-
)
ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002
Figure 3: Result of kriging with a spherical model of vari-
ogram.
Figure 4: Aerial image of the studied area of Brussels
((OEurosense).
3 ENERGY MINIMIZATION APPROACH
One method for image restoration with edge preserving is
based on energy minimization (Geman et al., 1992). In this
approach a solution is an image which minimizes a cost
function, also called energy. An expression for the energy
consists of two terms:
e data-fidelity term, which penalizes variations between
a surface and experimentally measured data
e regularization term, which imposes a roughness penalty.
The optimization of the energy is, in the general case, too
expensive. One generally chooses to introduce a marko-
vianity assumption which makes possible to ensure that a
minimum can be obtained as a sum of local terms (this
seems reasonable in our case, since it is probable that the
quality of reconstruction of a roof of a building does not
concern the geometry of other distant buildings). During
the reconstruction, one makes iterative calculations of the
surface so that a minimum of an energy is obtained. Sev-
eral elements are significant for the method: the defini-
tion of the neighbourhood considered around each point,
the definition of the terms of energy and the corresponding
potentials, and, finally, a method to decrease an energy.
3.1 Definition of neighbourhood
Let observed data samples be (z(z4, yx) ) at positions (xx, yx)
for k — 1,..., N. We are looking for samples u(i, j) on a
regular grid (7, 7).
The neighbourhood for the regularization term consists of
points inside a circle, which includes 8 nearest points on
the regular grid (Figure 5).
12 * +
*
* ** * *
* *
* * *
10 o o o o o
*
* *
*
8pk o o
*
*
6 o o
* *
*
4r o o
*
*
à
2 Oo o * o o ©
* *
* x x
*
* *
o —— 3
Figure 5: Definition of the neighbourhood of a pixel: it is
represented by all the points located inside the circle. The
stars represent the irregularly distributed original data, the
circles are the regular points which one wants to determine.
So the neighbourhood for the regularization term for each
point (i,j) is:
M (i, j) = {(', 5") : dist((i', 5), (3, 5)) <r}
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