I
h
= max{
I
–
h
,0}.
(4)
The
h
-dome transform is defined as (Vincent, 1993)
M
h
(
I
h
) =
I
–
ρ
I
(
I
h
)
(5)
where
ρ
I
(
I
h
)
refers to the morphological reconstruct of the
original image I using the mask from marker
I
h
. The regional
maxima refer to the pixels that correspond to the
h
-domes:
EX x
M I x
h
h h
( )
,
( ( )) ,
,
,
=
>
1
0
0
if
otherwise
(6)
where
x
refers to a pixel of the image domain.
Building Scale Detection
The available data include airborne multiple-spectral images
of Tokyo and Sapporo, Japan with a 0.2 m resolution, and the
DSM
derived through photogrammetry with a 0.5 m resolu-
tion (which is registered to the image). Although the spatial
accuracy of the
DSM
is not as good as the original images,
the signal is immune to color variations for a certain object
caused by shading. We use the
DSM
to detect the markers and
to generate the masks that correspond to the houses, and use
the multiple-spectral image to create the precise house extents
with the help of the masks.
The scales of the houses can be detected using granulom-
etry analysis. A granulometry is a kind of morphological
histogram (Serra, 1982). It is computed using a sequence (
Γ
S,n
)
of morphological
opening
s with a structuring element (
SE
)
S
of increasing size
n
.
MBI
(Huang
et al
., 2011) adopted the
opening-by-reconstruction in a granulometry analysis (the
top-hat by reconstruction for the differential morphological
profiles), and the mean or the maximum in the profiles was
used to indicate buildings since it represented the mean or
maximum local contrast degree. So did Pesaresi and Bene-
diktsson (2001). They all carried out granulometry analysis
pixel by pixel. We propose a method of constructing the gran-
ulometry histogram considering all the pixels in continuous
scales. The histogram reflects the main sizes of the objects in
the scene by obvious increases in the curve. The scales of the
buildings of close sizes then are detected accordingly. Unlike
Santibañez (2012) and Mura (2011), we define a granulometry
as the mapping of a sequence (
Γ
S,n
) of morphological
openings
by reconstruction
with a
SE
S
of increasing size
k
. That is:
Γ
S,n
= (
ν
(
ρ
I
(
ε
kS
(
I
))))
k
∈
[1,
n
]
,
(7)
where
kS
refers to the
S
dilated
k
-times, and
v
(o) refers to
the surface function (which returns the total height of the
non-zero pixels and the
volume
of o). We propose to imple-
ment the scale mining by the granulometry using
opening by
reconstruction
instead of
opening
. Both
opening by recon-
struction
and
opening
can remove the bright objects smaller
than the
SE
. The difference is that a bright object bigger than
SE
will remain unchanging in the opened by reconstruc-
tion image, while become smaller and more compact in the
opened image. In other words, the
opening by reconstruction
operation reflects the real area or volume change of the image
caused by the filtered objects.
In our case,
I
refers to the
nDSM
, which can be taken as a
surface. The buildings can be viewed as cylinders of some
certain sizes and heights above the flat ground. When the size
of
SE
rises to some value
n
, the buildings of sizes smaller than
n
will be filtered out by
openings by reconstruction
from the
nDSM
, no matter how high they are. The difference between
the volume of the original data and
Γ
S,n
exactly corresponds
to the total volume of the buildings of sizes smaller than
n
. If
there are many buildings of size
n
or some very high build-
ings of size
n
in the scene, there will be a big lifting in the
granulometry profile at scale
n
; and vice versa. There may be
no buildings on some scales. In this case the granulometry
may be flat at those scales. The difference between
Γ
S,n
and
Γ
S,n
–1
is called the pattern spectrum (
PS
), and is also known
as a differential morphological profile (Lefèvre
et al.
, 2007).
It represents the reduced volume between the two continu-
ous scales. Ideally, the value of
PS
at scale
n
refers to the total
volume of the buildings of size
n
. When there are many build-
ings of size
n
or there are some very high buildings of size
n
occurring in the scene, there will be a peak in the
PS
curve at
scale
n
. Therefore, the peak of the
PS
curve reflects the main
scale of the objects in the image. We propose to detect the
peaks of
PS
to get the main scales of the buildings.
Actually, the buildings appear more complex in the
nDSM
and more similar to hills or domes than cylinders. When the
size of
SE
rises to some value
n
, the buildings not only with
foot size of
n
but also with dome size of
n
will be filtered.
The practical granulomatry profile is not a step curve, but a
smoothed one. The
PS
curve generally also have noise.
Figure 2 gives an example of a typical granulometry profile
of a
DSM
. Figure 2a shows the
DSM
patch of Sapporo. Figure
2b and 2c shows the granulommetry profile and the
PS
curve
of the
DSM
respectively, where the
x
-axis corresponds to the
radius of the
SE
.
We can find that, there is a large increase of the granulom-
etry curve or a main peak of the
PS
curve within a range of
scales. It reflects that there are a lot of domes or buildings of
similar sizes in the scene. Since the
PS
curve is very rough,
the proper scale range is difficult to detect. Thus, we propose
a scale range detection method using the
PS
curve, described
as follows. The start of the range (
s
0
) is defined as the point of
(a)
(b)
(c)
Figure 2. Granulometry and PS of Dataset 1 using
opening by reconstruction
: (a)DSM, (b) Granulometry, and (c) PS.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
January 2016
23
