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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 Terms—digital 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 [1–5]. 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

(gp−fp),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((|fp−gp|+ 1)2−1) + 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((fp−gp)2+ 2|fp−gp|)

+λ|(∇f)p|+λpmax(fp−gp,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:

|fp−gp| ≈ dp(fp−gp)2

dp= (|ˆ

fp−gp|+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(fp−gp,0) ≈hp(fp−gp)2

hp=sgn(max( ˆ

fp−gp,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((fp−gp)2+ 2dp(fp−gp)2)

+λ(wx,p(∂xf)2

p+wy,p(∂yf)2

p)

+λphp(fp−gp)2

(7)

Equation (7) can be cast in the matrix-vector form as below:

J(n)(vf) =(vf−vg)>+ 2(vf−vg)>DT(vf−vg)

+λ(v>

fCx

>WxCxvf+v>

fCy

>WyCyvf)

+λp(vf−vg)>H(vf−vg),

(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 10−3.

Algorithm 2 DTM Extraction Algorithm

1: Input: g, λ = 5, nmax = 104, Ctoler ance = 10−3

2: vg←g,vf←g,λp←0.5λ,Tng ←0.5,ε←0.1

3: for n= 1 to nmax do

4: Update terrain indicator map:

5: vt=

1−min((vg−vf)/Tng,

1)

6: Update terrain elevations:

7: vˆ

f←vf

8: Construct Wx,Wy,T,D,R,H,A, and b

9: solve Avf=b

10: vf←min(vf, vg)force the f6gconstraint

11: Check for the convergence:

12: if kvf−vˆ

fk∞< Ctolerance then break the loop

13: end for

14: return fwhere f←vf

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:

N40◦49047.0700 E32◦52022.1700 to N40◦47054.9800 E32◦54049.6100

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.

5. REFERENCES

[1] Z. Li, C. Zhu, and C. Gold, Digital Terrain Modeling: Princi-

ples and Methodology, CRC Press, 2004.

[2] H.S. Lee and N.H. Younan, “DTM extraction of LiDAR returns

via adaptive processing,” IEEE Transactions on Geoscience

and Remote Sensing, vol. 41, no. 9, pp. 2063–2069, 2003.

[3] Keqi Zhang, Shu-Ching Chen, D. Whitman, Mei-Ling Shyu,

Jianhua Yan, and Chengcui Zhang, “A progressive morpho-

logical ﬁlter for removing nonground measurements from air-

borne LiDAR data,” IEEE Transactions on Geoscience and

Remote Sensing, vol. 41, no. 4, pp. 872–882, 2003.

[4] M. Grabner A. Klaus M. Unger, T . Pock and H. Bischof, “A

variational approach to semiautomatic generation of digital ter-

rain models,” in International Symposium Advances in Visual

Computing, pp. 1119–1130. 2009.

[5] Charles Beumier and Mahamadou Idrissa, “Digital terrain

models derived from digital surface model uniform regions in

urban areas,” International Journal of Remote Sensing, vol. 37,

no. 15, pp. 3477–3493, 2016.

[6] G. Sithole and G. Vosselman, “Experimental comparison of

ﬁlter algorithms for bare-earth extraction from airborne laser

scanning point clouds,” ISPRS Journal of Photogrammetry and

Remote Sensing, vol. 59, no. 1, pp. 85–101, 2004.

[7] Joachim Hohle and Michael Hohle, “Accuracy assessment of

digital elevation models by means of robust statistical meth-

ods,” ISPRS Journal of Photogrammetry and Remote Sensing,

vol. 64, no. 4, pp. 398–406, 2009.

[8] D. Mongus and B. Zalik, “Computationally efﬁcient method

for the generation of a digital terrain model from airborne Li-

DAR data using connected operators,” IEEE Journal of Se-

lected Topics in Applied Earth Observations and Remote Sens-

ing, vol. 7, no. 1, pp. 340–351, 2014.

[9] A. H. Ozcan and C. Unsalan, “LiDAR data ﬁltering and DTM

generation using empirical mode decomposition,” IEEE Jour-

nal of Selected Topics in Applied Earth Observations and Re-

mote Sensing, vol. 10, no. 1, pp. 360–371, 2017.

[10] L. Bruzzone J. Munoz-Mari and G. Camps-Valls, “A support

vector domain description approach to supervised classiﬁca-

tion of remote sensing images,” IEEE Transactions on Geo-

science and Remote Sensing, vol. 45, no. 8, pp. 2683–2692,

2007.

[11] G. Camps-Valls, D. Tuia, L. G´

omez-Chova, S. Jim´

enez, and

J. Malo, “Remote sensing image processing,” Synthesis Lec-

tures on Image, Video, and Multimedia Processing, vol. 12, pp.

1–194, 2012.

[12] Leonid I Rudin, Stanley Osher, and Emad Fatemi, “Nonlinear

total variation based noise removal algorithms,” Physica D:

Nonlinear Phenomena, vol. 60, no. 1, pp. 259–268, 1992.

[13] C. Ozcan, B. Sen, and F. Nar, “Sparsity-driven despeckling for

SAR images,” IEEE Geoscience and Remote Sensing Letters,

vol. 13, no. 1, pp. 115–119, 2016.

[14] F. Nar, A. Ozgur, and A. N. Saran, “Sparsity-driven change

detection in multitemporal SAR images,” IEEE Geoscience

and Remote Sensing Letters, vol. 13, no. 7, pp. 1032–1036,

2016.