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COMBINED RIDGE-STEIN ESTIMATOR IN EXTERIOR ORIENTATION FOR LINEAR
PUSHBROOM IMAGERY
Wang Tao, Zhang Yong-sheng, Zhang Yan
Department of Remote Sensing Information Engineering, Zhengzhou Institute of Surveying and Mapping, No. 66
Longhai Middle Road, Zhengzhou City 450052, Henan Province,China - wt4289@sina.com; zy4289@sina.com
Commission IV, WG IV/7
KEY WORDS: exterior, orientation, pushbroom, estimation, algorithm, accuracy
ABSTRACT:
The paper presents the combined ridge-stein estimator (CRS) for linear pushbroom imagery exterior orientation, which is severely
ill-conditioned for the strong correlation among exterior orientation elements of linear pushbroom imagery. The estimator is a new
biased method combining the ridge estimator and the stein estimator. It can effectively change the ill-conditioned state of linear
pushbroom imagery exterior orientation process and achieve optimum estimation values through applying different scale
compression to each least squares estimation component. Its performance is evaluated using one 10-meter SPOT 1 panchromatic
image and one 2.5-meter SPOT 5 panchromatic image. Experimental results show that the combined ridge-stein estimator can
effectively overcome the strong correlation among exterior orientation elements and reach high reliability, stability and accuracy. It
is within one pixel accurate for ground directional points and within one and a half pixels accurate for ground check points.
1. INTRODUCTION
The linear pushbroom imagery is widely used in remote sensing
mapping for its stable geometry and good image quality, such
as SPOT, MOMS-02, IRS-1C/D and IKONOS images.
However, the strong correlation among exterior orientation
elements of this kind image induces normal equation heavily
ill-conditioned (Gupta, 1997) and severely affects the reliability,
stability and precision properties of exterior orientation.
Aiming at solving this problem many researchers have put
forward different methods, including the incorporation of high-
correlated elements, the separate and iterative solution of line
elements and angle elements, the fictitious error equation, the
ridge estimator (Huang, 1992; Zhang, 1989), the stein estimator
(Zhang, 1989) and so. But there lie various shortcomings
among these methods. The incorporation of high-correlated
elements method only works well when the photography state is
near to vertical photography. The separate and iterative solution
of line elements and angle elements method isn't rigorous in
theory and the orientation precision and iterative times depend
on the accuracy of initial exterior orientation elements. The
fictitious error equation method adds great workload and
demands much ancillary data, such as orbit parameters, satellite
photo data, and etc. The ridge estimator (Guo, 2002) and the
stein estimator are both biased methods and can achieve better
results than those unbiased estimation methods listed before.
However, the two methods still need improvements. The ridge
estimator has not a unique solution for it is non-linear to its
estimation parameter. The stein estimator applies the same scale
compression to each least squares estimation element without
considering the fact that each element has different sized error.
Therefore, in this paper the combined ridge-stein estimator
(CRS) (Gui, 2002) is proposed up for linear pushbroom imagery
exterior orientation. The CRS estimator is new biased estimator
and has never been applied to photography before.
727
In the following, after a brief introduction of the CRS estimator,
focus is put on its application in the linear pushbroom imagery
exterior orientation, and then experiments are performed using
one 10-meter SPOT 1 panchromatic image and one 2.5-meter
SPOT 5 panchromatic image, and finally conclusion is drawn.
2. DEFINITION OF THE COMBINED RIDGE-STEIN
ESTIMATOR AND ITS PROPERTIES
In the adjustment for the unknown X the following
observation equation can be established, and mean square error
(MSE) is adopted to assess the accuracy of the estimation value
X . The smaller the MSE is, the more accurate the estimation
value is.
V—AX LUE )s0Co(V)zoiP^ (1)
MSE(&)- o2 YA, Y. (ee )- x, ) Q)
where A = coefficient matrix (n row X t column)
L = constant matrix (n row X 1 column)
7 — residual matrix (n row X 1 column)
n = the number of observed values
t = the number of unknowns
O, = variance of unit weight
À, = the ith eigenvalue in the eigenvalue matrix
The least squares estimation, the ridge estimation and the stein
estimation for the parameter X are respectively:
