Distance as the "Cost" in Euclidean Space for 
similarity assessment. Because the distances 
behave as SIN function of angle between the fea- 
ture vectors in case the distance is small, it 
offers a sensitive measure for the distinction of 
close similarity in the primitive feature space, 
In the Least Squares Matching Method, though the 
approach of matching is totally different, the 
basic idea is similar (Min. Zvv <--> Min. dis- 
tance), namely matching with very high accuracy to 
the degree of subpixel; it requires, however, a 
very good conjugacy prediction (coarse DEM). The 
commonly used Normalized Cross Correlation method 
provides a correlation function behaviour as a COS 
function [LO,1991]. This function has a flat peak 
at maximum similarity; therefore, to interpolate 
this function to get subpixel accuracy is hardly 
worth the effort. 
C) When we use the unknown parameters to establish 
a mathematical model (Observation Equations) which 
can precisely describe the phenomena of observa- 
tion/ sampling (e.g. the back mapping of intensity 
from image space to object space), the Least 
Squares Method minimizes the differences between 
the description model with actual model to deter- 
mine the value of unknowns ( when the observation 
equation is linear ), or to update the estimated 
values of unknowns iteratively (when the observa- 
tion equations are non-linear). Every pixel 
involved in matching can provide one observation 
equation; if redundant observations exist (the 
number of observation equations is more than the 
unknown parameters, i.e. the  over-determined 
problem), the Least Squares Method is a good 
method to determine the value of unknown parame- 
ters even when the errors of observations are not 
a Normal Distribution. The principle of the 
Maximum Likelihood Method is identical with the 
principle of Least Squares Adjustment, if we have 
assumed that the errors of observations are a 
Normal Distribution. But unlike the Maximum 
Likelihood Method of estimation, the method of 
Least Squares does not require the knowledge of 
distribution from which the observations are drawn 
for the purpose of parameter estimation; however, 
for testing of hypotheses, we would require the 
knowledge of the distribution  [Bouloucos,1989]. 
d) Least Squares Window Matching has been used in 
the two-step approach whereby matching in image 
space is performed to get the conjugated position 
first, then the Space Intersection is used to 
reconstruct the object surface [Ackermann,1984]. 
Its window size is limited to 20x20 pixels or 
30x30 pixels; if the window is smaller than this, 
reliability is decreased, but if it is larger than 
this, accuracy will be poor because matching is 
performed on image space, and the geometry model 
is simplified  [Rosenholm,1987a]. The improved 
approach unifies these two steps into one, and 
matching on object space. Back mapping of image 
intensity into object space to get object reflec- 
tion D(X,Y)(image inversion) is done by referring 
the coarse object surface Z(X,Y), then perform 
matching on object surface and refine the coarse 
object surface iteratively. In addition to this, 
two functions in the object space, i.e. the object 
surface Z(X,Y) and the object reflectance D(X,Y) 
are simultaneously determined (considered) in one 
solution with Least Squares Adjustment; this is 
the reason why we consider this to be a more 
rigorous method than the earlier methods. 
e) By referring the coarse object surface Z(X,Y), 
there are two ways to back mapping of the image 
intensity into the object surface in order to get 
the estimated object reflection D(X,Y). One is 
called Directed Pixel Transformation: it starts 
with pixel position (x,y) in the image; the 
Collinearity Equation with orientation parameters 
of its scanner is used to intersect the coarse 
object surface Z(X,Y) to get the position of 
corresponding groundel X,Y, and transfer the 
intensity of this pixel to it. However, the prob- 
lem is that the groundels distribution on object 
surface after back mapping present a random pat- 
tern and they are different in different multi- 
view images also; another shortcoming is that the 
height of these random position points needs to be 
interpolated to grid DEM but lose information. 
136 
The other way is called Indirected Pixel Trans- 
formation: we start with a coarse DEM grid 
(X,Y,2), and the Collinearity Equation with orien- 
tation parameters of the scanner is used to get 
the corresponding pixel position (x,y). Another 
problem arising from SPOT's Push Broom scanner is 
that the orientation parameters of each scan line 
are different (for a frame camera, it is the same 
in the whole image); therefore, if we don't know 
which set of orientation parameters we should use 
for transformation, we can not use the 
Collinearity Equation to get the corresponding 
pixel on the image and transfer its intensity 
(after resampling) to that grid point. If we can 
solve this problem, the Indirected Pixel Trans- 
formation is better than Directed Pixel Transform- 
ation, as the weighted average of pixel inten- 
sities which are obtained from multi-view images 
according to the same grid point, can be used as 
the estimated object reflection D(X,Y), and the 
weight is assigned according to the slope of ray. 
f) The Least Squares Matching method is capable to 
handle any number of images over two, e.g. the 
Triplet, as well as images scanned in various 
spectral bands simultaneously. It increases both 
reliability and accuracy of the result [Shibasaki 
& Murai,1988]. 
g) The traditional method uses windows of pixels 
for matching to determine a single point (usually 
the middle point) only, but Object Space Least 
Squares Matching uses a window of pixels for 
matching to determine multi-points in a grid 
pattern DEM in one solution. If preprocessing 
provides prior knowledge about the quality of 
matching windows (e.g. the gradient of intensity), 
we can assign different weights accordingly (e.g. 
give the high contrast pixels a larger weight) in 
the Least Squares Adjustment. This means that we 
require the high contrast pixels to offer a larger 
contribution for the decision making which helps 
to avoid making the wrong decision in the homo- 
geneous part of the image. Thus, a combination of 
advantages from Feature Based Matching and Area 
Based Matching can be obtained. DEM determination 
executed in this way is thus called Multi-Point 
Matching, and offer higher reliability [Rosenholm 
,1987b]. 
h) Robust Estimation techniques can be applied in 
Least Squares Adjustment to get rid of noise as a 
gross error of sampling (observation). 
i) The Least Squares Method provides theoretical 
quality estimation of the matching result based on 
statistics theory. It also offers useful informa- 
tion for cleaning gross errors in DTM data in the 
postprocessing stage as quality control, as well 
as for using the DTM data in GIS [Day & Muller 
£1988]. 
j) The grid DEM is separately generated patch by 
patch with the quality estimation. The aggrega- 
tion of the whole DEM can be done, e.g. by means 
of the Finite Element Method, with a quality 
improvement of DEM in the last stage. [Xiao et 
a1.,1988]. 
4. SUMMARY 
4.1 Summary of the Matching Algorithms and the 
Selection of Approach 
There are many matching algorithms that can be 
used; we summarize the relevant algorithms, and 
indicate those selected and applied in this system 
with the mark of ***, 
(a) Information for Matching: 
*** Intensity-Based 
*** Feature-Based (Property-Based) 
(b) Criterion for Similarity Assessment: 
* Angle between Matching Vectors: 
COS Function ---> 
Less Accuracy/ High Reliability 
*** Distance between Matching Vectors: 
SIN Function(in small distance) ---» 
High Accuracy / Low Reliability 
(c 
** 
** 
** 
** 
** 
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