ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002
dist((z, y); (i, 3)) = (z a; i? + (y m i)?
The radius r is chosen to encompass to the 8 nearest neigh-
bours on a regular grid.
The neighbourhood for the data-term is:
P(i,j) — (za, yx) : dist((zx,yx), (53)) € r)
Thus for every position (i,j) we obtain a set of indexes
of data points, say Æ (à, j), which are placed in the neigh-
bourhood of (i, 7):
K (i, j) = {k : (Tk, Yr) € P(i,3)}
3.2 Expression for the cost function
The form of the energy generally consists of 2 terms, data-
fidelity term and regularization-term. For our problem, we
have chosen the following cost function:
z(x2r:4r)—u(1,j)
Fes 2kG) V (ies)
5j \ ta) mes) P EE
u(? ,j )—u(4j
where 1 and q are potential functions, and the multiplier
a gives a weight to the regularization term. The first sum-
mation (on ¢ and 7) will be made on all the points of the
regular grid. The second summation (on the set K) is done
on all the points of the irregular grid inside the circle of
neighbourhood of the current point i,j. The third summa-
tion is done on the 8 neighbors of the current point i,j. The
solution % will be a surface which minimizes the cost func-
tion:
à — arg min F(u) (2)
3.3 Potential functions
The choice of the potential functions q and «y is supposed
to lead us to the best solution which is determined by fea-
tures of altimetric reconstruction in urban environment. Let
us express some common-sense remarks about urban envi-
ronment.
1. A lot of surfaces are horizontal (or about): streets,
pavements, terraces, gardens and yards, etc.
2. Other surfaces are flat, but oblique, in particular the
sides of the roofs.
3. Many discontinuities have to be found in the verti-
cal ortho-photographic projection which we want to
make of the city, the frontages in particular. Never-
theless, these surfaces can give measurements which
do not correspond the real model because of the angle
of scanning of the laser which can hit the frontages or
cling on convex objects: balconies, canopies, etc.
4. Finally, a small number of objects do not correspond
to any of these models. It happens for vegetation, ve-
hicle surfaces of car parks where these vehicles are
gathered in a very dense way.
A lot of studies have been done to determine potential func-
tions for filtering images while recovering edges (Nikolova,
2000; Charbonnier et al., 1997; Bouman et al., 1993). Some
potential functions are convex, some are not. They can be
smooth or non-smooth at the origin. We limit our interest
to four choices for potential functions. Each of them, ex-
cept for total variation, has a parameter to tune (denoted by
B).
Huber function (Figure 6(a)):
p(t) = &1(d « 8) - (8? -- 280 — 8D1( > 8) (3)
where I(p)=1 if p is true and I(p)=0 otherwise. This func-
tion is supposed to preserve slopes on the surface.
Total variation function (Figure 6(b)):
e(t) = 18]. (4)
Since this function is non-smooth at zero, it causes steplike
zones on the surface (Nikolova, 2000).
Figure 6: (a) Huber and (b) total variation function.
Generalized Gaussian function (Figure 7(a)):
pl) = [ef 1<B<2 (5)
Truncated quadratic function (Figure 7(b)):
plt) ^ min(t^, B). (6)
qais ov nn
Figure 7: (a) generalized Gaussian and (b) truncated
quadratic function.
3.4 Optimization algorithm
In order to find an optimum of the cost function several
algorithms may be used (Li, 1995). Conjugated gradient is
one of them. It is useful when the cost function is convex.
Since not all the potential functions are convex in our case,
we can use stochastic optimization methods. One of them
is Iterated Conditional Modes (ICM) (Li, 1995) algorithm
which has the following steps.
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