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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