PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
March 2014
261
Nested Regression Based Optimal Selection
(NRBOS) of Rational Polynomial Coefficients
Long Tengfei, Jiao Weili, and He Guojin
Abstract
Although the rational function model (
RFM
) is widely applied
in photogrammetry, the application of terrain-dependent
RFM
is limited because of the requirement for numerous ground
control points (
GCPs
) and the strong correlation between the
coefficients. A new method,
NRBOS
, based on nested regres-
sion was proposed to select the optimal
RPCs
automatically
and to gain stable solutions of terrain-dependent
RFM
using
a small amount of
GCPs
. Different types of images, including
QuickBird,
SPOT5
, Landsat-5, and
ALOS
, were involved in
the tests.
NRBOS
method performed better than conventional
methods in estimating
RPCs
, and even provided a reliable
solution when less than 39
GCPs
were used. Additionally,
the test results showed that the simplified
RPCs
are almost
as accurate as the vendor-provided
RPCs
. Consequently,
in favorable situations such as when the orientation pa-
rameters of the satellite are not available or are not suffi-
ciently accurate, the proposed method has the potential
to take the place of the regular terrain-independent
RFM
.
Introduction
The rational function model (
RFM
) with 78 rational polyno-
mial coefficients (
RPCs
) is completely a mathematical model,
which approximately describes the imaging process in photo-
grammetry and remote sensing. Terrain-independent
RFM
con-
stitutes a comprehensive reparameterization of the rigorous
sensor model, and is widely applied in high resolution image
products. Terrain-dependent
RFM
, on the other hand, is hardly
used because of the requirement for numerous observation
data (ground control points (
GCPs
)), and the strong correlation
between the coefficients. Actually, terrain-dependent
RFM
provides a useful approach to rectify remotely sensed images
without knowing the position and orientation information
of specific sensor. However, as 78
RPCs
of the
RFM
are strong-
ly correlated, stable, and precise solutions of the
RPCs
are
difficult or even impossible to achieve (Lin and Yuan, 2008).
The objectives of this research are to find a robust approach to
estimate
RPCs
and to make use of the terrain-dependent
RFM
.
Many studies have been carried out on the topic of
RFM
during the recent decades.
OGC
(1999) has normalized the
range of the image and object space coordinates of
RFM
to −1
to +1, and effectively enhanced the condition number of the
normal equation matrix. Tao and Hu (2000) have studied the
RFM
comprehensively and have proposed to strengthen the
solution of
RFM
using Tikhonov regularization and the L-curve
method. Yuan and Lin (2008) have compared the results of
several methods for solving
RPCs
including ridge trace meth-
od, L-curve method, empirical formula method, and general-
ized ridge estimate method; they have verified the validity of
L-curve method. In addition, Levenberg-Marquardt method
(Tao and Hu, 2001) and singular value decomposition method
have been applied to solve
RPCs
(Fraser
et al
., 2006). Among
all these methods, the ridge estimation method (especially the
L-curve method) is the most widely used approach. However,
there are still some problems in solving
RPCs
using the exist-
ing methods. For example, ridge estimate is a biased estimate,
which requires numerous
GCPs
to solve
RPCs
.
Solving
RPCs
is a problem of multiple regression analy-
sis. The problem of multicollinearity results in an ill-posed
normal equation, and the ordinary least squares (
OLS
) estima-
tion does badly in achieving a stable and reliable solution.
In order to solve the problem of multicollinearity, “variable
selection (Draper
et al
., 1966)” and “ridge estimation (Hoerl
and Kennard, 1970)” are usually used to improve the
OLS
.
Variable selection can simplify the original model by selecting
a subset of variables from the original set of variables which
give the most significant response to the regression, and there-
fore the multicollinearity of the model can be reduced after
the variable selection (Guyon and Elisseeff, 2003). Both of
the two methods have drawbacks. Variable selection provides
interpretable model but can be extremely variable because it
is a discrete process. Ridge regression is a continuous process
that shrinks coefficients and hence is more stable. However,
it does not set any coefficients to 0 and hence does not give
an easily interpretable model (Tibshirani, 1996). In addition
to the methods mentioned above, in statistics, some improve-
ments of ridge estimation method (Bashtian
et al
., 2011;
Jurczyk, 2012; Kibria and Saleh, 2012; Park and Yoon, 2011)
have been proposed in recent years. However, they still need
a large number of observation data.
Variable selection is a non-deterministic polynomial
(
NP
) problem whose search space is very large (Amaldi and
Kann, 1998). During the past decades, hundreds of methods
have been proposed to solve this problem, including genetic
prediction, decision tree prediction, Bayes prediction, least
square prediction, and support vector machine prediction
(Guyon and Elisseeff, 2003), and all these methods need a
huge
computation. Greedy search strategy can greatly reduce
the search space and thus improve the efficiency of the algo-
rithm. The greedy search methods are mainly divided into
three categories: forward selection, backward elimination, and
stepwise regression analysis. Nonetheless, there are common
disadvantages in these methods. Not all the possible combi-
nations of variables are taken into account, and the results
which greatly depend on the evaluation criterion are usually
not the optimal solution. In addition, the variable transfor-
mation, such as principal component analysis, partial least
The Institute of Remote Sensing and Digital Earth (RADI),
Chinese Academy of Sciences, No.9 Dengzhuang South Road,
Haidian District, Beijing 100094, China (
.
Photogrammetric Engineering & Remote Sensing
Vol. 80, No. 3, March 2014, pp. 261–269.
0099-1112/14/8003–261
© 2014 American Society for Photogrammetry
and Remote Sensing
doi: 10.14358/PERS.80.3.261
