ISPRS Commission III, Vol.34, Part 3A ,Photogrammetric Computer Vision“, Graz, 2002
E{(X,Y) -n - f u*y)n? -— r°)}= ( — f(u°))Ar, + f (u°)Ar, T
You a
ou [2 A od ad |
with
u* 2 X cosa? - Y sina? - d?
as the approximate signed distance of a point to the edge. This
equation can be formulated for all laser points near an edge. The
gradients of the reflectance strength are derived from the
analytical edge response function rj*f(u)(r;-ro). This approach
has several advantages over the standard least squares image
matching:
e As the partial derivative df/du equals zero for points at a
distance of larger than R from the edge location, these
points will not directly effect the estimation of the edge
location, but only contribute to the estimation of the
reflectance values on either side of the edge. The edge
location is primarily determined by the points with
footprints that actually cross the edge. The gradients at
those positions are properly modelled by the edge response
function and thus do not cause a bias in the estimation of
the edge location.
Since the gradients are derived analytically, they are not
affected by the (high amount of) noise in the reflectance
data. Maas [2000] noted that the low signal-to-noise ratio in
the coefficients of equations for matching height data
caused an overestimation of the matching precision. The
usage of an analytical edge model will allow a more realistic
estimation.
e The observation equations can be formulated for the
original laser points and do not require computations on an
additional data structure, like a TIN.
In order to obtain accurate results, one should, however, select
long edges for the matching. This is required because of the
high noise level in reflectance data, but also since only few
points may fall within a distance of R from the edge. The
amount of these points depends on the ratio between the point
distance and the footprint size and on the orientation of the edge
with respect to the scan lines. In bad configurations only very
few or even no points may contribute to the edge location
estimation. This should, however, then result in a very high
value of the estimated edge location precision. By checking the
propagated reflectance variances those edges can be selected
that have a good location accuracy.
The edge location equations above were formulated such that an
edge is located in a single laser data set. If the assumption can
be made that the systematic errors in the laser data do not cause
a rotation of an edge, the edge fitting can also be done
simultaneously in two or more point sets, using the same edge
orientation Q for all point sets in which the edge is visible. This
may further improve the offset estimation between strips.
Initial values are required for all four edge parameters ry, r;, 0,
and d. They can easily be obtained by low level image
processing of the gridded reflectance data. Figure 8 shows
detected long edges that were obtained in the lower image of
figure 5 by a straight line growing algorithm on a median
filtered image. Statistical tests on the fit of the reflectance data
to the edge model should be used to eliminate those edges that
can not be modelled properly by this model. This can be the
case if the edge is slightly curved or if other objects near the
edge violate the assumption that the reflectance strength is
homogeneous on both sides of the edge.
Figure 8: Extracted lines on a median filtered reflectance
image.
As for matching height data, one should avoid reflectance edges
near occlusions or height jumps. By examining the height data,
this can be verified easily.
4. CONCLUSIONS
In this paper it was shown that the determination of offsets
between laser altimetry datasets can not be solved reliably by
standard least squares image matching algorithms. This holds
for height data as well as for reflectance data. Height jump
edges in laser altimetry data show the same behaviour as edges
in binary imagery. Their location may show a bias which
depends on the edge length and the orientation of the edge with
respect to the scan line direction.
For matching height data it was advocated to also use linear and
planar features besides points for which offsets can be
determined in three dimensions. Because of the limited width of
linear features a model based fitting may often be required.
Reflectance data may be suitable for the estimation of
planimetric offsets, even though its noise level is often fairly
high. In order to avoid biases in the estimation of edge
locations, the response of a laser beam to a grey value edge
needs to be modelled. The usage of such an edge response
function also enables the computation of noise free gradients,
which results into a better convergence behaviour and a more
realistic estimate of the edge location precision.
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