ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision‘‘, Graz, 2002
The behaviour of edges in real reflectance imagery is
somewhere in between the ideal case and the case of the
infinitely small footprint. The different ways in which a
reflectance image may be generated from the reflectance
strengths of the laser beams is visualised in figure 6. The left
picture shows a grey value edge and different locations of three
successive footprints in scan lines perpendicular to the edge. In
the top scan line the second and third footprint are at the same
distance from the grey value edge. The reflectance strengths in
the footprints are used to derive the grey values of the
rectangular areas that are represented by the footprint. The
resulting grey values are depicted in the right picture. In the top
scan line the grey value edge exactly coincides with the edge
between the second and third pixel of the reflectance profile. In
the following scan lines the footprints are gradually shifted to
the right. Although the pixel of the second footprint in the
second scan line is partly in the bright area, the footprint is still
completely in the dark area. Therefore, the pixel is assigned a
low reflectance value. In the reflectance image in the right
picture, this leads to a reconstructed edge position that has a
bias to the right. This bias increases in the following scan lines
until the footprint is tangent to the grey value edge. In the next
few scan lines the footprint captures intensity information from
both sides of the edge and the pixels in the reflectance image
obtain mixed grey values. An unbiased estimate of the edge
location is again obtained in the scan line where the centre of
the footprint is located on the edge. Shifting the scan lines
further to the right a pattern symmetric with the upper half of
the pictures appears.
Figure 6: Left: footprints on scan lines across a grey value
edge. Right: resulting pixel grey values and
reconstructed edge locations. See text.
As can be derived from figure 6, the maximum bias in the
location of the edge in the reflectance image equals half the
point spacing minus half the size of the footprint. For the above
example of a laser scanner with a point distance of 1.3 m and a
footprint size of 0.3 m, an edge location bias of up to 0.5 m can
occur. For DEM data acquisition with a point distance of 4.0 m
and a footprint size of 0.6 m, the bias may be even 1.7 m.
3.3 Matching edges in reflectance data
When matching reflectance images the matching bias may even
be twice as large since the edge location bias may be in opposite
direction in the overlapping strips. In order to minimise the bias
one should try to select long edges as the bias tends to decrease
with the edge length, although this does not hold for all edge
orientations. For edges parallel to the scan lines, the edge length
does not influence the bias in the edge location [Fórstner 1986].
A - 378
The derivation of gradients from the grey values in the
reflectance images nearly always will lead to errors in the edge
location as illustrated in figure 6. In order to obtain a better
estimate of the edge location a model is required for the
reflectance of a laser beam on a grey value edge.
Let the position of an edge be described by
X cos + Ysina zd
with at as the edge orientation and d as the distance of the edge
to the origin of the coordinate system. The signed distance u of
a point (X, Y) to the edge is then given by
u=Xcoa+Ysina-d
Let the edge orientation be chosen such that u is negative for
points on the dark side of the edge. If the footprint radius
equals R, the footprints of all points with u « -R are completely
located on the dark side of the edge and the footprints of all
points with u > R are completely located on the bright side. For
footprints in between, the footprint is intersecting the edge
(figure 7a). The fraction of the footprint on the bright side can
be defined as a function f{u) of the unsigned distance u (and the
footprint radius R) (figures 7a and 7b).
-R | 0 R u
Figure 7: (a) Footprint located at unsigned distance u from a
grey value edge. (b) Fraction of footprint on bright
side of the edge.
Mathematically, f(u) is defined by
0 us-R
1 uw) u u \
f(u) 341——] arcco an 1-| — -R«u«R
c
1 uzR
This function can now be used to model the expected
reflectance strength of a laser beam near an edge. Let ry and r;
denote the reflectance strength in the dark and bright area
respectively. The expectation of the reflectance strength r(X, Y)
within a footprint on the location (X, Y) is then given by
Er (X,Y)}=n + f (y —1,)
This equation can be regarded as the non-linear observation
equation. Linearising around the approximate values 7’ and »’
of the unknown reflectance values on either side of the edge and
the approximate edge location parameters «^ and d^ yields the
linear observation equation for the estimation of the edge
location
