]
Where:
𝑚11 = cos∅cos 𝑘
𝑚12 = sin ω sin ∅ cos𝑘 + cos ω sin 𝑘
𝑚13 = −cos ω sin∅ cos 𝑘 + sin 𝜔 sin 𝑘
𝑚21 = − cos∅sin 𝑘
𝑚22 = −sin ω sin ∅ sin k + cos𝜔 cos 𝑘
𝑚23 = cos 𝜔 sin ∅sin 𝑘 + sin 𝜔 cos 𝑘
𝑚31 = sin ∅
𝑚32 = −sin ω cos∅
𝑚33 = cos 𝜔 cos∅
c. Form the linearized observation equations. Note that a pair of these equations is
formed for each control point, and thus a total of eight equations results. These
can be represented in matrix form as
𝐵∆= 𝜀 + 𝑉
Where
𝐵 =
[
𝑏11 𝑎
𝑏12 𝑎
𝑏13 𝑎
𝑏21 𝑎
𝑏22 𝑎
𝑏23 𝑎
𝑏11 𝑏
𝑏12 𝑏
𝑏13 𝑏
𝑏21 𝑏
𝑏22 𝑏
𝑏23 𝑏
𝑏11 𝑐
𝑏12 𝑐
𝑏13 𝑐
𝑏21 𝑐
𝑏22 𝑐
𝑏23 𝑐
𝑏11 𝑑
𝑏12 𝑑
𝑏13 𝑑
𝑏21 𝑑
𝑏22 𝑑
𝑏23 𝑑
−𝑏14 𝑎
−𝑏15 𝑎
−𝑏16 𝑎
−𝑏24 𝑎
−𝑏25 𝑎
−𝑏26 𝑎
−𝑏14 𝑏
−𝑏15 𝑏
−𝑏16 𝑏
−𝑏24 𝑏
−𝑏25 𝑏
−𝑏26 𝑏
−𝑏14 𝑐
−𝑏15 𝑐
−𝑏16 𝑐
−𝑏24 𝑐
−𝑏25 𝑐
−𝑏26 𝑐
−𝑏14 𝑑
−𝑏15 𝑑
−𝑏16 𝑑
−𝑏24 𝑑
−𝑏25 𝑑
−𝑏26 𝑑
]
∆=
[
𝑑𝜔
𝑑∅
𝑑𝑘
𝑑𝑋 𝐿
𝑑𝑌𝐿
𝑑𝑍 𝐿]
𝜀 =
[
𝐽𝑎
𝐾𝑎
𝐽𝑏
𝐾𝑏
𝐽𝑐
𝐾𝑐
𝐽 𝑑
𝐾𝑑]
𝑉 =
[
𝑉𝑥 𝑎
𝑉𝑦 𝑎
𝑉𝑥 𝑏
𝑉𝑦 𝑏
𝑉𝑥 𝑐
𝑉𝑦𝑐
𝑉𝑥 𝑑
𝑉𝑦 𝑑
]
Calculation of the coefficients corresponding to point A is shown as an example.
Compute r, s, and q terms for point A.
𝑟 = 𝑚11( 𝑋𝐴 − 𝑋 𝐿) + 𝑚12( 𝑌𝐴 − 𝑌𝐿 ) + 𝑚13( 𝑍𝐴 − 𝑍 𝐿 )
𝑠 = 𝑚21( 𝑋𝐴 − 𝑋 𝐿) + 𝑚22( 𝑌𝐴 − 𝑌𝐿 ) + 𝑚23( 𝑍𝐴 − 𝑍 𝐿)
𝑞 = 𝑚31 ( 𝑋𝐴 − 𝑋 𝐿) + 𝑚32( 𝑌𝐴 − 𝑌𝐿) + 𝑚33 ( 𝑍𝐴 − 𝑍 𝐿)
Compute linearized collinearity equation for point A
𝑏11 =
𝑓
𝑞2
[ 𝑟 (−𝑚33∆𝑌 + 𝑚32∆𝑍) − 𝑞(−𝑚13∆𝑌 + 𝑚12∆𝑍)]
𝑏12 =
𝑓
𝑞2
[ 𝑟 (cos∅ ∆𝑋 + sin 𝜔 sin ∅ ∆Y − cos 𝜔 sin ∅ ∆𝑍) − 𝑞(−sin ∅ cos 𝑘 ∆𝑋 +
sin 𝜔 cos∅cos 𝑘 ∆𝑌− cos 𝜔 cos∅cos 𝑘 ∆𝑍)]
𝑏13 =
−𝑓
𝑞
( 𝑚21∆𝑋 + 𝑚22∆𝑌 + 𝑚23∆𝑍)
𝑏14 =
𝑓
𝑞2
( 𝑟𝑚31 − 𝑞𝑚11)

7. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
𝑏15 =
𝑓
𝑞2
( 𝑟𝑚32 − 𝑞𝑚12)
𝑏16 =
𝑓
𝑞2
( 𝑟𝑚33 − 𝑞𝑚13)
𝑏21 =
𝑓
𝑞2
[ 𝑠 (−𝑚33∆𝑌+ 𝑚32∆𝑍) − 𝑞(−𝑚23∆𝑌 + 𝑚22∆𝑍)]
𝑏22 =
𝑓
𝑞2
[ 𝑠 (cos∅ ∆𝑋 + sin 𝜔 sin∅ ∆Y − cos 𝜔 sin ∅ ∆𝑍) − 𝑞(−sin ∅ cos 𝑘 ∆𝑋 +
sin 𝜔 cos∅cos 𝑘 ∆𝑌− cos 𝜔 cos∅cos 𝑘 ∆𝑍)]
𝑏23 =
𝑓
𝑞
( 𝑚11∆𝑋 + 𝑚12∆𝑌 + 𝑚13∆𝑍)
𝑏24 =
𝑓
𝑞2
( 𝑠𝑚31 − 𝑞𝑚21)
𝑏25 =
𝑓
𝑞2
( 𝑠𝑚32 − 𝑞𝑚22)
𝑏16 =
𝑓
𝑞2
( 𝑠𝑚33 − 𝑞𝑚23)
𝐽 = 𝑥 𝑎 − 𝑥0 + 𝑓
𝑟
𝑞
𝐾 = 𝑦 𝑎 − 𝑦0 + 𝑓
𝑠
𝑞
Remaining coefficients, corresponding to points B, C, and D, are calculated in a similar
manner.
d. Form and solve normal equations, where ∆, 𝐵, and 𝜀 have been substituted for 𝑋,
𝑌, and 𝐿, respectively. The solution is
∆= ( 𝐵 𝑇
𝐵)−1( 𝐵 𝑇
𝜀)
e. Add corrections. (Note: Angle corrections are in radians.)
𝜔 = 00
+ 𝑑𝑤 (
1800
𝜋
)
∅ = 00
+ 𝑑∅ (
1800
𝜋
)
𝑘 = 𝜃 + 𝑑𝑘
𝑋 𝐿 = 𝑇𝑋 + 𝑑𝑋 𝐿
𝑌𝐿 = 𝑇𝑌 + 𝑑𝑌𝐿
𝑍 𝐿 = 𝑇𝑍 + 𝑑𝑍 𝐿
f. Second iteration: Repeat step b, using updated approximations
- Space Intersection By Collinearity
In space intersection, two photos (in a stereo pair) with known exterior orientation
parameters are used to determine the spatial position of any object point in the
overlap area since the two rays to the same object point from the two photos must
intersect at the point.
The procedure is known as space intersection, so called because corresponding rays
to the same object point from the two photo must intersect at the point. To calculate

8. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
the coordinates of points A by space intersection, collinearity equations of the
linearized. Note, however, that since the six elements of exterior orientation are
known, the only remaining unknown in these equations are 𝑑𝑋𝐴, 𝑑𝑌𝐴, 𝑑𝑍𝐴. These are
corrections to be applied to initial approximations for object space coordinates
𝑋𝐴, 𝑌𝐴 , 𝑍𝐴 respectively, for ground point A. The linearized forms of the space
intersection equations for point A are
𝑏14 𝑑𝑋𝐴 + 𝑏15 𝑑𝑌𝐴 + 𝑏16 𝑑𝑍𝐴 = 𝐽 + 𝑉𝑥𝑎
𝑏24 𝑑𝑋𝐴 + 𝑏25 𝑑𝑌𝐴 + 𝑏26 𝑑𝑍𝐴 = 𝐽 + 𝑉𝑦𝑎
Figure 3. Space intersection with a stereopair of aerial photos
- Self-Calibration
Self-calibration is a computational process wherein calibration parameters are
included in the photogrammetric solution, generally in a combined interior-relative-
absolute orientation. 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 common form of the augmented collinearity equations is given as
𝑥 𝑎 = 𝑥0 − 𝑥̅ 𝑎( 𝑘1 𝑟𝑎
2
+ 𝑘2 𝑟𝑎
4
+ 𝑘3 𝑟𝑎
6) − (1 + 𝑝3
2
𝑟𝑎
2)[ 𝑝1(3𝑥̅ 𝑎
2
+ 𝑦̅ 𝑎
2) + 2𝑝2 𝑥̅ 𝑎 𝑦̅ 𝑎] − 𝑓
𝑟
𝑞
𝑦 𝑎 = 𝑦0 − 𝑦̅ 𝑎( 𝑘1 𝑟𝑎
2
+ 𝑘2 𝑟𝑎
4
+ 𝑘3 𝑟𝑎
6) − (1 + 𝑝3
2
𝑟𝑎
2)[2𝑝1 𝑥̅ 𝑎 𝑦̅ 𝑎 + 𝑝2( 𝑥̅ 𝑎
2
+ 3𝑦̅̅̅̅ 𝑎
2)] − 𝑓
𝑠
𝑞
Where
𝑥 𝑎, 𝑦 𝑎 = measured photo coordinates related to fiducials
𝑥0, 𝑦0 = coordinates of the principal point
𝑥̅ 𝑎 = 𝑥 𝑎 − 𝑥0

9. Name: Muhammad Irsyadi Firdaus
Student ID: P66067055
Course: Digital Photogrammetry
𝑦̅ 𝑎 = 𝑦 𝑎 − 𝑦0
𝑟𝑎
6
= 𝑥̅ 𝑎
2
+ 𝑦̅ 𝑎
2
𝑘1, 𝑘2, 𝑘3 = symmetric radial lens distortioncoefficients
𝑝1, 𝑝2, 𝑝3 = decentering distortion coefficients
𝑓 = calibrated focal length
𝑟, 𝑠, 𝑞 = collinearity equation terms