ul 2004
(8)
, [ (9)
IN
he CRS
on. The
ers and
ter and
ze pixel
point in
(10)
1 in the
1ich can
central
(11)
ientation
slements
“exterior
Y Z /
so?
, Y, and
| M, is
ly, small
1e strong
normal
1ditioned
close to
t square
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
estimation value X,, become very large and least square
estimator is no more an optimum estimator.
The ridge estimator and the stein estimator succeed in reducing
I
the first item 0, > 1/2. together with the total MSE value In
i=2
equation (2) at the expense of increasing the second item
2
SS ((%, J A | . In equation (4) the ridge estimator alters the
i=l
ill-conditioned state of the normal equation by using
A" PA + KI to take place A" PA . By adding a small positive
value K, , the eigenvalue A, which is near to zero goes up to
À, + K, and the least square estimation component AT sun 18
compressed toward the origin point on the scale of
À, /(4, + K,). Consequenly, the first item in MSE together
with the total MSE drop greatly and the accuracy of estimation
value is enhanced. The stein estimator reduces the MSE value
and improves the estimation accuracy through compressing
t= value on the fixed scale c . However, because the ridge
estimator is not a linear function to its estimation parameter K ,
it hasn't a unique and optimum solution approach, which results
in its unreliability and instability. The common approaches for
speculating the ridge estimator include Horel-Kennad method,
Horel-Baldwin method, Lawless- Wang method and Mcdonald-
Galarneau method. The stein estimator compresses each
component of the least square estimation value on the same
scale ¢ without considering the fact that each component has
different sized error. Therefore, the components with big errors,
ie. the components corresponding to infinitesimal eigenvalues,
don’t get adequate compression and the precision of the stein
estimator is limited.
In the CRS estimator (4 PA 4 D)( A' PL ^ dl)! is substituted
for A' PA . The eigenvalue A, rises to A, (A, + 1)/(A, * d)
and the least square estimation component X ,,... is compressed
LS(i)
toward the origin point on the scale of (4 + d (A, rk 1). For
the infinitesimal eigenvalue, the new eigenvalue is near to
A, /d , augmented tens, hundreds, or maybe thousands of times.
At the same time the component corresponding to the
infinitesimal eigenvalue, i.e. the component with large error, is
compressed on the scale of near to d , while the component
corresponding to the large eigenvalue, i.e. the component with
minor error, remains nearly unchanged. Hence, it can be
concluded that the CRS estimator changes the ill-conditioned
state of the normal equation more significantly and more
rationally. Moreover, it can be also testified that the CRS
estimator is the most accurate in MSE meaning by comparing
MSE values of the CRS estimator, the ridge estimator and the
stein estimator.
3.2 Calculation Process
The exterior orientation process for linear pushbroom imagery
by the CRS estimator can be divided into several steps:
Step 1: Computation of the initial values of exterior orientation
elements X,.
729
Step 2: Computation of the least square estimation value X
and its corresponding canonical value Y. :
First, coefficient matrix A , residual matrix V should be
constructed, and then normal equation is established, and finally
X is computed according to equation (1).
Step 3: Computation of the CRS parameter d .
The parameter d should be computed by equation (7). Because
7
the parameter d and the valuc Y. (d) are inter-determined,
the solution for parameter d is an iterative process. For the first
time, Y. is used to replace f s) in equation (7) to
calculate the first iterative value d, , and then d, is used to
calculate f.) inversely; for the next time Y. (d,) is
used to calculate d, and d, is used to calculate I. (d): the
process will continue on until la; -d;, ZE (ES is the
tolerance).
Step 4: Computation of the CRS estimation value A ed)
according to equation (6). The superscript i denotes the ith
circulation time in the whole process.
Step 5: Updating the
X, =X +X. (A).
Step 6: Judgement is put on X ,.,.. (d) to see if it is smaller than
exterior orientation elements
the threshold value. If X es) is smaller than the threshold
value, X is the ultimate exterior elements. If X sd) is
bigger than threshold value, the next circulation process will
repeat from step 2. Iterative circulation won't end unless
X. = (d) is smaller than the threshold value.
4. EXPERIMENTAL RESULTS AND EVALUATION
4.1 Test Data Sets
The test fields comprise two data sets. The first data set is a
panchromatic SPOT 1 1A image taken in northern China in
1986, named as Image 01. The image contains 6000 76000
pixels and covers 607 60 ground square kilometers, the northern
and northern-eastern parts of the image are mountains, the
central part is urban city and the other parts are predominantly
rural with land use. The ground elevation range varies from
about -50 to about 550 meters. The sun angle azimuth is 140.6°,
the sun angle elevation is 64.60°, the principal distance of the
sensor is 1082 millimeters, the film pixel size is 13 microns,
and the ground resolution is 10 meters. 57 well-defined control
points were measured from 1:50,000-scale maps, among them
18 control points are used as directional points and the left
points are used as check points. Figure 1 illustrates the coverage
area of the image with the ground points distribution. Triangles
represent directional points and circles represent check points.
