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Sparsity-Driven Digital Terrain Model Extraction

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We here introduce an automatic Digital Terrain Model (DTM) extraction method. The proposed sparsity-driven DTM extractor (SD-DTM) takes a high-resolution Digital Surface Model (DSM) as an input and constructs a high-resolution DTM using the variational framework. To obtain an accurate DTM, an iterative approach is proposed for the minimization of the target variational cost function. Accuracy of the SD-DTM is shown in a real-world DSM data set. We show the efficiency and effectiveness of the approach both visually and quantitatively via residual plots in illustrative terrain types.
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SPARSITY-DRIVEN DIGITAL TERRAIN MODEL EXTRACTION
Fatih Nar1, Erdal Yilmaz2, Gustau Camps-Valls3
1Konya Food and Agriculture University, Konya, Turkey; 2Zibumi Studios, Ankara, Turkey
3Image Processing Lab (IPL), Universitat de Val`
encia, Val`
encia, Spain
ABSTRACT
We here introduce an automatic Digital Terrain Model (DTM)
extraction method. The proposed sparsity-driven DTM ex-
tractor (SD-DTM) takes a high-resolution Digital Surface
Model (DSM) as an input and constructs a high-resolution
DTM using the variational framework. To obtain an accurate
DTM, an iterative approach is proposed for the minimization
of the target variational cost function. Accuracy of the SD-
DTM is shown in a real-world DSM data set. We show the
efficiency and effectiveness of the approach both visually and
quantitatively via residual plots in illustrative terrain types.
Index Termsdigital surface model, digital terrain
model, sparsity, variational inference
1. INTRODUCTION
A Digital Terrain Model (DTM) is an elevation map of bare
ground where man-made objects (buildings, vehicles, etc.) as
well as vegetation (trees, bushes, etc.) are removed from the
Digital Surface Model (DSM) [1]. In Fig.1,grepresents sur-
face elevations hence DSM, frepresents terrain elevations
hence DTM, and trepresents terrain vs non-terrain classifica-
tion (t= 1 for terrain regions, t= 0 for non-terrain regions).
Fig. 1. DSM versus DTM.
DTMs are useful for extracting man-made and vegetation
objects, extracting terrain parameters, precision farming and
forestry, planning of new roads and railroads, visualization
and simulation of the 3D world, modeling physical phenom-
enas, such as water flow or mass movement, rectification of
aerial photography or satellite imagery, and many other Ge-
ographic Information Systems (GIS) tasks [15]. However,
This work was supported by the Scientific and Technical Research Coun-
cil of Turkey (TUBITAK), Grant Number: TUBITAK-BIDEB-2219. GCV
was funded by the European Research Council (ERC) under the ERC-CoG-
2014 SEDAL project (grant agreement 647423). Authors would like to thank
A. Ozgur and M. Ergul for their help during the preliminary investigation.
manual preparation of a DTM using ground measurements is
expensive and time consuming [2]. Certainly, the definition
of DTM is often elusive and controversial. Thus, automatic
extraction of a DTM from an automatically obtainable DSM
is a reasonable and often preferred alternative, even though it
poses important challenges to be addressed [6,7].
Several approaches to derive DTM exist in the literature.
In [2], a modified linear prediction technique followed by
adaptive processing is proposed for DTM extraction. In [3],
a progressive morphological filter is developed to preserve
ground while removing non-ground objects. An alternative
approach was presented in [4], where a variational approach is
proposed for the semiautomatic generation of the DTM. More
recently, in [8], the most contrasted connected-components
are extracted to generate DTM from LiDAR data, while in [5],
the DSM is segmented into uniform regions and interpolation
is applied between selected regions. Lately, in [9], 2D empir-
ical mode decomposition is proposed for DTM generation.
In this work we propose, a methodology based on the vari-
ational approach introduced in [4]. Our proposed methods
follows an iterative procedure that ‘peels the onion’ according
to a target cost function under sparsity-preserving constraints.
Accuracy of the derived DTM will be shown in a real-world
DSM data set, and analyzed both qualitatively and quantita-
tively in illustrative terrain types.
The remainder of the paper is organized as follows. sec-
tion 2briefly reviews the proposed method used in this work.
Section 3first describes the dataset collected, and then gives
an empirical evidence of performance both visually and quan-
titatively. We conclude in section 4with some remarks and an
outline future work.
2. PROPOSED DTM EXTRACTION METHOD
DTM can be constructed from a DSM by interpolating the
elevation values in the non-terrain cells using the elevations
of the nearby terrain cells [5]. However, manual delineation
of the cells (as terrain versus non-terrain) is a tedious task [2]
error-prone, and automatic classification is challenging [10,
11]. On top of all this, determining the elevation values for
the non-terrain cells is an ill-posed scattered data interpolation
problem, where it is also sensitive to errors in the terrain non-
terrain boundaries [1].
Inspired by [4], to handle the above mentioned issues, we
propose the minimization of a similar variational cost func-
tion, yet by using a novel iterative approach and numerical
solver for the construction of DTM. The pseudocode is given
in Algorithm 1: firstly the DTM (f) is initialized with ele-
vations of the DSM (g), then a terrain indicator map (t) is
updated which is followed by an update of the terrain eleva-
tion values in an iterative manner. The algorithm is iterated
with the previous solution until it convergences or a maximum
number of iterations nmax is reached. In this study, we use a
regular grid format for representing the DSM and the DTM,
where each grid cell stores a floating number for its elevation
value.
Algorithm 1 DTM Extraction Pseudo-code
1: Input: g, nmax
2: Initialize: f(1) g
3: for n= 1 to nmax do
4: Update terrain indicator map t(n)using f(n)and g
5: Update terrain elevations f(n+1) using t(n),f(n),g
6: Check for convergence using f(n)and f(n+1)
7: end for
8: return f
If DSM is smoothed, then elevations of non-terrain ob-
jects will become lower. However, this simple approach also
leads to an increase in the elevations for the terrain regions
(see Fig. 2, top). In order to prevent this problem, smooth-
ing can be applied onto the DSM using the prior knowledge
f6g. This prior knowledge can be included in the mini-
mization functional as an inequality constraint, and thus can
be combined into a smoothing operation by the minimization
of a cost function which will prevent the height increase in
terrain regions. (see Fig. 2, middle). In Fig. 2, solid blue line
is surface (g) where dotted red line is smoothed surface (f).
Fig. 2. DSM versus DTM.
If we define fas the smoothed version of the surface g,
then the terrain indicator map for each cell can be defined as
below (see Fig. 2, bottom):
tp= 1 min 1
Tng
(gpfp),1,(1)
where pis the cell index number, tis the terrain indicator
map, gis the existing DSM, fis the smoothed DSM (rough
DTM), Tng is a terrain threshold (set to 0.5for simplicity).
In this study, the proposed variational cost function that is
minimized to obtain terrain elevations (f) by smoothing the
surface elevations (g) using the prior (f6g) and the terrain
indicator map (t) as following:
J(f) = X
p=1
tp((|fpgp|+ 1)21) + λ|(f)p|
w.r.t. fp6gp,
(2)
where pis the cell index number, tis the terrain indicator
map, gis the existing DSM, fis the DTM to be obtained, λ
is a positive value determining smoothing level, and is the
gradient operator. The first term is the data fidelity term that
ensures keeping fsimilar to gby using an 1-norm penalty
when the difference between fand gis small, and an 2-norm
as the difference gets larger. The second term is the total vari-
ation (TV) regularization term that implies a penalty on the
changes in image gradients using an 1-norm, thus preserving
details while enforcing smoothness [12]. Higher smoothing
effects are obtained by an increasing the λvalue. The con-
straint, fp6gp, prevents terrain elevations being higher than
surface elevations, as common sense dictates. Here, tindi-
cates a fuzzy membership (06tp61) such that tp= 0 for
non-terrain cells and tp= 1 for terrain cells. As tpgets closer
to 0, data fidelity term vanishes and only TV-regularization
(TV diffusion) term remains, thus cost function acts as a scat-
tered data interpolator. As tpgets closer to 1, data fidelity
term becomes active and surface is preserved more.
2.1. Minimization of the cost function
After doing algebraic manipulations and taking the constraint,
fp6gp, into the cost function using the penalty method with
λpas penalty multiplier, equation (2) becomes as below:
J(f) = X
p=1
tp((fpgp)2+ 2|fpgp|)
+λ|(f)p|+λpmax(fpgp,0)
(3)
In equation (3), maximum function (max) returns zero
penalty if fp6gpand it returns a penalty proportional to
λpotherwise. λpshould be increased as the smoothing (λ)
increases, thus we set λp= 0.5λ.
Although equation (3) is convex, absolute and max func-
tions are non-differentiable which makes the minimization
difficult. Inspired from [13,14], we set ˆ
fpas a proxy for fp
to be able to approximate non-differentiable terms in equa-
tions (4), (5), and (6). First, the absolute function in the data
fidelity term is approximated as below:
|fpgp| ≈ dp(fpgp)2
dp= (|ˆ
fpgp|+ε)1,(4)
where εis a small positive constant. In this study, ε= 0.1
is used for all the experiments. Second, the absolute value of
the gradient operator is approximated as:
|(f)p|=|(xf)p|+|(yf)p|
wx,p(xf)2
p+wy,p(yf)2
p
wx,p = (|(xˆ
f)p|+ε)1
wy,p = (|(yˆ
f)p|+ε)1
(5)
Finally, the max-function is approximated as:
max(fpgp,0) hp(fpgp)2
hp=sgn(max( ˆ
fpgp,0))dp
(6)
where sgn is the sign function. The approximated cost func-
tion in equation (7) is accurate around ˆ
fpso it must be solved
in an iterative manner [14], where nis the iteration number.
This cost function has a different data fidelity term and numer-
ical minimization approach and it is also iterative comparing
to two-phase solution proposed in [4].
J(n)(f) = X
p=1
tp((fpgp)2+ 2dp(fpgp)2)
+λ(wx,p(xf)2
p+wy,p(yf)2
p)
+λphp(fpgp)2
(7)
Equation (7) can be cast in the matrix-vector form as below:
J(n)(vf) =(vfvg)>+ 2(vfvg)>DT(vfvg)
+λ(v>
fCx
>WxCxvf+v>
fCy
>WyCyvf)
+λp(vfvg)>H(vfvg),
(8)
where vg,vf, and vˆ
fare vector forms of gp,fp, and ˆ
fp;Dis
a diagonal matrix formed of dp;Tis a diagonal matrix with
entries tp,His a diagonal matrix formed of hp;Wx,Wyare
diagonal matrices with entries wx,p,wy,p; and Cx,Cyare the
Toeplitz matrices as the forward difference gradient operators
with zero derivatives at the right and bottom boundaries.
Equation (8) is quadratic; and hence taking its derivatives
with respect to vfand equating to zero yields its global mini-
mum. This leads to the below sparse linear system:
Av(n+1)
f=b
A=R+λ(Cx
>WxCx+Cy
>WyCy)
b= (R+λpH)vg,
(9)
where R=T(2D+I)as Ibeing the identity matrix. Here,
iteration number is nfor the A,R,T,D,H,Wx,Wyma-
trices and bvector unless it is explicitly stated.
2.2. DTM Extraction Algorithm
The DTM extraction method is provided in Algorithm 1. De-
tails on the terrain indicator map update and terrain elevations
update approaches are given therein.
In Algorithm 2, preconditioned conjugate gradient (PCG)
with incomplete Cholesky preconditioner (ICP) is used as an
iterative solver to solve the linear system at line 9, where the
maximum number of PCG iterations is set to 103and conver-
gence tolerance is set to 103.
Algorithm 2 DTM Extraction Algorithm
1: Input: g, λ = 5, nmax = 104, Ctoler ance = 103
2: vgg,vfg,λp0.5λ,Tng 0.5,ε0.1
3: for n= 1 to nmax do
4: Update terrain indicator map:
5: vt=
1min((vgvf)/Tng,
1)
6: Update terrain elevations:
7: vˆ
fvf
8: Construct Wx,Wy,T,D,R,H,A, and b
9: solve Avf=b
10: vfmin(vf, vg)force the f6gconstraint
11: Check for the convergence:
12: if kvfvˆ
fk< Ctolerance then break the loop
13: end for
14: return fwhere fvf
In [4], large smoothing factor was used to determine the
terrain indicator map, and then the algorithm was executed
again with a smaller smoothing factor. Alternatively, in our
approach, a small smoothing factor is used and the terrain
indicator map is iteratively updated, which leads to a better
preservation of details in terrain regions. Therefore, in Al-
gorithm 2, terrain elevations (f) are initialized as surface el-
evations (g) and then both the terrain elevations (f) and the
terrain indicator map (t) are iteratively refined.
Fig. 3. Evolution of terrain elevations (f) and terrain indicator
map (t) for the Algorithm 2on 1-dimensional data.
3. EXPERIMENTAL RESULTS
3.1. Data Collection and Characteristics
We applied the proposed method to Cerkes village dataset to
illustrate performance in a large terrain with wide variety of
features (i.e. flat regions, hills, rivers, buildings, utility poles,
cars, trees, etc.). This dataset covers 11 km2area which is ob-
tained using photogrammetry techniques, where raster image
has 5 centimeter pixel resolution and DSM has 5 centimeter
pixel-spacing. Coverage of Cerkes village (at Cankiri city of
Turkey) dataset as a bounding box is given as below:
N4049047.0700 E3252022.1700 to N4047054.9800 E3254049.6100
3.2. Visual results
Fig. 4shows rasters (top), DSMs (middle), and extracted
DTMs (bottom) of 3 subregions in the Cerkes village dataset.
As seen in Fig. 4, man-made objects and vegetations are suc-
cessfully removed from the terrain and these regions are also
interpolated smoothly. Thus, it can be noted that the proposed
method is able to extract bare earth successfully.
Fig. 4. Cerkes data: (a) raster, (b) DSM, (c) extracted DTM.
3.3. Numerical evaluation
A numerical evaluation was conducted using residual his-
togram for Cerkes village dataset (in 11 km2) where mean
residual is 0.24 cm, median residual is 0.1cm, and mean
squared error is 1.19 cm. The residual histogram in Fig. 5
shows that the proposed method performs well for a real
world data. Note that, frequencies of residuals are shown in
log10-scale to prevent zero residual dominating the plot.
Fig. 5. Residual histogram of the proposed method.
4. CONCLUSIONS
In this study, we presented an automatic DTM extraction
method that iteratively estimates terrain indicator map and
terrain elevations. Experiments show that proposed method
can produce an accurate DTM for the given high-resolution
DSM where wide variety of non-terrain objects exist on the
terrain with various slopes. Future work will consider adding
asymmetry constraints and doing more experiments in re-
gions showing additional characteristics.
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... On the other hand, there are different methods based on finding the ground points to interpolate the remaining surface. These methods mostly depend on rulebased techniques that uses slope-based methods, morphological approaches (Weidner & Förstner, 1995), multi-directional scanlines (Perko, Raggam, Gutjahr, & Schardt, 2015), the cloth simulation (Zhang, et al., 2016) and the sparsity driven algorithms (Nar, Yilmaz, & Camps-Valls, 2018). Even there is a big progress achieved in automated DTM extraction methods, there are still some misclassifications in features such as cliffs, hills or small rocks. ...
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