The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part BI. Beijing 2008
Unfortunately, the relative nature of the LiDAR intensity signal
does not allow for a general parameterization of the intensity
values for pavement surfaces and pavement markings, and thus,
there is no absolute threshold that would separate the two areas
Therefore, first the distribution of the intensity signals in the
search window should be analyzed to determine an optimal
threshold for separating pavement and pavement marking points.
In our approach, the point, where the curve of the pavement
surface points levels out, was selected as a threshold, and
subsequently used for extraction of the pavement marking
points. The points extraction based on this threshold could
result in errors, such as marking points are omitted or pavement
points are included. Therefore, further checks are needed,
which is accomplished by curve fitting and matching, described
below, where the availability of object space information, such
as curvature of the pavement markings, can be utilized. Figure 6
shows the pavement markings extracted for the area pictured in
Figure 3; the threshold was 180.
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(a) (b)
Figure 5. Changes of intensity values along pavement markings:
LiDAR point locations overlaid on optical image (a) and
intensity values (b).
Figure 6. Pavement markings extracted by thresholding.
4. CURVE FITTING
The extracted pavement marking and GPS-surveyed points have
no point-to-point correspondence, and thus, a point-based
transformation is directly not applicable. However, their shape
can be matched, on condition that the two representations
provide an adequate description of the same linear feature. In
this case, the problem is simply how to match two free-shape
curves. In the following, the key steps of curve fitting are
presented, while the matching is discussed in the next section.
The purpose of curve fitting is twofold: first, it provides a
validity check for the pavement marking points extracted, and
second, it allows for modeling both pavement marking
descriptions as linear features, so they can be matched to each
other. The selected curve fitting method is an extended version
of the algorithm, originally proposed by Ichida and Kiyono in
1977, and is a piecewise weighted least squares curve fitting
based on cubic (third-order polynomial) model, which seemed
to be adequate for our conditions, such as linear features with
modest curving. To handle any kind of curves, defined as the
locus of points f(x, y) = 0, where f(x, y) is a polynomial, the
curve fitting is performed for smaller segments in local
coordinate systems, which are defined by the end points of the
curve segments. The primary advantage of using a local
coordinate system is to avoid problems when curves become
vertical in the mapping coordinate system. Obviously, the
fitting results as well as the fitting constraints are always
converted forth and back between the local and mapping
coordinate frames, for details, see (Toth et al., 2007).
The main steps of the piecewise cubic fitting (PCF) process are
shortly discussed below; the notation used in the discussion is
introduced in Figure 7. To achieve a smooth curve, the curve
fitting to any segment is constrained by its neighbors by
enforcing an identical curvature at the segment connection
points; in other words, PCF polynomial is continuous with its
first derivative at connection points x=s, x=t, etc. The equations
describing the 3rd polynomial and its first derivative are:
S k (x) = y s +m s-(x-s) + a s -(x-s) 2 +b s -(x-s) 3
slope = S k (x) = m s + 2 ■ a s • (x - s) + 3 • b s • (x - s) 2
Figure 7. Piecewise weighted least squares curve fitting method.
The core processing includes the following steps: 1) a s and b s ,
the coefficients of the second and third order terms of the fitted
curve for interval ‘f are estimated; consider the constant term
(ys) and the coefficient of the first order term (m s ) fixed, known
from the curve fitting from the previous segment. In the
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