Content uploaded by Dominik Kellner
Author content
All content in this area was uploaded by Dominik Kellner on Jun 29, 2016
Content may be subject to copyright.
Abstract—The online observation using high-resolution radar
of a scene containing extended objects imposes new
requirements on a robust and fast clustering algorithm. This
paper presents an algorithm based on the most cited and
common clustering algorithm: DBSCAN [1]. The algorithm is
modified to deal with the non-equidistant sampling density and
clutter of radar data while maintaining all its prior advantages.
Furthermore, it uses varying sampling resolution to perform an
optimized separation of objects at the same time it is robust
against clutter. The algorithm is independent of difficult to
estimate input parameters such as the number or shape of
available objects. The algorithm outperforms DBSCAN in terms
of speed by using the knowledge of the sampling density of the
sensor (increase of app. 40-70%). The algorithm obtains an even
better result than DBSCAN by including the Doppler and
amplitude information (unitless distance criteria).
I. I
NTRODUCTION
n the automotive field, high resolution radar is required
for roadside detection, lane prediction and classification of
objects to make an intelligent interpretation of a traffic scene
possible. In these systems, vehicles appear as laterally and
longitudinally extended objects caused by reflections from
the road surface. In radar systems, the complete bottom side
of the car can be seen, allowing an accurate recognition and
classification into different classes (car, van and truck). [2]
With high-resolution radar, an extended object results in
more than one observation. A clustering algorithm is
required to associate the single reflections (observations)
with an unknown number of different objects. Otherwise, the
following signal processing steps must deal with a large
amount of data. E.g., conventional tracking algorithms are
optimized for point targets. Dealing with a large amount of
observations per object decreases their speed significantly
and results in association problems. An additional step is
required to sort and merge tracks. [3]
These steps can be avoided by using an adequate
clustering algorithm. But most common cluster algorithms
are based on an equidistant sampling density (recorded
samples per unit distance). The same object results in the
same number and distribution of points independent of its
position inside the field of view.
Manuscript received January 24, 2012.
D. Kellner is with Departement of Measurement, Control and
Microtechnology, University of Ulm, Germany. Contact:
dominik.kellner@uni-ulm.de
J. Klappstein is with Daimler AG, Ulm, Germany
K. Dietmayer is with Departement of Measurement, Control and
Microtechnology, University of Ulm, Germany
In contrast, the observation of a target in a radar signal is
determined by an azimuth angle (θ) and the distance to the
sensor (r). Its position in Cartesian coordinates can be
calculated using trigonometric functions (Fig1-left). Both
variables have a fixed sampling resolution, with r in units of
length and θ in units of angle. The result is a non-equidistant
sampling in Cartesian coordinates resulting from the
trigonometric calculation of the x-y values. Therefore, the
minimal azimuth distance between two points increases with
range. Due to the fixed sampling resolution, a grid
representation consisting of range and azimuth cells
(Fig1-right) is easy to apply.
For the mentioned applications in the automotive field,
[2] proposes a 77 GHz radar with a resolution of 1° in the
azimuth direction and 1m in the range direction. This results
in a strong variation of the sampling density in Cartesian
coordinates, as seen in a simple example (Fig. 2).
The shape of the side of a car (length 5m, width 2m) at a
distance of 3 meters causes over 150 possible observations
compared to only 9 possible observations at a distance of 80
meters. Because of the fixed metric range resolution, the
number of affected range cells is equal while the number of
affected azimuth cells differs significantly. For the car at 3m,
59 azimuth cells are affected compared to only 3 cells for a
car at 80m.
The CFAR method presented in [5] is used to suppress
clutter and remove no-target points. The targets result in the
points used for the clustering algorithm.
This paper is outlined as follows: Section II gives an
overview of the most common and automotive-specific
clustering algorithms and their disadvantages compared to a
non-equidistant sampling density. In Section III, the standard
Grid-Based DBSCAN for Clustering Extended Objects
in Radar Data
Dominik Kellner, Jens Klappstein and Klaus Dietmayer
I
Fig. 1: Data output of the sensor (right) in r- and θ-grid and data
transformation in common Cartesian (x-y) coordinates obtained from
trigonometric functions
DBSCAN algorithm is analyzed in terms of clustering
high-resolution radar data. The grid-based DBSCAN
algorithm is presented in section IV and its advantages
compared to DBSCAN are discussed. In Section V, the
execution time on real data is compared. Finally, a
conclusion is presented in Section VI.
II. O
VERVIEW
C
LUSTERING
A
LGORITHM
In general, clustering divides the data into different classes
(clusters), with data in the same cluster showing a great
similarity. The differences for data in different clusters are
greater. In this paper, the similarity is based on the spatial
information of each point (x-y or r-θ).
There are two main categories of cluster algorithms:
hierarchical and partitioned. A good overview of algorithms
with regards to radar data is presented in [6]. Only the most
common algorithms and their disadvantages in terms of
clustering high-resolution radar are discussed here:
Hierarchical algorithms, such as CURE, BIRCH, and
CHAMELEON, can basically discover clusters of any shape
and size. But the algorithm’s complexity regarding space and
time is high. Furthermore, the merge and split process needs
a constant distance criterion, which is hard to estimate for a
non-equidistant sampling density. [7]
Some of the most common algorithms in clustering are the
k-mean based methods. They rely on prior knowledge of the
number of clusters present and each point in the dataset must
be assigned to one of the clusters. In radar datasets, there are
an unknown number of extended objects and reflection
centers, so this input parameter is hard to estimate [8]. Radar
datasets usually contain clutter, so even those points are
assigned to one of the clusters, deteriorating the performance
of the algorithm. There are density approaches to exclude
potential outliers, but they are time-intensive and can have
restrictions on the noise distribution. [9]
All methods which assume, for example, an elliptical
cluster, e.g. presented in [6], can’t be used due to the varying
object shapes. Pixel- and segment-based segmentation are
not suitable because neither the amplitude nor Doppler value
is significant enough to precisely cluster an object.
Furthermore, the sampling density might be too small for an
edge search [10]. DBSCAN, which uses a fixed density,
can’t be used due to the high variation in density. Variable
density modified DBSCAN algorithms like OPTICS [11] are
too time-consuming. SDDC [6], or the region growing
approach in [6] handle a non-equidistant density just between
features (like x- and y- value). But these algorithms fail if the
feature itself has a high density variation.
III. DBSCAN
F
OR AUTOMOTIVE
R
ADAR
A. DBSCAN
The general idea of density-based methods is to continue
growing a cluster as long as the density in the neighborhood
exceeds some threshold. This density criterion can be
described as follows: “for each point of a cluster the
neighborhood of a given radius has to contain at least a
minimum number of points, i.e. the density in the
neighborhood has to exceed some threshold” [1]. All points
in a cluster which meet this criterion are marked as core
points. A point is marked as a border point if it doesn’t meet
this criterion but is inside the search area of a core point. All
other points are marked as outliers. The procedure of
growing clusters and marking points remains unchanged for
all algorithms presented. Therefore, only the density criterion
is the subject of this paper.
B. DBSCAN on radar data
DBSCAN uses a fixed density threshold for the creation of
a cluster. This means, each observation is examined if there
are at least k observations within a radial search radius ε.
Applying DBSCAN on a dataset with a non-equidistant
sampling density as mentioned in section I (Fig 2.) induces
some disadvantages which are shown in Fig.3:
1) Presence of clutter
There is a high sampling density in Cartesian
coordinates close to the sensor, caused by the small
azimuth distance between possible observations. With
this large number of possible observations in the search
radius ε, even widely distributed clutter is clustered to
objects.
2) Limited Range
Objects which are far away from the sensor aren’t
clustered because the search radius ε is smaller than the
sampling resolution or the number of possible
observations is smaller than the amount of required
observations (k). In general, the number of possible
observations in the search radius decreases with the
range. The result is that small objects are only clustered
close to the sensor.
3) Separation resolution
The ability to separate two objects in the azimuth
Fig. 2: Difference in number of possible observations for a car with
dimensions 5x2m at a distance of 3m and 80m in front of a radar
sensor (∆r = 1m, ∆θ = 1°) in Cartesian coordinates (top) and in r-θ
grid (button)
direction varies significantly with the range. Close to
the sensor, the search radius includes a large number of
azimuth cells, so that a separation of two closely spaced
objects is not possible.
4) Object orientation
Range and azimuth direction are not treated equally by
DBSCAN. For close objects, the number of possible
observations in the azimuth direction is significantly
higher than in the range direction. Therefore, the
algorithm prefers small and long objects in the azimuth
direction and discriminates objects in the range
direction.
Each of the mentioned disadvantages can be avoided
separately by changing the input parameters k and ε. But they
have a strong negative influence on each other. E.g., if the
search radius is increased to cluster objects far away (2),
more clutter will be clustered together (1) and the separation
resolution of two objects decreases (3).
A possible improvement is the introduction of a variable
threshold for the number of observations: k(x,y). This
parameter must show a high dependency on the number of
possible observations inside the search radius. The threshold
k has to decrease in the range direction. The results would be
that close clutter isn’t clustered (1) anymore. Objects far
away (2), which have less possible observations in their
search radius, are clustered due to the significant smaller
threshold k.
Nevertheless, the separation resolution (3) and the
disparity of both directions (4) can’t be improved. Therefore,
it is necessary to introduce spatial-variable parameters for
both search radius and number of observations: ε(x,y) and
k(x,y). A simple and fast approach of a spatial density
criterion based on the r- and θ-grid (Fig. 1) is presented in
the following sections.
IV. G
RID
B
ASED
DBSCAN
A. Data representation in grid
DBSCAN needs the x- and y-position of each point to
calculate the distance between the examined point, and the
rest of the dataset to determine the points inside the search
radius ε. The modified grid-based DBSCAN algorithm is
based on neighborhood relations and can use the r- and
θ-cell-information as a grid (Fig 1-right).
To deal with the non-equidistant sampling density, it is
unavoidable to obtain spatial dependant variables for the
search area and for the number of required points. The grid-
based DBSCAN uses only the ratio c between the radial and
angular distance for each point. Since the range distance is
always constant, this parameter is sufficient to calculate the
spatial sampling density:
))sin()(sin(
2
1,,,1,
,
,−+
−+−
∆
=
jijijiji
ji
ji
r
r
c
θθθθ
(1)
with: i,j index of grid in r / θ - direction
ji
r
,
radial distance
r
∆
radial resolution (constant)
ji,
θ
azimuth angle
This is a sensor-specific ratio, which can be calculated in
advance and stored in a look-up-table. During the clustering
process, this local ratio is used to determine the optimal
number of relevant r- and θ-neighbors (search area), as
discussed in the following section.
B. Determination of spatial density criterion compared to
DBSCAN
As discussed in III) for DBSCAN, the major difference
between DBSCAN and grid-based DBSCAN is the density
criterion. In this section, the calculation of the density
criterion of DBSCAN is presented first and then the
modification for grid-based DBSCAN.
DBSCAN starts the process of testing this criterion on all
observations by determining the x-y values of all
observations (Fig. 4). Due to the fixed sampling, these values
are sensor-specific and can be calculated in advance, A
position-dependant search radius ε(x,y) and threshold k(x,y)
has to be determined at runtime for each current observation.
Then the Euclidean distance to all other observations is
calculated. In the final step, the number of observations
inside the search radius is compared to the threshold k(x,y).
Compared to the grid-based DBSCAN algorithm (Fig 5.),
only the ratio c is calculated in advance for each cell. At
runtime, this ratio is used to determine the search area in
cells. The area has a constant value in r-direction (constant
width h) and depends on c (width w(c)) only in θ-direction.
The threshold k is calculated as the simple percentage of
possible observations inside the search area.
Fig. 3: Effects of original DBSCAN algorithm on two close objects
(green, black), one object far away (red) and clutter (cyan)
Fig. 5: Calculation of the density criterion using grid-based DBSCAN
(left), with the search area specified by the parameters h and w and the
adaptive threshold k. The equal representation in Cartesian coordinates
is on the right side.
Fig. 4: Calculation of the density criterion using DBSCAN with an
adaptive search radius ε and adaptive threshold k in Cartesian coordinates
The advantages of grid-based DBSCAN is a simple
comparison in both directions to determine all observations
inside the search area. Compared to the calculation of the
Euclidean distance between all observations, it is
significantly less time-intensive. Furthermore, the data
format can be simple integers, compared to float values in
DBSCAN. In section V the computation time is examined on
real-world data.
C. Cluster size and separation of clusters
The only mandatory input parameter is the ratio of present
observations to possible observations for creating a cluster.
This parameter depends strongly on the sensor and the
desired cluster size.
If no other input parameter is set, the search area is equal
to a constant search radius in DBSCAN. Therefore, the
second input parameter f (default 1) adjusts the search
distance in θ-direction. The search distance in θ-direction
can be calculated as follows:
ji
ji
cf
cw
,
,
1
)( ⋅
=
(2)
For example, using the factor f = 2 results in a circular
search area for an equal r- and θ-distance. If the θ-distance
gets smaller than the r-distance, the search area becomes an
ellipse which gets narrower and narrower for points closer to
the sensor. The result is an improved separation of objects in
the θ-direction. An example for f = 2 is shown in Fig 6.
Typically, the search area in the r-direction includes one
point in each direction. This is the minimum value to cluster
extended objects. A larger value is normally not applied
since the r-distance is quite large. But with parameter g,
more than one possible observation in the r-direction can be
taken into account. Parameter g adjusts the number of
examined r-cells. Then the search distance in the θ-direction
can be calculated as follows:
ji
ji
cf
g
cw
,
,
)( ⋅
=
(3)
D. Advantages of DBSCAN in terms of the clustering result
Regarding the problems mentioned in III-A with respect to
the modifications of the density criterion in the grid-based
version (Fig 7):
1) Presence of clutter
Clutter close to the senor isn’t clustered due to a
percentage-based threshold and a small search area in
the azimuth direction.
2) Limited Range
The search area always contains at least one cell on
each side and in each direction. Furthermore, the
number of required observations is a percentage-based
threshold and therefore depends on the possible
observations in this area.
3) Separation resolution
Fig. 6: Influence of different sampling density in the θ-direction of the
search area (expressed by w – number of azimuth cells in each direction)
of grid-based DBSCAN (f = 2). If the sampling resolution is equal in both
directions (r and θ), the result is a constant search radius (a). If the point is
closer to the sensor, the θ-distance and thus ratio c decrease (b-d).
A main advantage is that the search area is adjusted to
the local sampling density. This results in a variable
separation distance of two objects. For objects close to
the sensor, the separation ability (in the azimuth
direction) of the modified DBSCAN can be significant
higher compared to far objects. This ability depends on
the input parameter f.
4) Object orientation
The search distance in r- and θ-direction is independent.
The ratio of the search distances depends on the local
resolution at the examined point and on the parameter f.
This means that the points inside the search can be
determined independently for θ- and r-direction. This
results in high execution efficiency.
V. E
XPERIMENTAL
R
ESULTS
A. Clustering results on real data
A real data set is used to show the different cluster results
of DBSCAN (Fig. 8) and grid-based DBSCAN (Fig. 9). The
dataset contains clutter (1), one pedestrian (2), three vehicles
(3-5) and a crash barrier (6). The data set is chosen to
demonstrate the disadvantages of DBSCAN clustering of
non-equidistant data.
The DBSCAN results (Fig. 8) show that the clutter close
to the sensor (1) has clustered together. The pedestrian (2)
next to the car (3) can not be separated and the two form a
cluster. The reason for this is that minimal distance of both
objects is app. 0.8m, whereas the search distance is larger
(ε = 1m). A car (5) heading towards the sensor results in an
object orientation in the r-direction. DBSCAN is not able to
cluster the car since its search radius covers more possible
observations in the azimuth than in the radial direction. The
furthest object (6) is not clustered, due to the small number
of possible observation compared to the fixed threshold of
DBSCAN.
Compared to grid-based DBSCAN (Fig. 9) the clutter (1)
is marked as outliers. The pedestrian (2) next to the car (3)
can be separated. The ratio c in this point is 0.23, so the
search distance can be calculated using equation (2), which
results in a search distance of 2 cells (= 0.4m) in each
azimuth direction. Grid-based DBSCAN is able to cluster the
car (5) and the barrier (6) properly. Since the search area
contains only one cell in each direction, even objects in
radial direction (5) and thin objects can be clustered (6).
Furthermore, the search radius always contains the
neighboring points and the threshold depends on the possible
observations in the search area.
The execution time of grid-based DBSCAN using an
ordinarily personal computer decreases by 43%. A detailed
Fig. 7: Effects of the grid-based DBSCAN algorithm on two close
objects (green, black), one object far away (red) and noise (cyan)
Fig. 8: Clustering results with original DBSCAN (ε = 1m, k = 6)
showing outliers (pink triangle) and 3 clusters (red, green, black) for
clutter (1), a pedestrian (2) next to a car (3), two other vehicles (4-5)
and a barrier (6)
Fig. 9: Clustering results for grid-based DBSCAN (f = 2) showing
outliers (pink triangle) and 5 clusters (red, green, black, blue, cyan)
for clutter (1), a pedestrian (2) next to a car (3), two other vehicles
(4-5) and a barrier (6)
analysis of the execution time is presented in the next
section.
B. Execution Efficiency
In this section the runtime of both algorithms is compared
using real data. To obtain a significant test, the parameters of
the grid based DBSCAN algorithm are chosen to have the
same search area as DBSCAN (f = 1, g = 1). Further, a part
of the data is chosen that does not contain front clutter and
has a limited range distance, so that all algorithms have the
same cluster result. The data shows different objects
recorded with our image radar and contains 125, 250 and
500 points. The only difference between DBSCAN and grid-
based DBSAN is another calculation of the density criterion,
the rest of the algorithm is identical. The third algorithm is a
speed optimized version. Instead of calculating the distance
between all points, it has a look-up-table representing the
grid where each point is registered. With a simple look-up of
the search area, the algorithm can determine the indices of all
points in the search area. The disadvantage is that for a large
database, the look up table increase its size by a factor O(n
2
),
whereas the other algorithm increases by only a factor O(n).
The corresponding execution time on an ordinary personal
computer in Matlab was determined for the three data sets.
The grid-based DBSCAN has a significant decrease in
execution time compared to the original DBSCAN algorithm
(125 points: -60%, 250: -44%, 500: -58%). The fast
implementation of the modified algorithm decreases the
execution time for large datasets even more (125: -56%, 250:
-61%, 500: -69%). The results are shown in Fig. 10.
VI. C
ONCLUSION
This paper presents a density-based algorithm to cluster
high-resolution radar data. It not only outperforms the classic
DBSCAN algorithm in terms of execution time with app.
40-70%, but increases the separation resolution of two
objects. In addition, it is robust against clutter and a
non-equidistant sampling density. Especially in
high-resolution radar systems, the sampling density is highly
non-equidistant and using common cluster algorithms results
in a number of disadvantages. With its optional input
parameter, the improved clustering is flexible and can be
adjusted not only for the sensor but also for the desired
cluster size or separation resolution of two close objects.
Since a unitless distance criterion is used, the algorithm
can be enhanced by the velocity or amplitude information. A
constant parameter, similar to g for the r-direction, has to be
introduced to adjust the search area for this new feature.
Another possibility is to fill the r-θ-grid with the amplitude
values. Then the density criterion could be a comparison of
the mean amplitude in the search area to an amplitude
threshold.
R
EFERENCES
[1] M. Ester, H.-P. Kriegel, J Sander, and X. Xu, “A density-based
algorithm for discovering cluster in large spatial databases with
noise”, Proc. 1996 Int. Conf. Knowledge Discovery and Data
Mining. Portland, USA. Aug. 1996, pp.226-231.
[2] R. Schneider, and J. Wengen, “High resolution radar for automobile
applications”, Advances in Radio Science 2003, Volume 1, 2003 pp.
105-111.
[3] V. Leonhardt, G. Wanielik, and S. Kälberer “A region-growing based
clustering approach for extended object tracking”, Radar Conference
IEEE 2010, Washington DC, USA., May 2010
[4] R. Schneider, “Automotive Radar – Status and Perspectives”,
Compound Semiconductor Integrated Circuit Symposium, 2005.
CSIC ’05. IEEE, Nov 2005.
[5] H. Rohling, “Radar CFAR Thresholding in Clutter and Multiple
Target Situations”, IEEE Transactions on Aerospace and Electronic
Systems, July 1983.
[6] F. Pauling, M. Bosse, and R. Zlot, “Automatic Segmentation of 3D
Laser Point Clouds by Ellipsoidal Region Growing”, Australasian
conference on robotics and automation (ACRA), Sydney, Autralia,
December 2-4, 2009.
[7] K. Qin, M. Xu, Y. Du, and S. Yue, “Cloud Model and hierarchical
clustering based spatial data mining method and application”, The
international archives of the Photogrammerty, Remote Sensing and
Spatial Information Sciences. Vol. XXXVII. Part B2, Beijing 2008.
[8] D. Comaniciu, and P. Meer, “Mean Shift: A robust approach toward
feature space analysis”, IEEE Transactions on pattern analysis and
machine intelligence, vol 24. no. 5, May 2005.
[9] D. Rajesh, “Characterization and detection of noise in clustering”,
Pattern Recognition Letters 12, Nov 1991.
[10] M. Ankerst, M. Breunig, H-P. Kriegel, and J. Sander “OPTICS:
Ordering points to identify the clustering structure”, Proc. ACM
SIGMOD’99 Int. Conf. on Management of Data, Philadelphi PA,
USA., 1999.
[11] Z. Li, and X. Wang, “High resolution radar data fusion based on
clustering algorithm”, Radar Conference IEEE 2010, Washington DC,
USA, May 2010.
Fig. 10: Execution time for DBSCAN, grid-based DBSCAN and a
fast implementation of grid-based DBSCAN, normalized on
grid-based DBSCAN for 125 points
