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.