9-11 Nov. 1999
et surface with superglacial
2 2 is extracted along the
poundaries and the dashed
perglacial lakes and com-
sreenland ice sheet (Fig-
rface fitting is the logical
shape, ice sheets are so
es can directly be recog-
as roughness and slope.
analysis of the elevation
1d derivatives renders the
oth, therefore the deriva-
nall (for example see the
tween the solid arrows).
of bumps contouring the
ple around dashed arrow
zhness resulting large and
re 2(b)-(c)). To suppress
tion profile was smoothed
; of increasing width. The
elevation profiles was in-
'endering the lake bound-
determination of surface
' laser waveform. For hor-
re the roughness scale is
| surface roughness within
m the pulse width of the
his approximation is valid
ce and for large footprint
ore complicated surfaces
and Jordan, 1988] might
n floor as an anisotropic,
to recover second order
ation, characteristic wave
»ension of seafloor topog-
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
| M
OA A N Aa + LA mmm pee Az
e ETT E
10 20 (km)
Figure 2: Object recognition from surface elevation and roughness
on laser altimetry profile in the area depicted in Fig. 1. Solid arrows
mark lake boundaries and the dashed arrow points to a ripple zone.
(a) Surface elevation from airborne laser profiling, (b) first, and (c)
second derivatives, and (d) second derivative of elevation profile
smoothed by convolution with a Gaussian kernel.
4 Error model of ALR
The 0.1-0.2 m accuracy of ALR provided by the
service providers [Flood, 1999] and investigators (e.g.,
[Krabill et al., 1995]) refers for flat or gently sloping surfaces,
low to medium flight height, and for objects with high surface
reflectivity. In this "ideal’ case, the height accuracy consists
of a 0.05-0.15 m fairly constant error from GPS and laser
ranging (for GPS baselines shorter than say 100 km), and an
error of ca. 0.5-2 cm per 100 m of flying height for typical
attitude errors and a scan angle of 30 ? [Baltsavias, 1999].
[Schenk et al., 1999] examines the resulting surface errors
for horizontal and sloped surfaces. His simple error analy-
sis demonstrates that the errors are complicated functions of
topography, flight direction and the systematic positioning
and attitude determination errors.
No comprehensive analysis has been attempted yet for quan-
tifying the error of ALR for complex, 3D surfaces. It is known
that larger errors tend to appear along vertical and steep ob-
ject boundaries since the same planimetric error manifests
in larger elevation error on steep slopes than on flat sur-
faces. There is also an increase in ranging error where abrupt
changes in elevation and surface normal occur within the laser
footprint. This error depends on several factors, such as the
ranging principle used by the laser system, the distribution
of the elevations within the footprint, and the background
noise. [Gardner, 1992] uses closed form equations of the laser
return waveforms to describe the increase of the ranging er-
ror on sloping or random rough surfaces caused by the pulse
broadening. His result is supported by the observations of
[Kraus and Pfeifer, 1998] who report that with increasing ter-
rain slope and roughness, the height accuracy deteriorates to
0.5-1 m for 1000 m flying height.
These results suggest that the surface error of ALR has non-
Gaussian distribution. Larger errors, possibly outliers are ex-
pected for example on steep slopes and along object bound-
aries.
5 Robust segmentation algorithm
In this paper we present an autonomous, statistically robust,
sequential estimator (RSE) approach to simultaneously pa-
rameterize and organize laser surfaces. Details of the proce-
dure, including the mathematical background, has been pub-
lished in [Boyer et al., 1994]. The approach is recommended
for the segmentation of noisy, outlier-ridden (not Gaussian)
laser points into smooth surface patches. Unlike most existing
techniques, this method creates complete surface hypotheses.
5.1 Mathematical model
We assume that surface elevations z; = (x4, yi) are measured
by ALR at n arbitrary locations. Then, a general linear re-
gression model fitting a (generally nonlinear) function of p
parameters can be written as:
Zi = 1;101 +... + Tipp + €; (1)
for íi — 1,..,n, where ej; represents a disturbance (error,
noise) in the observation. For example, with p = 6,xi1 =
1,042 = UT DL mS STE Tao PA and
with 01 = a,02 = b, and so on, we can fit a function of the
form:
fr y) = a + be. + oy + doy + ex’ + fff (2)
In matrix notation, eq. 1 is represented as follows:
z=X0+e (3)
where z is an n-vector of observations on the dependent vari-
able (elevation), X is an n x p matrix of observation on the
explanatory variable (the horizontal coordinates of the obser-
vations and their polynomials) having rank p, 0 is a p-vector
of parameters to be estimated (parameters of fitted surfaces),
and e is an n-vector of disturbances (errors in elevation)
Only planar and biquadratic surfaces can be fitted in the cur-
rent implementation of the algorithm. However, the algo-
rithm is general and can be, in principle, extended to any
number of parameters. Based on the arguments presented in
Section 4 we assume that the error distribution of ALR data
is non-Gaussian and contaminated by outliers around object
boundaries. Since heavy tailed error distributions are reason-
ably represented by a t(Student)-distribution, a t-distribution
having degree of freedom f and scaled by a parameter o was
chosen as a stochastic model.
5.2 Preprocessing of ALR data
The preprocessing involves the computation of the laser foot-
print position in a global, geographic coordinate system. De-
tails on how to establish the relationship between the laser
range and the attitude and position of the aircraft are given
for example in [Vaughn et al., 1996]. We omit here these
details and assume that the positions of the laser footprints
in a Cartesian coordinate system are made available for the
segmentation.
Interpolation The RSE algorithm like most other segmen-
tation algorithms has originally been developed for close-
range applications and it works on range images rather than a
