28 (and
ameter
, of a
sample
les are
ve the
pically
DEM
of the
LSM
of the
s. The
ly?) of
M ad
Ih(x,y).
', from
Il local
normal
um of
il as a
f small
sed in
OCesses
ces g-f
omials,
ons
Gy).
(1)
vertical
1s. The
cture
(2)
nates of
g three
1 (2) as
is their
n,,n,,3
rs of p
ensions
t only
in their
Another
hat the
als at a
ation of
one 2-D polynomial takes n1-1 or n2-1 (depending on the
order of summations in (2)) additions per point vs n1*n2
multiplications and additions per point if evaluated at
random locations. These savings are realized in the
reshaping process of the LSM tie point grid, enabling the
speed of over 10,000 tie points/sec. ELSM and GLSR
exploit these "fast" properties as discussed next.
* The fast solution of the 4-D parameter array A enables
the real-time ELSM triangulation of the orientation
parameters using the gridded LSM "tie point values" of
local dh,dx,dy estimates and their full 3x3 weight matrix.
A brute-force solution of the orientation parameters and
the m merged elevation parameters is prohibitive as its
operation count of each iteration would be about op=
(3pn1n2 + m)3. In the example of p= 4 DEM models,
n1=n2=4 and m=1,000, op = 1,1923. ELSM exploits the
fast solution techniques of the 4-D array parameters A,
allowing m to become so large that the valid tie points
overpower the effect of outliers in the estimation of the
orientation parameters. This is achieved by an array
reformulation of the adjustment of independent models.
* Merged DEM values are considered as the vertical
coordinate parameters of an independent model adjustment.
The observables g are processed in the post wise order of p
values at a given post. This allows the elimination of the
unknown (merged) elevation parameter resulting in the
reduced normals of the orientation polynomials. After all
posts are processed, the solution of the orientation
normals has the count of p(3n1n2)3 =4 (3x4x4)3 in each
iteration of array relaxation. This idea makes the array
algebra applicable for practical problems that otherwise
would prevent the use of the “fast” solutions, (Rauhala,
1986). ELSM exploits the array relaxation such that the
effect of the covariance terms among the p sets of
orientation parameters is moved to the right hand side of
the normals. A fast convergence is achieved when the
number of tie points is increased such that the effect of
outliers is overpowered by the *good" DEM values.
* Back substitution of orientation parameters for the
"merge" or the solution of the vertical coordinate
parameters has two "fast" solutions. As in the traditional
model adjustment, the coordinate parameters are merely
weighted averages of the transformed coordinates of each
model. By coinciding the tie point density with the 3x3-
7x7 post window size of LSM, the merge takes place as a
by-product of updating the reference window value in the
final ELSM iteration of the orientation polynomials. The
Second, more general, solution with the finite element
constraints is achieved by GLSR.
* The solution of the array parameters A can be further
speeded up by sacrificing some rigor in the stochastic
(statistical) error model. This sacrifice is minor in
comparison to reducing the functional math model of
dh,dx,dy to three averaged shifts or to the 7-parameter
transformation of absolute orientation, (Rosenholm and
Torlegard, 1988). The very fast solution consists of
preserving the rigor of the stochastic model in the partial
solution along the x-direction. The full (point variant) 3x3
323
LSM weight matrix of the local tie point observations
makes the three polynomials correlated. This weight
matrix is applied in the 1-D partial solutions of each line,
each requiring the matrix inversion of order 3n] or about
27 n]? operations. This is followed by the "corner
turning" (partial solution over the second index of the 4-D
array) or unweighted polynomial regression along the y-
direction, involving only one matrix inversion of the order
n2. This very fast solution was discarded after some
practical experiments. Its quality and robustness could not
compete with the adopted rigorous baseline.
By reducing the values of nj,n2 in (2) into 1-2, the
traditional orientation models are recovered as special cases
of the adopted baseline. Higher degree polynomials also
compensate for the systematic deformations or shear
errors. Similar polynomials are used in the generic math
models of softcopy workstations to approximate the
rigorous nonlinear models of image geometries. These
systems can handle the real-time transforms from the
input-to-output space based on the raw support data. Thus,
their small corrections dA can be considered as the main
parameters of the ELSM triangulation. The refined
support data are found by adding the estimates of the
ELSM polynomials dA to the polynomials of the raw
support data. The input of GLSR can then be taken from
the original input space (whose output served as the input
of ELSM). This requires that ELSM should be able to
handle up to the bi-cubic polynomials of n17n2-4 of the
typical softcopy systems employing fast image rectifiers.
3. TEST RESULTS OF ELSM
Two test areas were processed. Each area had three input
DEMs produced by the automated DEM technique of
Global Least Squares Matching (GLSM) from three SAR
stereo pairs. À manually measured ground truth DEM of 5
m spacing was used as the reference in 4-5 pull-in
iterations. The averaged posts of the LSM sample
windows were used as the reference in the last two
iterations. In practice, the ground truth DEM is replaced
by some information for the absolute orientation of the
relative ELSM solution where any input DEM can serve
as a temporary reference.
The ELSM algorithm outputs the statistics of each
iteration, including the weighted unit error of each
iteration and the solution of parameter corrections.
Residual statistics are summarized after the final iteration
for each input DEM. letting the operator to “see” the
locations of outliers or disagreements among all data sets.
The analysis of the tests confirmed the high parallax
mensuration accuracy of 0.2-0.4 pixel of a similar
experimental EO test of the GLSM technology,
(Hermanson et.al, 1993). This resulted in the ELSM
sigma r.m.s. difference of 1-2 m for the individual SAR
DEMs of a good 2-ray stereo geometry.
Both test areas had such a poor imaging geometry
(equivalent to a poor base-to-height ratio in EO stereo) for
two of the input DEMs that their contributions in the
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B2. Vienna 1996
