Block Adjustment with Drift Compensation
for Rational Function Model
Hongbo Pan, Siyuan Zou, and Liyi Guan
Abstract
Bias compensation has been the most widely used sensor
orientation method for high-resolution satellite images (
HRSIs
).
However, the theoretical precision of affine compensation and
drift compensation is still unclear. In this research, a stereo
pair-based block adjustment model for
HRSIs
is developed, in
which the block adjustment can be simplified to orientation
in steps. The impact of the number and arrangement of tie
points on precision is described, and it is proved that when
affine compensation is applied, the cofactors grow exponen-
tially with the number of strips. After eliminating the sample-
related compensated parameters, the theoretical precision
of the drift compensation model becomes independent of
lateral overlap. With four strips of
ZY3
images, the experi-
ment shows that the drift compensation model can achieve
homogeneous accuracy with four ground control points at the
corners of the entire block; nevertheless, the geolocation er-
rors of an affine compensation model differ from strip to strip.
Introduction
Bias compensation is widely used for sensor orientation of
high-resolution satellite images (
HRSIs
), especially when only
the rational function model (
RFM
) is available (Fraser and
Hanley 2003; Fraser
et al.,
2006; Hong
et al.,
2015; Teo, 2011).
Owing to its general form and high replacement accuracy,
the
RFM
is considered the standard sensor model for
HRSIs
.
It is impossible to compensate for the errors of the exterior
orientation parameters (
EOPs
) directly, because the rational
polynomial coefficients lack physical meaning (Di
et al.,
2003;
Xiong and Zhang, 2009). However, the errors of the
EOPs
are
highly correlated to each other and to the errors of the inner
orientation parameters (
IOPs
) (Orun and Natarajan, 1994). Bias
compensation in the image space can correct these errors for
HRSIs
with very narrow field of view (
FoV
) angles, because
errors of
EOPs
and
IOPs
can be modeled in the image space
(Grodecki and Dial, 2003).
Here, the development of bias compensation is reviewed
briefly. The bias compensation method was proposed to refine
the
RFM
for the Ikonos satellite (Fraser and Hanley, 2003; Gro-
decki and Dial, 2003). According to the compensation param-
eters, bias compensation can be divided into a shift model,
drift model, conformal model, affine model, and quadratic
polynomials. The shift compensation model is sufficient for
Ikonos images within a 50-km strip length (Grodecki and Dial,
2003). The drift compensation model is required for Quick-
Bird and ZiYuan-3 images (Noguchi and Fraser, 2004; Pan
et
al.,
2016). The affine model and quadratic polynomials in the
image space are used for Catorsat-1 (Yilmaz
et al.,
2008) and
QuickBird images in some cases (Tong
et al.,
2010). Among
these models, affine compensation is used extensively for
HRSIs
because of its superior accuracy and the limited number
of ground control points required (
GCPs
) (Teo, 2011). However,
most previous work focused on the accuracy of single stereo
pairs. In this case, the number of
GCPs
is not a critical issue,
and there is more concern regarding the function models
(Jeong and Bethel, 2014).
The theoretical precision of affine compensation has rarely
been studied. Lutes and Grodecki (2004) proposed a simpli-
fied azimuth-elevation block adjustment model to analyze the
accuracy of Ikonos blocks. The model was based on the shift
compensation model. With this model, Fraser
, et al.
(2006)
analyzed the impact that the accuracy and number of
GCPs
has
on precision. Puatanachokchai and Mikhail (2008) developed
an eigen approach in which original coefficients are updated
in the adjustment to study the covariance matrix of the
RFM
.
Förstner
et al.
(2013) derived an analytical expression of the
error covariance in adjustable parameters for the replacement
sensor model, with which optimal geopositioning and trian-
gulation are obtained. However, theoretical precision was not
discussed with the adjustable parameters. Topan and Kuto-
glu (2009) analyzed the error propagation based on first and
quadratic polynomials about the object coordinates. However,
there has been no further investigation on the theoretical pre-
cision of the drift and affine compensation models, which play
a prominent role in network design and parameter selection.
GCPs
are still needed to improve geolocation accuracy, even
though direct georeferencing can achieve several pixels of
precision for
HRSIs
. Block adjustment is a key technique for
improving the geometric accuracy for vendors (Massera
et al.,
2012; Yang
et al.,
2017). Because of the small convergence
angle between strips and the approximately parallel rays of
the overlap area in the same strip, the block adjustment of
HR-
SIs
suffers from an ill-conditioned normal matrix. To build the
geometric constraint for scenes in the same strip, during block
adjustment, error compensation is carried out for the entire
strip instead of for each scene (Cao
et al.,
2017; Kim
et al.
2007; Rottensteiner
et al.
2009; Zhang
et al.
2014). However,
when multiple strips are involved in block adjustment, there
are discrepancies in the overlap area of adjacent strips (Zhang
et al.
2015). In some cases, the geopositioning errors differ
from strip to strip (d’Angelo, 2013; Passini and Jacobsen,
2006). As shown in the generation of the Australian geograph-
ic reference images (Ravanbakhsh
et al.,
2012), it is expensive
and time-consuming to settle four to six
GCPs
in each strip of
ALOS PRISM images. To the best of our knowledge, only a
few studies have successfully bridged the entire block with tie
points. Grodecki and Dial (2003) utilized the shift compensa-
tion model to achieve a homogeneous distribution of variance
in planimetry and height throughout the entire block with the
RFM
. Pan (2017) proposed a drift model of the pitch and roll
angle for
ZY-3
after eliminating the yaw angle, with which two
GCPs
in theory could correct the errors of the entire block.
School of Geoscience and Info-Physics, Central South
University, #932 Lunan Lu, Changsha, 410083, P.R. China
(
).
Photogrammetric Engineering & Remote Sensing
Vol. 84, No. 12, December 2018, pp. 791–799.
0099-1112/18/791–799
© 2018 American Society for Photogrammetry
and Remote Sensing
doi: 10.14358/PERS.84.12.791
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
December 2018
791
