International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, 
X. > (A PA)" qd PL (3) 
X (K) 2 (A! PA - KI)! A' PL (4) 
Xo. (€)=c(A PAY ATPL (5) 
where X, 7 least squares (LS) estimation value 
X,(K) 7 ridge estimation value 
X . (c) = stein estimation value 
Stein 
I = unit matrix 
K =the ridge parameter vector (K, > 0) 
C = the stein parameter( ¢ > 0) 
If each element in vector K is equal, the ridge estimator is 
regarded as the ordinary ridge estimator; else the ridge 
estimator is deemed as the generalized ridge estimator. The 
second item in equation (2) is constantly zero for the least 
square estimator since it is an unbiased estimator. But for the 
ridge estimator and the stein estimator the second item is no 
longer zero since they are biased estimators. 
The combined ridge-stein estimator can be represented below: 
Xo (d)=(A"PA+ 1)" (A"PL+dD)X,, 
: (6) 
- Q(A* I) (^*dl)Q' X, 
where — Q - eigenvector matrix 
A = eigenvalue matrix 
d = the CRS parameter, 0 <d <1 
The CRS estimator possesses three favorable characteristics in 
comparison with the ridge estimator and the stein estimator. 
(1) The CRS estimator is more accurate than the ridge 
estimator and the stein estimator. 
From the viewpoint of statistics a good estimation value should 
have a minor MSE. The MSE value of the CRS estimator is 
smaller than the ridge estimator and the stein estimator when 
d is optimum, therefore, the CRS estimator is more accurate. 
(2) The CRS estimator is superior to the ridge estimator and the 
stein estimator in reliability and stability. 
To get correct and accurate estimation results, the ill- 
conditioned state of the normal equation must be changed. The 
CRS estimator alters the ill-conditioned state much more 
significantly than the ridge estimator and the stein estimator, 
therefore, its reliability and stability are more excellent. 
(3) The solution for the CRS estimator is unique and simple. 
Since the CRS estimator is a linear function to its estimation 
parameter, the optimal value for d is ascertained when 
11 
MSE(X ,. ) - 0. 
d= Poti TTT (7) 
: $ d (d) = ó, Y 6: + À, p (d) 
e (A «ly ES oA Al) 
where Y. (d) is the canonical value for X o5 (0) „which 
is imported to facilitate and simplify computation. 
Part B4. Istanbul 2004 
foy (8) 
Y. (d) 2 Q'X (d) S (A1) (^ dI)A" (4Q) L (9) 
3. THE CRS ESTIMATOR'S APPLICATION IN 
LINEAR PUSHBROOM IMAGERY EXTERIOR 
ORIENTATION 
3.1 Theoretical Basis 
This paragraph is mainly concerned with how to apply the CRS 
estimator in linear pushbroom imagery exterior orientation. The 
linear pushbroom imagery has multiple perspective centers and 
each scan line has an unique set of perspective center and 
rotation angles. The collinearity equation between a image pixel 
on the ith scan line (x;, 0) and its corresponding object point in 
the object spcae (X, Y, Z) is as follows: 
AX X) Y) 7 Z) 
! " a(X- X )-b(Y-Y)*e(Z-Z,) (10) 
Qo ,0(X-X) -(Y-Y)*e(Z-Z.) 
7" a(X-X)-bY-Y)*«Z-Z). 
  
The y coordinate along the flight direction is implied in the 
position and attitude of the satellite at a given time, which can 
be linearly related to the location and attitude of the central 
linear array as follows: 
9,79, *eky 
0-0, &y 
K,m/k,* My a1) 
Xj-X 4 + X y 
y, eV i+ & y 
Zelt 
so 
where f= focal length 
A Z ; dis: gas ati 
Oy Os Kor X os Yıosl,y = exterior orientation 
elements of the central scan line 
^ Y . . + 
9,0. K Xu 454, ^ exterior orientation elements 
of the ith scan line 
a;, bj, c; ^ elements of rotation matrix 
&, dx, s, X ; *, X — variation rates of exterior 
orientation elements. 
Each scene has 12 exterior elemets ( Q9, , €, , K,, X unda du 
&, dA, i&, XX, Y, &). Among them X, and ,, Y, and 
@, are highly correlated. A small change in @, is 
indistinguishable from a small change in X , . Similarly, small 
changes in Y, and @, can’t be differentiated either. The strong 
correlation induces the coefficient matrix of the normal 
equation A" PA singular, the normal equation ill-conditioned 
and some eigenvalues A, in the eigenvalue matrix A close to 
zero as well. Accordingly, the MSE value of least square 
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