Space resection is commonly used to determine the exterior orientation parameters (which refers to position and orientation related to an exterior coordinate system) associated with one or more photos based on measurements of ground control points (GCPs). space resection is a nonlinear problem, existing methods involve linearization of the collinearity condition and the use of an iterative process to determine the final solution using the least-squares method. The process also requires initial approximate values of the unknown parameters, some of which must be estimated by another least-squares solution.

1. BENHA UNIVERSITY
FACULTY OF ENGINEERING, SHOUBRA
GEOMATICS DEPARTMENT
REPORT ON
SPACE RESECTION
BY COLLINEARITY EQUATIONS
BY
AHMED YASSER AHMED MOHAMED NASSAR
ID: 20 SECTION: 1

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ABSTRACT
Space resection is commonly used to determine the
exterior orientation parameters (which refers to position
and orientation related to an exterior coordinate system)
associated with one or more photos based on
measurements of ground control points (GCPs). space
resection is a nonlinear problem, existing methods involve
linearization of the collinearity condition and the use of an
iterative process to determine the final solution using the
least-squares method. The process also requires initial
approximate values of the unknown parameters, some of
which must be estimated by another least-squares solution.

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SPACE RESECTION
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 X,Y,Z ground coordinates are
known . The parameters include the three coordinates of
the exposure station and the three elements of angular
rotation. Three object points are normally enough to solve
the problem, but with more points redundancy occurs and
the least-squares method is used.
COLLINEARITY CONDITIONS
The collinearity condition states that for any given photo
the exposure station, any object point, and the
corresponding image point should lie on a straight line. For
space resection with collinearity, the coordinates of a
minimum of three object points and the corresponding
image coordinates are required. The problem is to find the
six exterior orientation parameters: the coordinates of
exposure station (XL, YL, ZL) and the angular orientation
elements (ω, Φ, κ).

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SPACE RESECTION BY COLLINEARITY
Two vectors are collinear if one is a scalar multiple of the
other.
where k is a scale factor. The components of â in the image
space coordinate system, and  in the object space system
are:
Figure 1. Geometry of space resection using collinearity condition

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The two vectors are now referred to a common coordinate
system by transforming  from the object space system to
the image space with the help of the orientation matrix M.
By expanding M, performing the multiplication and
dropping subscripts. â and  results in the equation:
The scalar k, which is different for each ray in a bundle, can
be eliminated by dividing the first two equations with the
third one. The final form equation is obtained as:

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Let the final form equation re-written as:
where
By Linearizing the final form equation using Taylor
theorem and an iterative method is used to determine the
final solution.
The linearized equations in terms of the omega-phi-
kappa system (ω, Φ, κ) in simplified form are given by:

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PHOTOGRAMMETRIC APPLICATIONS OF THE SPACE
RESECTION
Some of the applications of space resection are:
• Fixing ground coordinates by intersection from single
photo after solving the space resection problem.
• Photo Triangulation, using multiple photos.
• Camera calibration.
• Head-Mounted Tracking System Positioning.
• Object Recognition.
• Ortho photo rectification.
OTHER APPLICATIONS OF THE SPACE RESECTION
Some of the applications in other fields are:
• Machine Construction, Metalworking, Quality Control.
• Mining Engineering.
• Objects in Motion.
• Shipbuilding.
• Structures and Buildings.
• Traffic Engineering.
• Biostereometrics.

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References
Derenyi E.E., (1996). Photogrammetry: The Concepts. Department of Geodesy and Geomatics
Engineering, University of New Brunswick, Fredericton, New Brunswick, Canada.
Elnima, Elgasim. (2013). A solution for exterior and relative orientation in photogrammetry, a
genetic evolution approach. Journal of King Saud University - Engineering Sciences. 27.
10.1016/j.jksues.2013.05.004.
Easa, Said. (2010). Space resection in photogrammetry using collinearity condition without
linearisation. Survey Review. 42. 40-49. 10.1179/003962609X451681.
Wolf, P. R., Dewitt, B. A., & Wilkinson, B. E. (2014). Elements of Photogrammetry with Application
in GIS. New York: McGraw-Hill.