A Stepwise-then-Orthogonal Regression (STOR)
with quality control for Optimizing the RFM of
High-Resolution Satellite Imagery
Chang Li, Xiaojuan Liu, Yongjun Zhang, and Zuxun Zhang
Abstract
There are two major problems in Rational Function Model
(
RFM
) solution: (a) Data source error, including gross error,
random error, and systematic error; and (b) Model error,
including over-parameterization and over-correction issues
caused by unnecessary
RFM
parameters and exaggeration of
random error in constant term of error-in-variables (EIV) mod-
el, respectively. In order to solve two major problems simul-
taneously, we propose a new approach named stepwise-then-
orthogonal regression (
STOR
) with quality control. First,
RFM
parameters are selected by stepwise regression with gross error
detection. Second, the revised orthogonal distance regression
is utilized to adjust random error and address the overcorrec-
tion problem. Third, systematic error is compensated by Fou-
rier series. The performance of conventional strategies and
the proposed
STOR
are evaluated by control and check grids
generated from
SPOT5
high-resolution imagery. Compared with
the least squares regression, partial least squares regression,
ridge regression, and stepwise regression, the proposed
STOR
shows a significant improvement in accuracy.
Introduction
A satellite sensor model, which contributes to the precise
georeferenced and geopositioning (Jeong
et al.
, 2015; Li
et al.
,
2014; Tong
et al.
, 2010),
DEM
generation (Qayyum
et al.
, 2015),
and image matching (Zhang
et al.
, 2006), describes a meaning-
ful relationship between the object space coordinates and the
corresponding image coordinates. The broadly used geometric
models can be roughly divided into the rigorous physical sen-
sor model and the generalized sensor model.
A rigorous physical sensor model is used for modeling the
physical imaging process of a specific satellite sensor. Since
different satellite sensors with different image processing
require specific physical sensor models, the rigorous physi-
cal sensor model becomes more complex and cost for user. By
contrast, the generalized sensor model is a simple mathemati-
cal description of photogrammetric exploitation. The general-
ized sensor method usually includes grid interpolation model,
rational function model (
RFM
), and universal real-time model.
Since its successful application in Ikonos (Dial
et al.
, 2003;
Fraser
et al.
2003), QuickBird (Li
et al.
2007, Tong, Liu and
Weng 2010), SPOT (Tao
et al.
2001), ALOS PRISM (Hashimoto,
2003), IRS-P6 (Nagasubramanian
et al.
, 2007),Ziyuan1-02C (Ji-
ang
et al.
, 2015), ZY-3 (Wu
et al.
, 2015), GF-1(Wu
et al.
, 2016),
and other high-resolution satellite imageries (
HRSI
),
RFM
has
been adopted to replace physical sensor models in photogram-
metric mapping and becomes a standard way for economical
and fast mapping from high-resolution satellite imagery.
RFM
, related the object-space (
Latitude
,
Longitude
,
Height
)
coordinates to image-space (
Line
,
Sample
) coordinates, is a
form of a ratio of two cubic polynomials with 78 Rational
Polynomial Coefficients (
RPCs
). The least-squares regression is
firstly employed to estimate the optimal
RPCs
(Grodecki
et al.
,
2003; Tao and Hu, 2001; Tong
et al
, 2010). However, owing
to the strong correlation between the 78
RPCs
and a limited
accurate result in
RPCs
estimation, various developments such
as the solutions, accuracy, and numerical stability of direct
RFM
have been achieved. Generally, for the sake of numerical
accuracy, the image- and object-space coordinates are normal-
ized to (−1, +1)(Tao and Hu, 2001). Singular value decom-
position (
SVD
) method has been applied to solve
RPCs
(Fraser
et al.
, 2006; Li
et al.
, 2009), since a design matrix is likely
to be close to singularity in
HRSI
data. In respect of ill-posed
problem caused by strong correlation among 78
RPCs
, several
methods have been proposed to solve the ill-posed normal
equation including the ridge estimation strategy, Levenberg-
Marquardt algorithm, and the artificial intelligence. The ridge
estimation strategy, a revised biased estimation based on the
least-squares regression, is a widely used method. Combined
with the L-curve method, the ridge estimation strategy can
address the ill-posed equation well and obtain stratifying
RPCs
easily (Yuan
et al.
, 2008). In spite of the accurate
RPCs
obtained, the automatic determination of the optimal regular-
ization parameter of ridge estimation is rather hard to obtain.
With regard to the shortcoming of ridge estimation, a stepwise
regression for ill-posed problem by removing all of the un-
necessary parameters based on scatter matrix and elimination
transformation strategies has been employed. With the F-sta-
tistic as an evaluation criterion, the parameter that contributes
to F-statistic more would be selected the necessary parameters
in
RFM
(Zhang
et al.
, 2012). Simultaneously, a method named
the Levenberg-Marquardt algorithm has been adopted to sub-
stitute the least squares regression and solve the
RPCs
(Zhou
et
al.
, 2012). The Levenberg-Marquardt, specialized in dealing
with ill-posed problem, combines the steepest decent method
and the Gauss-Newton method and inherits the global-search
of gradient descent as well as the local-fast-converge of Gauss-
Newton. Furthermore, another solution combined with matrix
orthogonal decomposition, Levenberg-Marquardt algorithm,
and compute unified device architecture high-performance
computing technique has been employed (Wu and Ming,
Chang Li and Xiaojuan Liu are with the Key laboratory
for Geographical Process Analysis & Simulation, Hubei
Province, and College of Urban and Environmental Science,
Central China Normal University, Wuhan 430079, China
(
).
Yongjun Zhang and Zuxun Zhang are with the School
of Remote Sensing and Information Engineering, Wuhan
University, Wuhan 430079, China.
Photogrammetric Engineering & Remote Sensing
Vol. 83, No. 9, September 2017, pp. 611–620.
0099-1112/17/611–620
© 2017 American Society for Photogrammetry
and Remote Sensing
doi: 10.14358/PERS.83.9.611
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
September 2017
611
