ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
points located in the neighborhood of point /; and 7; (/=1, ….,m) 
are its corresponding candidate matches. 
In order to link the matching results of the neighboring grid 
points to each other, we define the following compatible 
coefficient function C(i,j;k,]) which quantifies the compatibility 
between the match /;——/; and a neighboring match /,—1;: 
C ns ——— 
exp(Ap" / B) (1) 
Ap —(x, -x)- (x, - X,) 
In equation (1), Ap expresses the difference of the x-parallaxes 
in point 7; and its neighboring point /,. The bigger the Ap, the 
smaller the compatibility. This corresponds to a smoothness 
constraint on the image matching results. 7 is a value quantified 
by the texture information and it is defined as inversely 
proportional to the minimum of four gray value variances 
(horizontal, vertical, and two main diagonals) at the image 
window around the point /;. Normally, if one point is located in 
rich texture areas or at linear features, this value is small. This 
value can be treated as the weight of the smoothness constraint 
and it provides the possibility to control the continuity of the 
terrain surface. f is a constant value and is set to 400 
experimentally. 
In the relaxation scheme, the so-called global consistency of 
matching can be achieved by an iterative scheme where the 
probabilities P(i,j) are updated by the following rule: 
where 
peg pH = Te Do, J) 
2, P" G9.) Q) 
where @9G, N="TT1 Ww POQG. DC PED 
I,€Q(1;) I-l 
C(ij;kl) is the compatible coefficient function defined as 
above, Q(I;) is the neighbourhood of point 7; (can be its 8 or 24 
neighboring points), and n is the iteration number. The quantity 
Q""(i,j) expresses the support the match II; receives at the p 
iteration step from the matches /,—], in its neighbourhood Q(/;). 
The iteration. scheme can be initialized by assigning the 
normalized correlation coefficient to P/(;,;) and, ideally the 
process will terminate when an unambiguous match result is 
reached, that is when each point /; is matched with one 
candidate with probability 1, the probabilities for all other 
candidate matches for this point being zero. In practice we 
terminate the process if any one of the following 2 conditions 
holds: 
* For each grid point /;, one of the match probabilities P(i,j) 
(j=1,...,m) exceeds 1-& where & «« 1 (for example, we set the 
value of eto 0.1). 
* The pre-defined number of iterations has been reached. 
When the iterative procedure is terminated, the match which 
gains the highest probability P(,j) (j=1,...,m) is selected as the 
actual match. 
This method is performed by using the stereo pairs, which can 
be a combination of the forward and nadir view or the nadir and 
backward view TLS images. For speeding up the processing, 
reduction of the search range and the gain of higher reliability, a 
multi-resolution data structure, i.e. the image pyramid is used. 
The matching scheme is performed as follows: 
e Construct the image pyramid with a pre-defined number of 
levels 
e Perform the standard relaxation matching with the initial 
values on the highest level of the pyramid 
e Transfer the results from the previous pyramid level to the 
current level as the starting initial values 
e Perform the standard relaxation matching on the current 
pyramid level using the initial values obtained in step (3) 
e Check if the matching has reached the finest pyramid 
level. If it has not, advance the matching by one level and go 
to step (4). If it has reached the finest level, terminate the 
matching procedure 
The advantage of this method is that it can achieve reasonable 
matching results even in areas of little or no texture. Its 
disadvantage is that the matching results only have pixel-level 
accuracy. The relaxation matching results are further refined by 
the following modified MPGC and GCMM procedures. 
Another disadvantage is its failure to match all 3 TLS images 
simultaneously. Also, the grid matching results cannot express 
the terrain precisely in case the terrain is steep and rough. Under 
these conditions, some well-distributed feature points can 
compensate for this disadvantage. 
4.3 Modified Geometrically Constrained Multi-Point 
Matching 
A multi-point matching algorithm was suggested by 
Rosenholm, 1986 and further developed by Rauhala, 1988 and 
Li, 1989. Gruen, 1985b also proposed a conceptually similar 
method as multi-patch matching. The standard multi-point 
matching algorithm is image-based and uses the simultaneous 
computation of parallaxes in grid points which are connected 
with bilinear finite elements describing the parallax differences. 
Additional weighted continuity constraints on parallaxes are 
used to strengthen the connections between the grid points. The 
critical points of multi-point matching are the selection of the 
number of nodes in the grid mesh, the size of the image patch 
and the weight of the smoothness constraints. 
Our modified algorithm is an extension of the standard multi- 
point matching, using parallaxes in two directions and 
integrating the geometric constraints derived from the TLS 
sensor model. It can be used to match grid points on three TLS 
raw images simultaneously and provide the pixel and object 
coordinates simultaneously. 
In the case of matching of TLS images the geometric constraints 
are derived from the collinearity equations by Taylor expansion 
and result in 
  
  
  
ol, OE, OE, OE, E. 
y, 2—— Av +— Au +— Ars ay AZ Et) 
; ov Ou OX oY oz er 3) 
al, OE, oF, oF, OF, tes 
WV, Bede AY ot mm App ote ee AY fi AY pei AL (VE) 
: Ov Qu OX oY oZ ^ 4 
These two equations can be treated as weighted observation 
equations in GCMM. (Az, Av) are the unknown x-shift and y- 
shift in pixels, which relate the common unknowns (parallax in 
x and y direction), appearing in the gray level observations of 
the standard 2D multi-point matching through the following 
equations: 
Xs TX = Px (4) 
VEUT Dy 
(x, yj) and (x, y,) are pixel coordinates of the template and 
search image respectively, p, and p, are parallaxes in x and y 
direction. 
The weighted geometric constraints link the parallaxes in three 
TLS images and in principle, they force the matches for each 
grid point to move along their respective epipolar curves. 
One important issue is how to construct these weighted 
observation equations, because the computation of the 
derivatives in equation (3) needs the functions of the six exterior 
orientation elements with respect to the pixel coordinate ». Due 
to the small pull-in range of multi-point matching, we use the 
grid matching results as initial values. So the correct match can 
be achieved in a very small image search window, that means 
we can treat the exterior orientation values as quadratic 
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