Fig. 4: Building of a constraint edge.
3.6 Discussion
The presented method satisfies the requirements noted in
chapter 2. The order-criterion works independently of the
coordinatesystem and is (nearly) independent of the shape of
the surface. The incremental way of building the triangulation
supports dynamical editing and progressive sampling. The
method is qualified to be implemented only with the use of
local algorithms.
Unfortunately there are two problematic aspects of the method,
which may lead to errors. The first one is the locality of the
order-criterion. This criterion works well in the neighbourhood
of an edge and with moderate surfaces. If the surface is ben-
ding strongly, the criterion will possibly fail. Figure 5 shows a
situation, when the order-criterion says left, but the point
should apparently lie right of the edge k.
Fig. 5: Wrong result of the order-criterion
Another problem is the estimation of the surface-normals.
Especially when only a few points have been inserted already,
the surface is badly represented by these points. Hence the
normals, estimated with these points, do not correspond suffi-
ciently with the actual normals.
These problems can be solved by applying a verification of the
locating-step. This test can be used to detect gross errors of
measurement, as well as to expose wrong locatings.
4 CORRECTION AND SMOOTHING OF LINE-
NETWORKS
Due to the data capturing, points along lines are measured with
more or less accuracy. The unfiltered connection of such a
sequence of line-points would reproduce the real course of the
line just with low or even unacceptable quality. Therefore
measured points are to be considered only as noisy line-sup-
porting-points (LSPs).
By smoothing the course of the LSPs, gross measuring errors
can be found and eliminated. The adjustment of LSP-
sequences, based on a mathematical well-defined type of curve,
enables us to replace the often very extensive measuring data
by other suitable data allowing an unique reconstruction of the
lines. This circumstance is very advantageous because of the
reduction of data-amount.
For the application dealt with, curves consisting of joined cubic
polynomials (Spline-curves) are of advantage. In this case the
course of the curve is uniquely defined by the chosen type of
interpolation (e.g. Osculatory-, Akima-,”Spline-interpolation),
the type of parametrisation (e.g. chordal, centripetal, equidi-
stant) and the location of the (spline-)knots (SKs) between the
separate polynomials (Forkert, 1994). Very long lines and,
furthermore, more or less expanded networks of several lines
can occur in practice. Considering manipulation of data and
computing time, it is therefore necessary to use such a kind of
curve by which only the LSPs within a small surrounding area
‚(around the SK-interval to be calculated) have any influence
concerning the course of the curve. This demand also helps to
avoid disturbing oscillations of curves due to the position of
LSPs laying far away with regard to the part of the curve being
calculated at the time. The estimation of the spatial position of
the SKs is therefore done based on the Osculatory-interpola-
tion.
The possibility of a correct stochastic interpretation of the
smoothed course of the LSPs is guaranteed by applying an
adjustment following the method of least squares. The correct
adjustment of line networks (free of gaps) is a precondition for
modelling surfaces with high quality. Therefore, if there exists
a junction or a crossing of several lines, it is not only deman-
ded each line to be smoothed separately, but furthermore that
such a line-net-knot (LNK) itself gets a unique position
without any contradiction. (In many cases, a LNK is not captu-
red directly by a measured LSP but has to be calculated by
intersection.)
In general, line-networks contain more than one LNK and can
get very extensive if there are artificial objects to be recon-
structed (like traffic buildings, machine parts or urban areas).
Such big line-networks have to be divided - as far as possible -
into small ‘subnets’ in order to be adjusted independently. In
consequence, the principle of a strict adjustment of the whole
net of lines cannot be followed any longer. This blemish
however has nearly no effect if there are arranged overlapping
areas wherein the LSPs have to be taken into account for both
adjustments whenever two subnets (SNs) join together. De-
pending on the type of interpolation used, an overlapping area
will occur every time a line had to be cut at one end of a SN.
Its extension (number of SK-intervals) depends on the type of
interpolation used. Within the overlapping areas the adjusted
SKs have to be calculated in a way that no gaps remain in the
whole line-network after smoothing is finished. A SN cannot
be cut if there exists a further LNK within that overlapping
area.
{ay 5
0—— 0 independently adjusted SNs with SKs
o— —— — whole line-network free of gaps
Fig.6: Overlapping area of two already adjusted SNs
to be ,,sewed" together without gaps.
410
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
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