togrammetry 
f Surveying 
cal Recti- 
Collinea- 
University 
COMPARISON OF PRECISION AND RELIABILITY OF POINT COORDINATES 
USING DLT AND BUNDLE APPROACH 
By: M.Sc. Lars-Äke Edgardh 
Department of Photogrammetry 
The Royal Institute of Technology 
S-10044 Stockholm 
SWEDEN 
Commission number: III 
ABSTRACT 
The direct linear transform is a popular alternative to 
the Bundle adjustment method as it does not require 
calibrated cameras and the transform parameters can 
be computed directly from a linear function. However, 
the method has some drawbacks in precision and 
reliability compared to Bundle adjustment. 
This paper reports a comparative evaluation of the 
two methods. The methods are evaluated using a 
theoretical test where control points, check points and 
control points have been simulated. Precision of the 
two methods have been calculated using data from 
control point calculations. Internal reliability have 
been estimated using data from the calculation of 
parameters. 
DLT has been evaluated using both an iterative and a 
linear approach of the parameter calculations. Bundle 
adjustment has been evaluated in two ways: internal 
and external orientation parameters as unknowns, 
and only external orientation parameters as unknown. 
The evaluation shows that Bundle adjustment gives a 
better precision and internal reliability compared to 
DLT when 6 control points are used. When the 
number of control points is increased, the difference 
decreases in both precision and internal reliability. 
Key words: Photogrammetry, non-metric, accuracy 
1. INTRODUCTION 
Bundle adjustment and the Direct Linear Transform 
(DLT) are the most commonly used methods when 
calculating point coordinates from image data. This 
paper reports a theoretical study of the two methods 
focused on precision and estimation of internal relia- 
bility. 
Bundle adjustment is based on the collinearity equa- 
tions, where the physical reality is modelled in a 
straightforward way. The collinearity equations form a 
perspective transform which mathematically describes 
that the object point, the perspective centre of the 
camera and the measured image point ideally lie on a 
straight line. The transform includes parameters for 
interior and exterior orientation. Exterior orientation 
parameters determine the position of the camera in 
terms of position coordinates and rotation of the 
image relative to an object space coordinate system. 
35 
DLT is a projective transform, where the transform 
parameters are not directly interpretable in terms of 
interior or exterior orientation. One of the advantages 
of DLT is that the parameters can be calculated without 
any initial approximations. This has made the method 
popular to use in cases where only non-metric came- 
ras are available, i.e. cameras without calibration or 
without fiducial marks. 
11 Bundle adjustment 
The collinearity equations are defined by: 
Tx 
X= X = CJ (1) 
T 
y-yp"^ ONE (2) 
and 
Ti 5 fy(X-X) * 14(Y- Y + 797 (2 - 2e) (3) 
+ 
I 
y = 112 (X-Xc) + 122 (Y- Ye) + 132 (Z- Ze) (4) 
N -2 rj (X- X) + 135 (Y-Yo) + 133 (Z- Ze) (5) 
where x and y are the measured image coordinates in 
the comparator coordinate system and X, Y, Z are the 
object coordinates in an object space coordinate system. 
The collinearity equation contains in total nine un- 
known parameters where: X,, Y,, and Z, are the coor- 
dinates of the perspective centre in the object space 
coordinate system; x, and y, are the coordinates of the 
principal point in the image system; c is the principal 
distance of the camera; r4,,..,r33 are elements of the 
rotation matrix describing the rotation of the film 
plane into the object space. The nine rotation matrix 
elements are functions of three independent angles ©, 
® and K, which describe the rotations around the X, Y, 
and Z axes. 
The collinearity equations are non-linear and must be 
linearized. Initial approximations are needed for both 
the resection problem, i.e. estimation of inner and 
outer orientation elements, and the intersection pro- 
blem, i. e. calculation of object coordinates of new 
points. 
12 . DIT 
In DLT, comparator coordinates are expressed in terms 
of object coordinates and eleven transform para- 
meters.