c
1
on
c
2
with multiple piecewise affine functions estimated on
a Delaunay triangulation (Figure 4) given by the correspond-
ing points (Step 1). Indeed, the affine transformation (for the
single tile used in the previous section) is only an approxima-
tion for images with weak geo-localization accuracy.
The choice of an affine transformation on a triangulated
network ensures that the union of interpolated functions is
continuous. In addition, the transformation can be efficiently
evaluated by numerically stable methods. Let
x
2
i
and
x
1
i
(
i
= 1,
2, 3) be the un-deformed and deformed vertices of a generic
triangle by means of an affine mapping
x
2
i
=
L
(
x
1
i
); a corre-
sponding point
x
2
P
x
1P
inside the triangle can be written as
a unique combination of vertex points:
x
2
P
=
µ
1
x
21
+
µ
2
x
22
+
µ
3
x
23
=
µ
1
L
(
x
11
)+
µ
2
L
(
x
12
)+
µ
3
L
(
x
13
)=
L
(
x
1P
)
(4)
where the numerical coefficients can be evaluated as:
µ
1
2
22
23
21
22
23
1
1
1
1
1
1
=
=
det
det
det
P
x
x
x
x
x
x
/
x
x
x
x
x
x
x
1
12
13
11
12
13
2
21
1
1
1
1
1
1
P
det
det
=
/
µ
1
1
1
1
1
1
1
1
2
23
21
22
23
11
1
x
x
x
x
x
x
x
P
P
det
det
=
/
x
x
x
x
x
x
x
13
11
12
13
3
21
22
2
1
1
1
1
1
1
=
/
det
det
P
µ
1
1
1
1
1
1
1
21
22
23
11
12
1
=
/
det
det
P
x
x
x
x
x
x
/
det
x
x
x
11
12
13
1
1
1
(5)
and
µ
1
+
µ
2
+
µ
3
=1, with
µ
i
>0. This means that mapping can be
efficiently evaluated in the reference layer (
c
2
) and applied to
the other one (
c
1
) with a linear transformation. Coefficients
can be estimated for the different triangles of the network (a)
to provide an overall improvement of classification overlap
(in terms of spatial position), and (b) to reduce the effect of
variable off-nadir angles and weak geo-localization accuracy.
Resampling is not performed on images to improve
CPU
time.
As mentioned, the reference system is provided by classifica-
tion
c
2
, whereas
c
1
is warped using a set of piecewise func-
tions obtaining a new classification
c
1*
without alterations
of topological information. Geometrical constraints in the
triangulated networks (e.g., self-intersections) of images
I
1
and
I
2
can also be used to remove incorrect matches.
Finally, residual misalignments between segments can be
interpreted as spurious slivers visible after polygon intersection
with Boolean algebra. Slivers are located close to boundaries
and have an elongated shape that can be exploited for automat-
ed removal. Given a generic polygon made up of
n
pixels, a 2 ×
2 covariance matrix can be generated by estimating the follow-
ing geometry-based quantities (variances and covariance):
σ
σ
x
x y
i
x y
i
y
x y
i
x y
i
n
x
n
x
n
y
n
y
2
2
2
2
2
1
1
1
1
=
−
=
−
∑ ∑
∑ ∑
,
,
,
,
=
−
∑ ∑ ∑
2
1
1
1
σ
xy
x y
i i
x y
i
x y
i
n
x y
n
x
n
y
,
,
,
(6)
The identification of elongated features can be carried out by
using the ratio between the variances and the average side of the
object in term of pixels, obtaining the following density index:
δ
σ σ
=
+ +
n
x
y
2
2
1
(7)
where
δ
is a positive number that assumes small values (close
to zero) for very elongated objects, whereas squared elements
have a large density. The denominator of Equation 7 can be
interpreted as the average radius of the object.
Density can be used to separate slivers from real changes
detected from the intersection of variable classes. It is a rela-
tive index that does not depend on the spatial extension of
the object and can be used for images captured by different
platforms, including scale variations.
Case Study
Data Description and Object-Based Classification
This section illustrates the results of the method with a
dataset acquired over the city of L’Aquila (Italy). On 06 April
2009, the city was struck by an earthquake. Two hundred
ninety-seven people died, while both modern and historical
buildings underwent some severe damage. Remote-sensing
change detection plays a fundamental role in damage as-
sessment after natural disasters (Alexakis
et al.
, 2014), but a
rapid response is achievable only if the post-event image is
captured as soon as possible following the disaster, notwith-
standing a possible unfavorable acquisition geometry due to
the lack of geo-products such as accurate
DEM
s and on-site
measurements
Pre- (
I
1
: 04 September 2006, off-nadir angle 10.65°,
GSD
0.63 m) and post-event (
I
2
: 08 April 2009, off-nadir angle
5.60°,
GSD
0.62 m) QuickBird images were provided by Digi-
talGlobe as a standard product, that is, with a geolocation
accuracy of 23 m (CE90%). GCPs for accurate orthorectifica-
tion were not available. Pan-sharpening was used to produce
a four-band mosaic starting from panchromatic (0.6 m) and
multispectral (2.4 m) data. Overall, the nominal geospatial
error is significantly larger than the resolution of the pan-
sharpened images (23 m versus 0.6 m). Figure 5 shows the
matching and triangulation results for an image.
Object-based change detection was carried out with
specific algorithms developed for the city. The goal of this
paper is not a detailed description of the specific algorithms
developed to identify damaged buildings, which are only
briefly presented. The proposed case study is an example
used to demonstrate the advantage of the new method for
sliver removal.
Object-based classification for
I
1
and
I
2
was carried out
with three classes: vegetation (
V
), shadows (
S
), and object
(
O
). Complete classification results (i.e., for the whole im-
age) are given by the union of different categories:
c
1
= [
V
1
È
S
1
È
O
1
] and
c
2
= [
V
2
È
S
2
È
O
2
]. A preliminary segmentation was
used to detect small objects corresponding to vegetated areas.
Vegetation was identified by using the average normalized
vegetation index (
NDVI
). Objects classified as vegetation were
excluded from data processing. Next, a new segmentation
with a larger scale factor was used for the unclassified areas to
generate larger elements that include shadows, roads, parking
areas, buildings, and so on. Shadows can be extracted by con-
sidering their low average brightness. Finally, all the remain-
ing segments were classified as objects.
Figure 6 shows details for three historical buildings in
L’Aquila. Large damage (Figure 6a) is visible and casts an ad-
ditional shadow in the post-event image. It is evident that the
spatial error for classes
O
1
and
O
2
is very large for the weak
orientation parameters encapsulated in the delivered Rational
Polynomial Coefficients (Figure 6b).
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
February 2016
165
