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DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020231
DYNAMICAL SYSTEMS SERIES S
SEMI-AUTOMATIC SEGMENTATION OF NATURA 2000
HABITATS IN SENTINEL-2 SATELLITE IMAGES BY EVOLVING
OPEN CURVES
Karol Mikula∗, Jozef Urb´
an, Michal Koll´
ar and Martin Ambroz
Department of Mathematics, Slovak University of Technology
Radlinsk´eho 11, 810 05 Bratislava, Slovakia
and
Algoritmy:SK, s.r.o., ˇ
Sulekova 6, 811 06 Bratislava, Slovakia
Ivan Jarol
´
ımek, Jozef ˇ
Sib
´
ık and M´
aria ˇ
Sib
´
ıkov´
a
Institute of Botany, Slovak Academy of Sciences
D´ubravsk´a cesta 9, 845 23 Bratislava, Slovakia
Abstract. In this paper we introduce mathematical model and real-time nu-
merical method for segmentation of Natura 2000 habitats in satellite images
by evolving open planar curves. These curves in the Lagrangian formulation
are driven by a suitable velocity vector field, projected to the curve normal.
Besides the vector field, the evolving curve is influenced also by the local cur-
vature representing a smoothing term. The model is numerically solved using
the flowing finite volume method discretizing the arising intrinsic partial differ-
ential equation with Dirichlet boundary conditions. The time discretization is
chosen as an explicit due to the ability of real-time edge tracking. We present
the results of semi-automatic segmentation of various areas across Slovakia,
from the riparian forests to mountainous areas with scrub pine. The numerical
results were compared to habitat boundaries tracked by GPS device in the field
by using the mean and maximal Hausdorff distances as criterion.
1. Introduction. In this paper we present a semi-automatic method for image seg-
mentation of NATURA 2000 habitats in Sentinel-2 satellite images. In the section
2the main idea of our segmentation method is given. We design the velocity vec-
tor field and we prescribe the evolution equation for the segmentation curve in the
form of an intrinsic partial differential equation with Dirichlet boundary conditions.
Similar models are used in the context of dislocation dynamics in [5,10]. Section
3deals with the numerical discretization of the proposed segmentation model. The
comparison of numerically segmented areas and areas tracked by GPS device is
presented in section 4.
2010 Mathematics Subject Classification. Primary: 35R01, 65M08; Secondary: 35R37, 92F05.
Key words and phrases. Image segmentation, curve evolution, numerical method, Natura 2000,
satellite images, Sentinel-2.
∗Corresponding author.
This work was supported by projects APVV-16-0431, APVV-15-0522 and ESA Contract No.
4000122575/17/NL/SC.
1
2 KAROL MIKULA ET AL.
2. Mathematical model for semi-automatic image segmentation. In this
section we explain design of the segmentation model by using artificial images.
We use a suitable velocity vector field, based on the smoothed image information,
which drives the evolving curve automatically to the boundary of segmented habitat.
Such vector field is constructed by using the pre-smoothed image intensity gradient
as an edge indicator which is an input of the edge detector function. Then the
gradient of the edge detector function is projected to the normal of the curve and
the overall curve motion is regularized by using the local curvature of evolving curve
[3,4,7,8]. The final mathematical model is given by the corresponding nonlinear
intrinsic partial differential equation which is discretized and solved numerically by
the flowing finite volume method [9,7,1].
2.1. Construction of the velocity vector field. Let us consider a 2D scalar
function of image intensity I0:R2→R, given e.g. by a simple artificial example
plotted in Fig. 1left, and also the function Iσrepresenting a Gaussian smoothing
of the original image, see Fig. 1right.
Figure 1. An example of artificial greyscale 2D image (left) and
smoothed image (right).
In other words, I0(x) = I0(x1, x2) represents a scalar value of the image intensity
in a particular (pixel) coordinates x1and x2. In case of the Sentinel-2 optical data
we work, in general, with a combination of three chosen optical bands, so we consider
the image intensity as a vector function I0:R2→R3. However, the main ideas of
our semi-automatic segmentation approach can be simply explained in the scalar
case, so we do it in that way and emphasize arising differences only when necessary.
The smoothed greyscale image intensity function Iσfor the image in Fig. 1
can be also represented by a 3D graph as plotted in Fig. 2left. The boundaries of
segmented areas are usually represented by edges in the image and thus the edges are
the most important features which should attract the evolving segmentation curve.
The image edges can be identified by sharp changes of the smoothed image intensity
which is mathematically characterized by a large value of the norm of image intensity
gradient |∇Iσ(x)|. In Fig. 2right one can see a 3D graph of |∇Iσ(x)|calculated
for our illustrative image from Fig. 1right and its image intensity function plotted
in Fig. 2left.
SEMI-AUTOMATIC SEGMENTATION OF NATURA 2000 HABITATS 3
Figure 2. 3D graph of image intensity function Iσ(x) (left) and
graph of the image intensity gradient norm |∇Iσ(x)|(right), for
the smoothed image in Fig. 1right.
The norm of gradient |∇Iσ(x)|forms an input to an edge detector function g
defined by [11]
g(k, |∇Iσ(x)|) = 1
1 + k|∇Iσ(x)|2,(1)
where kis a so-called scaling factor. The values of g(k, |∇Iσ(x)|) are small along
the image edges and large elsewhere, see Fig. 3.
Figure 3. 3D graph of the edge detector g(|∇Iσ(x)|) for the
smoothed image in Fig. 1right.
Inspired by [3,4] we can define a velocity vector field v(x) by taking the gradient
of the edge detector function with minus sign,
v(x) = −∇g(|∇Iσ(x)|),(2)
and see in Fig. 4that such vector field points always towards the edges in the
image and thus can be used as a force driving segmentation curve always in a
correct direction, to the habitat border lines, see also Fig. 11 in section 4.
4 KAROL MIKULA ET AL.
Figure 4. A visualization of the vector field v(x) for image in
Fig. 1right. We see that arrows points to the edge in the image
from both sides.
2.2. The curve evolving in vector field. Let Γ be an open planar curve, Γ :
[0,1] →R2, Γ = {x(t, u), u ∈[0,1]}, depending on time tand let x(t, u) =
(x1(t, u),x2(t, u)) be a position vector of the curve Γ for parameter uin time t. In
the sequel, the curve will be discretized and represented by a set of points. An exam-
ple of an open planar curve discretization is displayed in Fig. 5, where xm
0,xm
1, ..., xm
n
are discrete curve points which correspond to the uniform discretization of the in-
terval [0,1] with spatial step h= 1/n at m-th time with time step τ. Due to the
Dirichlet boundary conditions we have that x(t, 0) = x(0,0) and x(t, 1) = x(0,1),
t > 0.
xm
0=x(0,0)
xm
n=x(0,1)
xm
i−1xm
i=x(mτ, ih)
xm
i+1 u= 0 u= 1
Figure 5. An open curve discretization (left) corresponding to the
uniform discretization of parameter u∈[0,1] (right).
2.2.1. Evolution driven by the vector field. The curve evolution driven by the vector
field v(x) is given by a nonstationary differential equation
xt=v(x),(3)
where xtdenotes a partial derivative of xwith respect to t. In our approach the
segmentation curve evolution begins from a uniformly distributed line segment,
defined by a user e.g. by mouse clicks, which is then moved automatically towards
the edge in the image, see Fig. 6for the discrete curve point evolution to the edge
in the artificial image from Fig. 1.
We can clearly see that such curve evolution brings suitable results for simple
images, however, it can be fairly inaccurate in more complicated cases and it needs
SEMI-AUTOMATIC SEGMENTATION OF NATURA 2000 HABITATS 5
Figure 6. Trajectories of points of a discrete segmentation curve
(red) evolved in the vector field vdriven to the image edge. The
final state of discrete segmentation curve is given by green points
localized on the image edge.
important modifications. Due to the discrete character of the vector field and
evolving curve as well, one of the problems arising is an accumulation of a discrete
curve points during the curve evolution and in the steady state. This phenomenon
is documented in Fig. 7, where more complicated objects were segmented by the
above simple approach, and lead us to a modification of the basic model (3).
Figure 7. Trajectories of points of a discrete segmentation curve
(red) evolved in the vector field vand their final position (green)
visualized over the original image. One can see a problem of non-
uniform distribution of points on evolving discrete segmentation
curve due to non-controlled tangential velocities in the vector field
v.
2.2.2. Ignoring of the uncontrolled tangential velocity component. In general, we can
describe the curve evolution given by Eq. (3) as a motion in normal and tangential
directions
xt=βN+αT,(4)
where αis a tangential velocity, T=xsis a unit tangent vector with sbeing
the arc-length parametrization of the curve Γ, ds =Gdu,G=|xu|,βis a normal
velocity and N=T⊥= ((x2)s,−(x1)s) is a unit normal vector.
Although the tangential component of velocity vector does not change the overall
curve shape, it only reparametrizes the curve and moves the points along the curve
in tangential direction, it can cause a non-uniform distribution of points on the
curve, see Fig. 7, and thus serious numerical errors in realistic situations. If we
ignore it, we can eliminate such undesired movements. Therefore, if we start from
uniformly distributed initial abscissa and move it only in normal direction, the
evolving curve will remain almost uniformly discretized, see Fig. 8. In such case,
6 KAROL MIKULA ET AL.
α= 0 in Eq. (4), and we can rewrite the curve evolution equation into the following
form
xt=λvNN,(5)
where λ > 0 is a parameter and the nonlinear term vNrepresents the projection of
velocity vector vonto the normal Nof the moving curve,
vN=v·N.(6)
Removing the non-controlled tangential part of velocity causes better distribution
of curve grid points as can be seen in Fig. 8.
Figure 8. Trajectories of points of a discrete segmentation curve
(red) evolved in the vector field vand their final position (green)
visualized over the original image. An improved distribution of the
curve grid points after removing the tangential component of the
velocity vector field vis obvious.
2.2.3. The regularization by curvature. The mathematical model (5) can be even
more improved and numerically stabilized by incorporating the local curvature in-
formation. Incorporating the curvature yields a sensitivity of the numerical dis-
cretization to a distance of neighbouring points, which ties together discrete points
of the evolving curve. In Fig. 9we illustrate a situation, when the image is less
smoothed and thus the velocity vector field is almost zero in some points of initial
abscissa. Clearly, by using numerically discretized equation (5), these points cannot
move, normal direction is changed and the evolution is not as one desires, see Fig.
9top. On the other hand, incorporating the local curvature influence, we smooth
the evolution, ties all points together in numerical discretization and get the result
plotted in Fig. 9bottom. So we modify the curve evolution equation (5) into the
following form
xt=λvNN+δkN,(7)
where δis a parameter and the term kNrepresents the so-called curvature vector.
From the Frenet equations we get for the curvature vector identities kN=Ts=
xss. Using this we obtain final intrinsic nonlinear partial differential equation for
evolution of the segmentation curve
xt=λvNN+δxss.(8)
The parameters λand δweight the vector field influence and the curvature influence.
SEMI-AUTOMATIC SEGMENTATION OF NATURA 2000 HABITATS 7
Figure 9. Trajectories of points of a discrete segmentation curve
(red) evolved in the vector field vand their final position (green)
visualized over the original image. Top: the curve evolution by
(5) in a less smoothed image when problem of crossing, accumu-
lating and not moving points may arise; bottom: such undesired
behaviour is removed by employing the local curvature influence
into the model (7).
3. Numerical discretization. Let us recall the intrinsic PDE (8) for the open
curve evolution and write it in the following form suitable for numerical discretiza-
tion
xt=δxss +wx⊥
s,(9)
where w=λvN. First, we perform the spatial discretization, which is based on the
flowing finite volume method [9,2,1].
Integrating (9) over the finite volume pi= [xi−1
2,xi+1
2], see Fig. 10, where xi−1
2
represents the middle point between the points xi−1and xiand ds is an integration
element of piecewise linear approximation of original curve, i.e.
xi−1
2=xi−1+xi
2, i = 1, . . . , n (10)
we get
xi+1
2
Z
xi−1
2
xtds =δ
xi+1
2
Z
xi−1
2
xssds +w
xi+1
2
Z
xi−1
2
x⊥
sds, (11)
where the values δand ware considered constant, with values δiand wion the
discrete curve segment piaround the point xi. We define hi=|xi−xi−1|, then the
measure of the segment piis equal to hi+hi+1
2. Using the Newton-Leibniz formula,
(10) and using approximation of the arc-length derivative xsby a finite difference
8 KAROL MIKULA ET AL.
xi
xi−1
xi+2
xi+3
xi−2
xi−3
xi+1
xi−
3
2
xi−
5
2
pi−1
xi−
1
2
xi+1
2
xi+3
2
xi+5
2
pi
pi+1
Figure 10. Visualization of the curve discretization [1] curve grid
points (red), discrete curve segments (different colors) and the mid-
points (black). Finite volumes pi−1,pi,and pi+1 are highlighted
by green, brown and yellow color. Note that piis not a straight
line given by xi−1
2and xi+1
2, but a broken line given by xi−1
2,xi
and xi+1
2.
we get semi-discrete flowing finite volume scheme
hi+hi+1
2(xi)t=δixi+1 −xi
hi+1
−xi−xi−1
hi+wixi+1 −xi−1
2⊥
,(12)
for i= 1, ..., n−1. In order to perform the time discretization, let us denote by mthe
time step index and by τthe length of the discrete time step. Let us approximate
the time derivative by the finite difference (xi)t=xm+1
i−xm
i
τ. Approximating both
the vector field term and the curvature term explicitly we obtain the fully discrete
explicit scheme
xm+1
i=xm
i+τδm
i
2
hm
i+1 +hm
ixm
i+1 −xm
i
hm
i+1
−xm
i−xm
i−1
hm
i(13)
+τwm
ixm
i+1 −xm
i−1⊥
hm
i+1 +hm
i
for i= 1, ..., n −1, where nis the number of the curve grid points. Due to boundary
conditions we have xm
0=x0
0and xm
n=x0
n. The parameter wm
iis given as follows
wm
i=λvN
m
i=λv(xm
i)·Nm
i=λv(xm
i).xm
i+1 −xm
i−1⊥
hm
i+1 +hm
i
.(14)
4. Numerical experiments. In the first numerical experiment, we show behaviour
of the model (7) in real data. As one can see in Fig. 11, the model (7) and its
numerical discretization (13) cause a regular, almost uniform distribution of grid
points during the evolution and in the segmentation result. Moreover, the numer-
ical scheme (13) can be efficiently implemented and allows real-time segmentation
of habitats.
We note here that in the segmentation of real data we use three optical channels
and thus we consider the vector image intensity function I0defined at the beginning
of section 2.1. However, the only change in the mathematical model (7) and its
numerical discretization (13) is that, instead of the norm of gradient of the scalar
SEMI-AUTOMATIC SEGMENTATION OF NATURA 2000 HABITATS 9
image intensity Iσin Eq. (2) we consider the average of norms of gradients of image
intensities in all three channels. Moreover, in our implementation, the Gaussian
convolution is realized numerically by solving the linear heat equation in one discrete
time step using the implicit scheme. The coefficients k, λ and δare chosen by
the user such that the semi-automatic method would give desired results. These
coefficients can be changed during the segmentation, which is natural for such real
time method.
Figure 11. A discrete segmentation curve evolving to habitat
boundary in a real 3-band Sentinel-2 optical image. The green
color shows trajectories of moving discrete curve points and blue
points represents the result of segmentation of this particular sec-
tion of the habitat border.
In Fig. 11 we present just one section of the segmentation curve and it illustrates
the procedure how a user performs semi-automatic segmentation. The user clicks
the mouse at some correctly chosen point on the habitat boundary and drag the
mouse along the expected habitat boundary. The algorithm always connects the first
clicked point with the last mouse position, constructs the straight line between them
and in real-time adjusts this line to the habitat border by using the numerical scheme
(13). When the user is satisfied with the detected borderline, clicks the mouse again
and that portion of segmentation is finished. By repeating the procedure the user
can continue along the borderline and eventually he clicks on the first point of the
first segment to close the boundary curve. We illustrate this consecutive procedure
in Fig. 12.
4.1. Comparison of discrete curves. After performing the semi-automatic seg-
mentation of habitats in Sentinel-2 data, we compare the segmentation results with
the GPS tracks obtained by botanists in the field. For this quantitative compar-
ison of two curves we use the classical (maximal) Hausdorff distance [12] and the
so-called mean Hausdorff distance, see e.g. [6], which are general tools for comput-
ing distance of curves, surfaces and even more complicated geometrical continuous
or discrete objects (sets). The mean Hausdorff distance dH(A, B ) is given by the
10 KAROL MIKULA ET AL.
Figure 12. An example of the semi-automatic segmentation
showing consecutive building of the segmentation curve (yellow),
the final result is on the bottom right.
SEMI-AUTOMATIC SEGMENTATION OF NATURA 2000 HABITATS 11
following formulae
δH(A, B) = 1
n
n
X
i=1
min
b∈Bd(ai, b),δH(B, A) = 1
m
m
X
i=1
min
a∈Ad(a, bi),(15)
dH(A, B) = δH(A, B) + δH(B, A)
2,(16)
where d(ai, b) is Euclidean distance of two points aiand bfrom the point sets
A={a1, a2, a3, ..., an}and B={b1, b2, b3, ..., bm}, and the maximal Hausdorff
distance dH(A, B), given as
dH(A, B) = max sup
a∈A
inf
b∈B
d(a, b),sup
b∈B
inf
a∈A
d(a, b).(17)
In order to test reliability and usability of the semi-automatic segmentation
method, together 24 areas of riparian forests - Natura 2000 habitat 91F0 Ripar-
ian mixed forests along the great river - were tracked by GPS device. Due to the
variable borders, they are suitable for testing the ability and performance of devel-
oped semi-automatic segmentation tool to detect accurately their shape. In case of
problems in the field, mainly due to flooded parts of forests where it was impossible
to walk around, some GPS tracks were corrected in the Google Earth Pro soft-
ware. The mean Hausdorff distance is in average 11.48 m which is very close to the
pixel resolution (10m) of Sentinel-2 data. This means that by the semi-automatic
segmentation we are able to detect habitat borders as accurately as the image reso-
lution allows. Maximal Hausdorff distance is in average about 58m, what represents
5-6 pixels. Some differences can be found only in areas with cotone zones where
tree dominated riparian forests are connected to surrounding meadows or fields by
shrub dominated zone, see Figs. 13 and 14.
Next, 18 areas of Natura 2000 habitat 4070* Bushes with Pinus mugo and Rhodo-
dendron hirsutum (Mugo-Rhododendretum hirsuti) were tracked in various moun-
tain ranges in Slovakia (Mal´a Fatra Mts., Z´apadn´e Tatry Mts., N´ızke Tatry Mts.,
Choˇcsk´e vrchy Mts. and Oravsk´e Beskydy Mts.). Bushes with Pinus mugo usu-
ally form large areas of diversified shape, discontinued by avalanche gullies, small
mountain creeks or by glacially formed moraines. Considering this fact, the correct
semi-automatic segmentation is a challenging task. Moreover, the field mapping
of habitat borders in high-altitude rugged terrain is very complicated and time-
consuming, so using the satellite image segmentation methods seem to be very ef-
ficient and promising way of monitoring this habitat. In general, we observed that
the mean Hausdorff distances of GPS tracked and semi-automatically segmented
borders of bushes with Pinus mugo areas are also close to the pixel resolution of
Sentinel-2 data, 13.9m in average of all 18 areas. The maximal Hausdorff distances
are in general bigger than those observed in the areas of riparian forests. Bushes
with Pinus mugo grow on large areas connected with the mountain spruce forests.
Some extreme values of the maximal Hausdorff distance are caused by the “ecotone
zone” where botanist in the field takes subjective decision about the habitat border,
for illustration of such problematic areas see Figs. 15 and 16.
5. Conclusions. In this paper we proposed a semi-automatic segmentation method
of Natura 2000 habitats in Sentinel-2 optical data. We discussed the segmentation
curve velocity vector field design and described the curve evolution numerical al-
gorithm using the explicit scheme allowing real-time boundary tracking. We also
12 KAROL MIKULA ET AL.
Figure 13. Semi-automatic segmentation (yellow) and GPS track
(light-blue) with almost exact overlap. The maximal Haussdorff
distance is 62.1m and the mean Hausdorff distance is 14.0m in this
case, which means that we obtained almost the pixel resolution
(10m) accuracy.
presented numerical experiments using Sentinel-2 optical images. The comparison
of areas obtained by the semi-automatic segmentation and by the GPS tracking in
the field shows that we can get the accuracy compared to the pixel resolution of
Sentinel-2 data. Some further improvements of the numerical model, e.g. asymptot-
ically uniform curve grid point redistribution, will be treated in the future together
with the implementation of the semi-implicit scheme. Here the real-time perfor-
mance of the method will be also a discretization quality criterion.
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SEMI-AUTOMATIC SEGMENTATION OF NATURA 2000 HABITATS 13
Figure 14. An example of a complicated border of the riparian
forest. We compare the semi-automatic segmentation (yellow) and
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employing the Sentinel-2 optical data.
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Received January 2019; revised September 2019.
E-mail address:mikula@math.sk
E-mail address:jozo.urban@gmail.com
E-mail address:michalkollar27@gmail.com
E-mail address:ambroz.martin.ml@gmail.com
E-mail address:ivan.jarolimek@savba.sk
E-mail address:jozef.sibik@savba.sk
E-mail address:maria.sibikova@savba.sk
14 KAROL MIKULA ET AL.
Figure 15. The locality with the highest, 413.3m, maximal Haus-
dorff distance between semi-automatically segmented and GPS
tracked curves among bushes with Pinus mugo tested areas, here
also the mean Hausdorff distance was the highest, 44.8m. On the
North-West habitat border, we can see the “ecotone zone” that
was included during field tracking (light-blue) and excluded by us-
ing the semi-automatic segmentation (yellow).
Figure 16. The locality dominated by Pinus mugo with the “eco-
tone zone” that was included during the field tracking (light-blue)
and excluded by using the semi-automatic segmentation (yellow).
The mean Hausdorff distance is 19.1m and the maximal Hausdorff
distance is 171.0m.
