International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
(a) (b)
Figure 2: (a) Black cells have already been visited whereas
gray ones are potential candidates for propagating (see Figure
1). Z eund local 18 calculated by averaging elevation over black
cells. (b) Illustration of the linear correction. Empty circles are
laser points. Black dash lines are the estimated local ground
Lio local Without linear correction whereas gray dash lines
are the estimated local ground 2 round local after linear correc-
tion
3.3 Post-processing
Let us introduced now an intermediate class called low non-
ground points. This class is a buffer with low vegetation features,
cars and sparse medium height micro-relief. À laser point pt be-
longs to this class if pt.z € [Sin + 0, Sin + 2m] where a is
the tolerance on ground points. The next step of our algorithm
consists of an iterative convergence toward a stable state of Sin
whereupon laser points will change their label depending on this
belonging to this intermediate class. Point label may change and
Sin 1s updated. The process carries on until convergence of the
algorithm (no longer label movement).
The last step consists of comparing the classified point cloud with
the final DTM (after deformation, section 3.4). Points lower than
$ y 4- 0.5m belong to the ground.
3.4 Deformable Model
The estimated ground surface S;n is of importance for classify-
ing laser points: the more accurate the surface, the more rele-
vant the classification. Nevertheless, the multiple pass filter will
force the continuity of the ground estimation. Topographic de-
tails will therefore be smoothed (80% overlapped neighborhood)
sidesteping major ground descriptive laser points. Seeing that
surveying micro relief is a major characteristic of airborne laser
technology, it is necessary to take these points into account when
estimating the true ground surface. Secondly, the resolution of
this surface is coarse mainly for computing time efficiency. Con-
sidering laser performances, we may fairly expect to have a final
hight resolution DTM with a micro detail description (modulo the
point density). We will therefore consider this surface Sin as an
initial input of a deformable model algorithm.
This method has similarities with active shape models, but we
will consider attractors belonging exclusively to the ground (fol-
lowing criteria of the classification algorithm). We will not
describe in this paper the whole theoretical framework of de-
formable models (Montagnat et al., 2000) (Fua and Leclerc,
1994) (Fua, 1997) (Metaxas and Kakadiaris, 2002), but only the
main hypothesis and the functions we used for airborne laser ap-
plications.
The energy of a deformable model is composed of several terms
including at least an intrinsic regularizing term £,eg and a data
term £54. The energy of the surface S is defined by:
EIS) = Ercg(S) + £i S)
Note that S must belong to the set of square integrable functions,
Vis Vici
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Figure 3: Vi; represents the neighborhood at grid cell (i, j)
whereas V;4.1,; is the next neighborhood extraction following the
propagation route. Empty circles are common laser points, which
will be consecutively processed. The ground estimation is per-
formed onto Sin(t, 7), Sin(t + 1,7). using points classified
as ground within a laser neighborhood V;,j, Vi+1,;j --
and be twice differentiable. We admit that the energy functional
is built such that its global minimum coincide with the expected
solution Sy:
S; min E(S)
The regularizing term has a stabilizer role since the data term is
usually very irregular and shows a large amount of local minima.
In our implementation, Sy is approximated by the minimum of
equation 4:
E(S) = min (wee + mes) (4)
DE dd
We used an Iterated Conditional Modes (ICM) algorithm (Li,
1995) (Zinger et al., 2002) for computing a local minimum. In
this context, S is discretized over a regular grid. The grid nodes
(4, j) are the movable DTM values. For each grid node, the cost
function is calculated for a large set of quantified values the sur-
face can have. We then attributes to the grid node the value which
minimizes the cost function. 3D laser points are treated as attrac-
tors and we define the data local energy £e») of Sn by
ext
2
Lj SG D -:0 if 289) exists :
5) = ( a( 33) a a , (5)
0 if not.
which is the Euclidean distance between the actual surface Sn
and the corresponding orthogonal laser attractor 263) Within the
ISM algorithm, the minimization is performed over the following
values of Sn
S553) zS$,-1( 53) Fóz ng €m € nama«
with Sno (4, j) = min(z0?, SE G, j)) (6)
Snmas (i) = max(zP, Ss (5.3)
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