COMPASS: A Probabilistic Indoor Positioning System

COMPASS: A Probabilistic Indoor Positioning System
COMPASS: A Probabilistic Indoor Positioning System Based on 802.11 and Digital Compasses
Thomas King, Stephan Kopf, Thomas Haenselmann, Christian Lubberger, Wolfgang Effelsberg
University of Mannheim
– Fakultät für Mathematik und Informatik –
Praktische Informatik IV
A5, 6
D-68159 Mannheim, Germany
COMPASS: A Probabilistic Indoor Positioning System
Based on 802.11 and Digital Compasses
Thomas King, Stephan Kopf, Thomas Haenselmann, Christian Lubberger, Wolfgang Effelsberg
Department of Computer Science
University of Mannheim
Positioning systems are one of the key elements required by
context-aware application and location-based services. This paper
presents the design, implementation and anaylsis of a positioning
system called COMPASS which is based on 802.11 compliant network infrastructure and digital compasses. On the mobile device,
COMPASS samples the signal strength values of different access
points in communication range and utilizes the orientation of the
user to preselect a subset of the training data. The remaining training data is used by a probabilistic position determination algorithm
to determine the position of the user. While prior systems show
only limited accuracy due to blocking effects caused by human
bodies, we apply digital compasses to detect the orientations of the
users so that we can handle these blocking effects. After a short
period of training our approach achieves an average error distance
of less than 1.65 meters in our experimental environment of 312
square meters.
Categories and Subject Descriptors
C.2.1 [Computer Systems Organization]: Network Architecture
and Design—Wireless communication; I.5.1 [Pattern Recognition]: Models—Statistical
General Terms
Algorithms, Design, Experimentation, Measurement
Positioning systems, location systems, 802.11, digital compass,
ubiquious computing, location-based services, context-aware applications
During the last couple of years we have seen great improvements
in downsizing computer hardware, increasing capacity of rechargeable batteries as well as the advent of wireless networks for the
mass markets. These technologies allowed the manufactures to
build mobile devices that can be carried around and have the same
performance as traditional computers several years ago. The benefit of mobile devices can be leveraged by so-called location-based
services or context-aware applications. Applications that act differently depending on the location or the context of the user or,
even better, proactively offer location dependent information to the
user are currently a hot topic in research and have been considered
as a promising market.
Nowadays, the Global Positioning System [10] is the predominant
outdoor positioning system. Although GPS works pretty well in
outdoor scenarios it suffers from obstacles such as walls or roofs
blocking the weak radio signals in indoor environments. So far,
there is no prevailing indoor positioning system available. Many
indoor positioning systems either require specialized hardware or
provide poor accuracy. However, 802.11 based positioning systems sally out to fill this gap because 802.11 compliant hardware is
inexpensive, already available in many buildings and first research
results show impressive accuracy for this kind of indoor positioning
802.11 based indoor positioning systems usually rely on a so-called
fingerprinting approach. Fingerprinting-based algorithms use a two
stage mechanism: an offline training phase and an online position
determination phase. During the offline phase, the signal strength
distributions collected from access points at predefined reference
points in the operation area (also called fingerprints) are stored together with their physical coordinate in a database. During the online position determination phase, mobile devices sample the signal strength of access points in communication range and search
for similar patterns in the database. The closest match is selected
and its coordinate is returned to the mobile device as a position
Users tend to carry mobile devices in front of themselves so that
they can see the displays and interact with the devices by clicking
on buttons or pressing keys. The human body consists to more than
50 percent of water and hence blocks the signal of 802.11 radios.
Thus, the signal strength may vary significantly dependent on the
user’s orientation [2] and therefore degrade the performance of the
We propose a novel 802.11 based positioning system that is not
affected by the orientation of the user. We utilize digital compasses
to detect the orientation of the user, and based on this information,
only fingerprints that share a similar orientation are selected during
the offline phase to determine the user’s position. Our probabilistic
positioning algorithm uses only this subset of fingerprint data to
determine the position of the user.
Signal strength [dBm]
Signal strength [dBm]
Orientation [degree]
Time [ms]
Time [ms]
(a) User is motionless.
(b) User is turning around.
Figure 1: An example of the blocking effects caused by a human body.
As far as we know no mobile device is by default equipped with a
digital compass. However, mobile devices can easily be enhanced
with a digital compass because the latest generation of digital compasses is integrated into a single chip and its energy consumption
is very low. Furthermore, digital compasses are cheaply available.
Our new approach leads to very encouraging results. We achieve
highly accurate position estimates with a small average error distance. The precision of the results make location-based services as
well as context-aware applications feasible in indoor scenarios. The
user’s orientation is further accessible as additional information in
these applications.
The remainder of this paper is structured as follows: We present a
summary of related work in the following section. In Section 3, we
describe our positioning system in detail. The experimental environment is reported in Section 4 and the results of our experiments
are presented and analyzed in detail in Section 5. Finally, we give
a conclusion and outlook in Section 6.
Location-based services as well as context-aware applications primarily require positioning information of the user to be functional [7]. Existing indoor positioning systems can be divided into
a group of systems that rely on specialized hardware and a set of
systems that use off-the-shelf products. The former group comprises systems like the Active Badge project or the Cricket system.
Active Badge utilizes badges that are capable of emitting infra-red
signals [15]. Ceiling-mounted infra-red sensors detect these signals and a central unit interprets the sensor’s data to determine the
room where the user is located. A successor of Active Badge is
the Active Bat project that used ultrasound time-of-flight information and multilateration to determine the position of users more precisely [16]. Similar to this project is the Cricket system that uses ultrasound and radio frequency receivers to locate specially equipped
mobile devices [11] [12]. RFID-Tags are used in the SpotOn project
to locate users by applying a simple radio propagation model [8].
In EasyLiving, cameras are used to detect users and to infer their
location based on the captured images [4].
The latter group of indoor positioning systems uses off-the-shelf
wireless network hardware (e.g. 802.11 compliant network hard-
ware). One of the first systems of this kind is the RADAR
project [2] [1], but similar systems have been proposed by other researchers as well (e.g. [14] [13]). The authors of RADAR already
experienced blocking effects of the human body to radio signals.
However, they have not come up with a solution to mitigate these
effects. While RADAR is a deterministic positioning system the
systems most recently proposed have embraced probabilistic models to estimate the location of the user. Probabilistic approaches
store the signal strength distribution from the access points and use
probabilistic algorithms to estimate the most likely position of the
user [6] [5]. HORUS is one of the projects that uses a similar probabilistic algorithm as our system [20] [17] [18] [19]. However, this
project does not address the problem of blocking effects caused by
the human body.
In this section, we present the 802.11 wireless channel characteristics and describe how our novel algorithm utilizes these properties
to improve the overall positioning accuracy.
Wireless Channel Characteristics
The 802.11 [9] wireless network protocol is inexpensive and widely
deployed in many offices, universities, and private properties.
Nowadays, most modern laptop computers and digital personal
assistants (PDAs) have built-in support of 802.11 and even most
cellphone manufacturers provide cellphones with 802.11 support.
802.11 uses the so-called industrial, scientific, and medical (ISM)
band at 2.4 Ghz or 5 Ghz depending on the substandard. However,
predominant are the 2.4 Ghz based substandards called 802.11b and
802.11g, hence, we only focus on these standards.
Client-side hardware is capable of measuring the signal strength of
access points in communication range to choose the best gateway
to the wired network. For this purpose, the 802.11 specification defines a mechanism called active scanning. Active scanning is a simple request-response protocol: a client broadcasts a ProbeRequest
packet and every access points in communication range replies with
a ProbeResponse packet. Each time a client receives a ProbeResponse packet the network card measures the signal strength and
forwards the power level value to the hardware driver. This information is accessible in all major operating system (e.g. Microsoft
Windows and GNU/Linux) by user-space programs. Our position-
ing system is implemented as such a user-space program because it
relies on the power level information provided by the network card.
2.4 Ghz is the resonance frequency of water so that the human body
absorbs these radio signals. This is one of the reasons why it is
difficult to predict the reception power measured at a mobile device
for a given distance from the access point. For instance, a person
who carries the mobile device may block the signal by standing inbetween the access point and the mobile device’s wireless network
We have analysed the blocking effect introduced by a human body
to the reception power: The signal strength from an access point is
sampled every 250 milliseconds while the user carries her mobile
device in such a way that she is able to see the display. The access
point is located five meters apart from the user. First, we measure
the signal strength for 200 seconds while the person carrying the
mobile device completely blocks the line of sight between the two
peers. We aggregate the signal strength measurements for 25 milliseconds and calculated the average and standard deviation. As
depicted in Figure 1(a) the reception power is relatively stable between 80 and 83 dBm and shows a standard deviation of less than
5 dBm. In a second step, we measure the signal strength every
45◦ while the person who is carrying the mobile device is turning
around. The line in Figure 1(b) shows the blocking effect of the
human body. The reception power increases nearly 15 dBm (see
135◦ ) in case of a direct line of sight between the receiver and the
access point. Even if the person only partially blocks the radio signals, the reception power still increases more than 5 dBm (see 90◦
and 225◦ ). Furthermore, we see from the figure that the standard
deviation increases if the line of sight is blocked by a human body
(compare 45◦ and 135◦ ).
Positioning Algorithm
The basic idea of our novel positioning algorithm is to sample the
signal strength for selected orientations at each reference point during the offline phase and combine a subset of these values to histograms in the online phase, so that an orientation specific signal
strength distribution can be computed and utilized to increase the
accuracy of position estimates. Orientation specific distributions
are calculated each time a user requests a position estimate. Only
those signal strength histograms that show a similar orientation as
the user are selected at each reference point (see Figure 2). These
histograms are combined and used to compute a position estimate.
To build a model of the real world, we overlay the operation area
with a grid of q reference points spaced 1.0 meters apart. At each
reference point we collect the access point’s signal strength distribution for eight orientations. Formally, we model a finite state
space S = {s1 , . . . , sn } whereas we use n reference points R =
{r1 , . . . , rq } and the orientations O = {0◦ , 45◦ , . . . , 315◦ } to define S = R × O. Each state represents a certain two-dimensional
coordinate with a certain orientation. The access point’s signal
strength distribution is aggregated by a bunch of measurements.
Each measurement M = {m1 , . . . , mk } contains the quantified
reception power of k access points. Two different measurements
may comprise a different number or set of access points. Furthermore, mi = (pi , idi ), i ∈ {1, . . . , k} is in fact a tuple containing
an access point’s reception power pi (pi ∈ {0, 1, . . . , 255}) and
unique MAC address idi as a string. The MAC address is required
to unambiguously identify a particular access point.
We use a reference vector ~π to model the probability that a mobile device is located at a certain reference point. Therefore, πi ,
i ∈ {1, . . . , q} represents the probability that the user is located at
reference point i. At the beginning, this vector is initialized with an
uniform distribution by assigning πi = 1/q.
Additionally, we need a definition for the similarity of states: Two
states a and b represented by coordinates and an orientations oa and
ob , respectively, are similar in respect of α iff |oa − ob | ≤ α. This
similarity measure is required by the online determination phase of
our algorithm to select a subset of valid states.
Offline Training Phase
During the offline phase, we train our algorithm by collecting a
number of measurements at each state and store the data as histograms in a database. The optimal number of measurements is
determined in Section 5.3.
Online Determination Phase
In the online determination phase, the user takes one or more samples containing the signal strength measurements Mi and her orientation Oi , i = 1, . . . , q. Our algorithm determines, based on these
data, an accurate position (or reference point) r∗ .
For a given threshold α, only the states that are similar to the user’s
current orientation are considered. For each reference point i and
access point j these similar states are merged by combining the
histograms. We calculate the mean µi,j and standard deviation σi,j
of this merged and normalized histogram and use these aggregated
values in the subsequent steps.
The probability to obtain the measurements M at reference point i
can be expressed by the conditional probability:
P (M |ri )
P (ma |ri )
Figure 2: Each asterisk represents a reference point with eight
signal strength histograms in an exemplary operation area.
Only the signal strength histograms with a similar orientation
as the user (green dot) are selected (red shaped areas) and used
by the positioning algorithm.
P (ma |ri )
P ((pa , ida )|ri )
Gi,id (pa ) + β
P255 a
m=0 Gi,ida (m) + β
Gi,ida (pa )
pa +1/2
pa −1/2
e−(x−µi,ida )
σi,ida 2π
Gi,ida (pa ) represents the probability to measure the signal strength
value pa of access point ida at reference point i. This probability
is calculated based on the distribution function determined by the
average µi,ida and standard deviation σi,ida . Furthermore, β is a
small value that artificially adds noise to the probability calculation.
During our experiments we set β = 10−44 .
These conditional probabilities are used to update the probability
vector ~π by applying Bayes’ Rule [3]
π ∗ P (M |ri )
Pn i
j=1 πj ∗ P (M |rj )
for each reference points i.
Our analysis (see Section 5.2) shows that there is often more than
one suitable candidate for a set of reference points. Surrounding
reference points are usually considered highly probable if the user
is located in the middle of these points. Therefore, the k most likely
reference points are averaged to estimate the users’ position r∗ by
1 X
k i=0
where max(i, ~π ) defines the ith largest value of ~π .
On the client, we used a Lucent Orinco Silver PCMCIA network
card supporting 802.11b. We collected the signal strength samples
on an IBM Thinkpad R51 running Linux 2.6.13 kernel and Wireless
Tools 28pre.
To obtain the orientation of a user we used the Silicon Laboratories
C8051F350 Digital Compass Reference Design Board. This device
provides a USB-to-Serial bridge to access the data and can be powered by the USB electricity supply. We calibrated the compass in
the middle of the operation area. In a closer area around the calibration point we measured a variation of 1◦ . However, variations
up to 23◦ were rarely detected at a few points of the testing area.
These measurement errors occured always close to electromagnetic
objects such as high voltage power lines and electronic devices.
We randomly selected 60 coordinates and orientations for the online phase. The only condition to select a point inside the testing
area as an online set point is that it is surrounded by four reference
points. Again, we collected 110 signal strength measurements for
each online set point, leading to 6,600 measurements in total. In
Figure 3 the online set points are marked by green dots.
The subsequent subsections briefly describe our experimental environment.
Local Test Environment
Data Collection
The grid of reference points applied to the operation area includes
166 points with a spacing of 1 meter (see the orange dots in Figure 3). During the offline phase, the signal strength was measured at
reference points for different orientations. We collected 110 signal
strength measurements at each reference point and for each orientation. This leads to 146,080 measurements for the offline phase.
We spent over 10 hours to collect all the data, however, we want
to point out that for a productive deployment of this positioning
system 20 signal strength measurements will be sufficient (see Section 5.3), cutting down the expenditure of time to less than 2 hours.
In this section, we evaluate the performance of each element of our
algorithm and discuss the impact of specific parameters on the performance of our positioning system. Afterwards, we compare the
performance of our system with the performance of the RADAR
We deployed our positioning system within the hallways of the A5,
6 building on the campus of the University of Mannheim. The
operation area is nearly 15 meters in width and 36 meters in length,
covering an area of approximately 312 square meters. The ground
plan of the testing area is visualized in Figure 3. The large hallway
in the left part of the map is connected by two narrow hallways that
are separated by rooms such as archives and a kitchen.
We developed a software application to analyze the performance
of various positioning algorithms in different environments. The
software logs every action and state (e.g. the signal strength) during
the offline and online phase so that we can replay the scenario,
and that we are able to change particular parameters we want to
investigate. We used this software to simulate the behavior of the
positioning algorithms.
We labeled the ground plan with markers which depict the grid
of the reference points (orange dots) and the online measurement
points (green dots). Additionally, the access points are marked by
red squares.
To measure the performance of different positioning approaches
we compute a cumulative distribution function (CDF) of the error
distances between the estimated positions and the real positions of
the user. Additionally, we calculate the average error distance.
Linksys / Cisco WRT54GS and four Lancom L-54g access
points. All access points support 802.11b and 802.11g. One
Lancom and all Linksys access points are located on the same floor
as our testing area, whereas three Lancom access point are located
in other places inside the building. The exact position of the access
points located inside the testing area is marked with red squares in
Figure 3.
Analysis of the Orientation
Our algorithm has two parameters that control the compass-specific
behavior: The number of orientations considered during the offline phase for each reference point and the angle α which determines similar states during the online phase. In the subsequent
paragraphs, we study the effect of these parameters.
In our first experiment we show that the performance of our algorithms increases if signal strength distributions are sampled during
Figure 3: Ground plan of the testing area. The orange dots represent the offline reference points and the green dots show the
randomly selected online measurement points. The access points are marked by red squares.
the offline phase for an increasing number of orientations. We selected four scenarios: In the first scenario only the signal strength
distribution of one orientation is considered (0◦ ) at each reference
point. The second scenario takes two orientations into account (0◦
and 180◦ ). Additionally, the signal strength distributions of the orientations at 90◦ and 270◦ are used in the third scenario and finally,
the signal strength distributions of eight orientations are analyzed
(0◦ , 45◦ , . . ., 315◦ ).
For each scenario, to build the training dataset we randomly select
20 of the 110 already collected measurements for each reference
point and each orientation. During the online phase, we randomly
pick three of the 110 collected samples for each online set point.
We set α = 185◦ , so that our algorithm is even able to proceed in
the single orientation case. Furthermore, we disable the averaging
part of our algorithm by setting k = 1 so that the pure performance
of orientation part of the algorithm is visible. We repeat the whole
procedure 1000 times to achieve stable results.
Figure 4 shows the CDF of the error distance for these four scenarios (we cutted error distances larger than 14 meters to emphasize on
the performance distinction of the different scenarios). The average
error distance drops from 2.95 meters in the single orientation case
down to 2.31 meters in the eight orientations scenario, resulting
in an accuracy improvement of 22 percent. Furthermore, we see
from the figure that each additional orientation leads to a diminishing marginal utility. For instance, the average error distance drops
more than half a meter if a second orientation is added compared
to an improvement of 4 centimeters in case four extra orientations
are added to the four orientations scenario.
In a next step, we investigate the effect of angle α on the error in
the eight orientations scenario. We repeat the evaluation procedure
as mentioned above, however, this time we change the value of α
stepwise from 25 to 100 degrees.
8 orientations
4 orientations
2 orientations
1 orientation
Error distance [m]
Figure 4: Relation between number of orientations and error
Our experimental analysis shows that the system achieves a minimal error distance for α = 69. This can be explained by the fact
that for small values of α only one signal strength distribution is
selected for each reference point. For instance, if the user’s orientation is 60◦ and α = 25◦ , only the states with an orientation of 45◦
are considered, whereas a better approximation would be a combination of the signal strength distributions of the 45◦ and 90◦ orientations. The accuracy degrades rapidly for large values of α because signal strength distributions for opposite orientations are also
merged into the reference signal strength distribution, thereby corrupting the orientation specific characteristic of the signal strength
For α = 69◦ the average error drops to approximately 2.05 meters, improving the accuracy by 12 percent compared to α = 185◦ .
To sum up, the usage of the compass-related components of our
algorithms reduces the average error distance to less than 2.05 meters, which corresponds to an accuracy improvement of more than
31 percent.
Averaging the Positions of the Most Probable Reference Points
In contrast to the previous section where we considered the most
probable reference point as the position estimate, we now consider
the average of the k most probable reference points as the user’s
position. In general, the user is surrounded by up to four reference
points whose averaged coordinates represent a good approximation
of the user’s position.
hand, the collected signal strength distributions should be stable
enough so that the positioning algorithm is able to accurately estimate the position of the user. We vary the training set size s between 3 and 100 to analyze how our algorithm performs with small
training set sizes. For each number of s we randomly chose s readings out of the 110 signal strength measurements already collected
in the data collection process and build the fingerprinting database.
In the online phase we set α = 69◦ , k = 5, and randomly select
three measurements for each online set point as input for our positioning algorithm. We performed this experiment 1000 times for
each value of s. The average error distance for each value of s is
shown in Figure 6.
Average error distance [m]
Average error distance [m]
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Training set size
Figure 6: Training set size versus average error distance.
Figure 5: Impact of the averaged number of reference points
(k) on the positioning accuracy.
We run an experimental analysis to find an optimal value of k.
For this, we repeat the same experiment as mentioned above, but
change the value of k from 1 to 10. As seen in Figure 5, the average error distance is minimal for k = 5. Smaller values of k
increase the average error distance up to half an meter, where the
error distance slowly increases for larger values of k. With a larger
number of averaged reference points more distant points are selected resulting in averaged coordinate that is moving away from
the real position of the user.
During our evaluation process, we also investigated other averaging algorithms (e.g. a weight-based averaging approach) but we
could not find an algorithm that outperforms our simple averaging
algorithm. The reason is that probability distributions differ significantly between different position requests and even a highly likely
reference point might be over half a meter away from the user’s
real position. Hence, a simple averaging with surrounding reference points increases the accuracy.
From these results we see that the average error distance is reduced
by more than half a meter due to the averaging.
Determination of the Training Set Size
The size of the training set is a critical parameter for the performance of the positioning algorithm. However, it determines the
lower bound on the time needed to collect data for the fingerprinting database. It takes 250 ms to collect data for a single measurement with our network card. A reduction of the number of measurements required by the algorithm reduces the hours of work to
collect data for the whole operation area significantly. On the other
The graph shows that the accuracy increases more than 10 centimeters if the training set size is increased from 3 to 20 measurements.
However, a further extension of the training set size only slightly
improves the accuracy. Due to minor variations the average error
distance oscillates around 1.625 meters for training set sizes larger
than 30.
We have chosen a training set size of 20 for most of our experiments because a further extension barely improves the performance
of our algorithm. Furthermore, the results in this paper should be
easily achievable in productive environments where it is critical to
minimize the time required to build the fingerprinting database.
Determination of the Online Set Size
In the following, we analyze the performance of our algorithm as
we vary the number of measurements from which the positioning
algorithms infers the user’s position. Our hardware requires 250 ms
to collect data for one measurement; the fewer measurements required to compute an accurate position estimate, the faster a position estimate is available. We repeat the same experiment as mentioned above, but this time we vary the number of measurements
from 1 to 10 before a position estimate is finally computed. We
perform this procedure 1000 times for each number of measurements. Our results are summarized in Figure 7.
The results show that an average error distance of less than 2.1 meters is achievable even if just one measurement is used. A second
measurement decreases the error distance more than 25 centimeters
(this corresponds to more than 17 percent). A thrid measurement
improves the accuracy more than 6 percent or in other words nearly
10 centimeters. As we see from the graph any additional measurement further improves the accuracy, however, with a diminishing
to only 55 percent in case of RADAR); it also produces a smaller
worst case error distance (less than 11 meters in contrast to more
than 15 meters). On average, our system achieves an error distance
of 1.65 meters in comparison to 2.26 meters in case RADAR is
used. These results show the power of our new approach.
Average error distance [m]
Number of measurements
Figure 7: Average error distance depending on the number of
online measurements.
marginal utility. In most of our experiments we use three measurements to compute a position estimate because three measurements
prove to be a reasonable trade-off between accuracy and time.
Performance Comparison
In the previous sections, we studied the impact of the compass first,
and separately investigated the effects of the different parameters
afterwards. In this section, we compare the performance of our
approach with the performance of the RADAR system [2]. We
have chosen the Multiple Nearest Neighbors algorithm as proposed
by Bahl et al. because this algorithm achieved the best performance
in their paper.
In this paper, we presented COMPASS, a positioning system based
on Wireless LAN and compasses. We have shown that the recognition of the user’s orientation can dramatically improve the accuracy of the positioning system. We used a probabilistic algorithm
to compute a probability distribution over the reference points and
presented a simple averaging algorithm to further improve the performance of the system.
We have deployed our positioning system in a building of the University of Mannheim to evaluate it in a real-world environment.
In our experiments, we achieved an average error distance of less
than 1.65 meters. We also investigated how different parameters
of the algorithms influence the overall performance of the system.
Finally, we compared the performance of our system with the performance of the RADAR system.
In the near future, we are going to work on a compass-enabled
tracking system to track users while they are walking around. We
believe, the user’s orientation will also help to improve the performance of this system because orientation is a well suited indicator
to recognize turnarounds of users.
Based on our system, location-based services as well as contextaware applications that require highly accurate position information can now be build. Within our research group at the University of Mannheim, we are currently working on such novel applications.
We would like to thank Alexander Biskop, Andreas Färber, and
Daniel Kölsch for their great support collecting data for the experiments. Furthermore, the authors acknowledge the support granted
by the Deutschen Forschungsgemeinschaft (DFG).
Error distance [m]
Figure 8: CDF of the performance of the COMPASS and
RADAR system.
To create equal conditions for both algorithms we use the data gathered from our experimental environment for the training and the
position determination phase. We set the offline set size to 20 and
randomly choose three readings for each online set point to compute the position estimates. This procedure is repeated 1000 times
to receive stable results.
Figure 8 shows the CDF for both algorithms. The COMPASS system outperforms the RADAR approach: our algorithm is not only
more accurate (e.g. the error distance in the COMPASS system is
less than two meters in 70 percent of all measurements compared
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