the noise. If we assume that the errors in the general regres-
sion model are independent and identically t-distributed, the
CAIC is given by
2
T
i
fa?
| + p(log(n) + 1)
(10)
where the first part describes the log-likelihood of the model
with p parameter, log(n)-- 1 is the cost of fitting an additional
parameter, n is the number of observations, and w; and r;
are the corresponding weight and residual for the ith obser-
vation. For a fixed data set it is easy to compare the CAIC
for different fitted surfaces. However, with a sequential ap-
proach the data set is not fixed. Since CAIC assumes a fixed
data set, we start the RSE algorithm in a local fixed win-
dow, and compute the parameter vector using IRLS. From
this estimate, we compute the initial CAIC. Then the RSE
expands the initial window, until it encounters outliers. To
compute the modified, asymptotically consistent Akaike in-
formation criterion (MCAIC) the CAIC is normalized by the
total number of observations in the final, maximal window
for each model:
2AIC z —2(0- fes. w;log[1 +
i=1
MOAIC = KA (11)
where K is an arbitrary constant, we use the number of points
in the initial window.
To select the best candidate model for each candidate model
the following procedure is used:
e For each seed point robust parameter estimates are
obtained for each candidate model.
e The initial value of CAIC is computed by using (10).
e Points are added sequentially by expanding the initial
window in four direction, updating the parameter esti-
mates by using RSE. A point is considered as outlier if
its weight falls below a threshold. If the ratio of out-
liers to total data points exceeds 75 percent on a side,
the window is not expanded further in this direction.
e For each model and its resulting window, the MCAIC is
computed by using (11). For this seed, the model min-
imizing MCAIC is selected as the best approximating
model.
5.6 Postprocessing
Expand The expand procedure is employed both as part
of the seed selection-model selection loop, and to establish
the final model parameters and the set of points supporting
that particular parameter vector. Once we have the best
approximating model for a surface (seed), we use this model
and let the surface grow over the entire scene. This process is
repeated for all surfaces at the appropriate seed points using
the best approximating models as obtained in the choose
step.
Prune The expand process may assign isolated points to a
surface. Those with fewer then 2 co-surface 8-neighbors are
assigned to the base surface.
Resolve For the end of the segmentation each laser point
should be assigned to one surface only. To resolve the am-
biguities a 5 by 5 neighborhood of each ambiguous point
is examined by calculating the average estimator weight for
each candidate surface. The surface yielding the maximum
average weight is selected.
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
Remove (re-prune) After the resolve step, some points
which were not isolated prior to resolution, may become iso-
lated. These points are removed by re-pruning.
Fill At this point the surface usually have pinholes where
points are assigned to the base surface within another surface.
This is the dual of the isolated point problem. The final
assignment of these points is based on their 8-neighborhood.
The final output of the segmentation is:
e 3D graph surface equations,
e 2D region boundary equations,
e fit error,
e other characteristics of surface patches.
5.7 Experimental results
Experiments on both synthetic data and real range imagery
are presented in [Boyer et al., 1994] to demonstrate the per-
formance of the RSE segmentation.
Figure 5-6 shows the result of one of these experiments. The
synthetic surface has two planar region with smooth tran-
sitions into a cylindrical joining region such that depth and
orientation is continuous everywhere (Figure 5(a)). Noise
and outliers were added to simulate a noisy set of range data
Figure 5(b). Surfaces with smooth joins are more difficult
to segment but the algorithms addresses this case reasonably
well. The recovered surfaces and the segmentation bound-
aries are shown in Figure 6.
(a)
Figure 5: (a) Smooth join surface and (b) surface contaminated
with added Gaussian noise and outliers
Internatior
(b)
Figure 6: Recovere
Plane2, (c) all three
6 I
Airborne laser scan
sition method for
at high a density
curate. However,
description of the
teristic, such as bre
