Screw Displacement-Based Bundle Block
Adjustment Considering Correlation
Qing H. Sheng, Xin Y. Zhang, Hui Xiao, and Hai Y. Guan
Abstract
In a traditional bundle block adjustment model that sepa-
rately determines the position and orientation elements of
each exposure station, inaccurate initial values of exterior
orientation elements (
EOPs
) lead to a strong correlation
relationship between rotation and translation matrices
defined by
EOPs
. To rectify this correlation problem, this
paper introduces a screw displacement based bundle block
adjustment (
SDA
) model, in which an imaging ray’s screw
displacements described by dual quaternion are used to
construct a collinearity equation. By
SDA
, the position and
orientation elements are determined jointly. To evaluate
the robustness and efficiency of
SDA
, a group of progres-
sive experiments were conducted on three types of data-
sets, including synthetic, close-range, and aerial images.
Experimental results demonstrate that
SDA
overcomes the
EOP
correlation problems and achieves reliable positioning
accuracies for large-angle oblique photographic structures.
Introduction
Bundle block adjustment recovers the complete exterior
orientation elements (
EOPs
) of each image, i.e., the (
X
S
,
Y
S
,
Z
S
)
coordinates of the exposure station and the (
ω
,
φ
,
κ
) rotations
and determines the object coordinates of points observed
in multiple overlapping images (Brown, 1976; Bethel
et al
.,
2001; McGlone
et al
., 2004; Hartley and Zisserman, 2004).
Advanced computer technologies and developed photogram-
metric techniques stimulate bundle block adjustment in a
wide range of applications, such as geometric calibration
(Furukawa and Ponce, 2008), digital elevation model (
DEM
)
generation (Kornus
et al
., 2006), and three-dimensional (3D)
photo-model reconstruction (Wong and Chang, 2004).
A collinearity condition equation, which specifies that the
object point, the corresponding image point, and the perspec-
tive center lie on the same line, is the basis of bundle block
adjustment (Habib
et al
., 2002; Zhang
et al
, 2015). Bundle
block adjustment determines the translation and rotation
parameters of the perspective center. Usually, the translation
parameters of the perspective center are represented by in
the form of a vector (Chasles, 1830). Multiple mathemati-
cal models, such as Direction-cosine matrix, Quaternion,
and Rodrigues, describe the
ω
,
φ
,
κ
rotations of the perspec-
tive center. A composition of three
ω
,
φ
,
κ
elementary rota-
tions based on different rotation angle systems describes a
direction-cosine model. The quaternion representation for
rotation is the only unique one showing no singularities (Xu
and Mandic, 2014). Rodrigues’ symmetric parameters (and
its parameter variants) are used in quaternion representa-
tion to resolve the normalization problem of Quaternion
representation. However, these mathematical models describe
only the
ω
,
φ
,
κ
rotations; no translation is involved (Phillips
et al
., 2015). In fact, the positions and orientations of an imag-
ing ray, which vary with time, are used as a whole in aerial or
close-range photogrammetry (Roberts, 1998). Therefore, sepa-
rately determining the position and orientation parameters
of the imaging ray inevitably causes a very high correlation
relationship between them, particularly in large-angle oblique
photographic situations, such as oblique and close-range pho-
togrammetry. In such cases, the significant correlation of the
position and orientation parameters leads to an approximate
linearity between the column vectors in a rotation matrix
(Gerke and Nyaruhuma, 2009). This approximate linearity
is equivalent to multi-collinearity in regression analysis. At
the presence of multicollinearity, optimal estimates by least
squares adjustment might not be obtained. Multicollinearity
even seriously causes morbid normal equations, resulting in
an incorrect determination of
EOPs
.
To remedy the high correlation problems of position and
position parameters, photographic structure was used in the
adjustment process, and factors, such as varying standard
deviation and correlation coefficient, were discussed with
experimental results and charts (Lichti
et al
., 2010; Mass,
2009). Relative control, a certain known geometric relation-
ship of unknown points, was applied in photogrammetric
processing to enhance the intensity of the photogrammetric
net and inspect the photogrammetric accuracy and reliabil-
ity (Feng, 2001). Geometric constraints include topological
constraints (e.g., point, line, and coplanarity constraints) and
object constraints (e.g., parallelism, perpendicularity, and
symmetry) (van den Heuvel, 1998). To restructure a scene
from single to multiple images covering different forms of
geometrical regularity, Grossmann and Santos-Victor (2005)
took advantage of image points of interest, as well as some
types of symmetry and geometrical regularity. Zhang (2012)
adopted near-planarity constraints in bundle block adjust-
ment to control error propagation and decrease parameter
correlation. Wang
et al
., (1995) overcame the high correlation
of
EOPs
using a combined ridge-stein estimator and finally
improved the accuracy, reliability, and stability of the photo-
grammetric task. Heikkinen (2004) refined image parameters
by using a circular image block adjustment model with circu-
larity and coplanarity constraints. Using rotation angles and
circularity conditions, Cao
et al
. (2006) estimated orienta-
tion parameters for self-calibration from turn-table sequence
images. Such constraints increase redundancy, thereby
emphasizing the independency of
EOPs
. However, because
some images to be processed might lack intrinsic photo-
graphic structures with geometric and object constraints,
these methods are not universal.
Qing H. Sheng and Xin Y. Zhang are with the Nanjing
University of Astronautics and Aeronautics, Nanjing
(
).
Hui Xiao is with the Nanjing Xiaozhuang University, Nanjing.
Hai Y. Guan is with the Nanjing University of Information
Science Technology, Nanjing.
Photogrammetric Engineering & Remote Sensing
Vol. 83, No. 11, November 2017, pp. 761–767.
0099-1112/17/761–767
© 2017 American Society for Photogrammetry
and Remote Sensing
doi: 10.14358/PERS.83.10.761
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
November 2017
761
