GENERATING OPTIMIZED MARKER-BASED RIGID BODIES

FOR OPTICAL TRACKING SYSTEMS

Frank Steinicke, Christian Jansen and Klaus Hinrichs

Institut f

ur Informatik, WWU M

unster, Einsteinstr. 62, 48149 M

unster, Germany

Jan Vahrenhold

Informatik XI, Universit

at Dortmund, 44221 Dortmund, Germany

Bernd Schwald

TWT GmbH, Information & Engineering Technologies, 73765 Neuhausen, Germany

Keywords:

Optical tracking, model-based object tracking, rigid bodies.

Abstract:

Marker-based optical tracking systems are often used to track objects that are equipped with a certain number

of passive or active point markers. Fixed conﬁgurations of these markers, so-called rigid bodies, can be

detected by, for example, infrared stereo-based camera systems, and their position and orientation can be

reconstructed by corresponding tracking algorithms. The main issue in designing the geometrical constellation

of these markers and their 3D positions is to allow robust identiﬁcation and tracking of multiple objects, and

this design process is considered to be an essential and challenging task. At present, the design process is based

on trial-and-error: the designer constructs a marker conﬁguration, evaluates it in a given setup, and rearranges

the marker positions within the conﬁguration if necessary. Even though single ready-made rigid bodies permit

sufﬁciently good tracking, it is not ensured that the corresponding arrangements of markers meet any quality

criteria in terms of reliability and robustness. Furthermore, it is unclear whether it is possible to add further

rigid bodies to the setup which are sufﬁciently distinguishable from the given ones.

In this paper, we present an approach to semi-automatically generate marker-based rigid bodies which are op-

timal with respect to the properties of the tracking system for which they are used, e.g., granularity, accuracy,

or jitter. Our procedure which is aimed at supporting the design process as well as improving tracking gen-

erates conﬁgurations for several devices associated with an arbitrary set of point-based markers. We discuss

both the technical background of our approach and the results of an evaluation comparing the tracking quality

of commercially available devices to the rigid bodies generated by our approach.

1 INTRODUCTION

The usage of optical tracking systems based on in-

frared (IR) light is becoming more and more com-

mon for virtual, augmented or so-called mixed re-

ality (MR) systems used in several application do-

mains. This is due to the fact that these systems

provide a large interaction space and quite high ac-

curacy, and in contrast to mechanical approaches no

wires disturb the interaction. Furthermore no interfer-

ences may occur as when using magnetic or ultrasonic

technology. Nowadays IR-based optical tracking sys-

tems exist as prototypes in research institutes (Kato

and Billinghurst, 1999; Dorfm

uller-Ulhaas, 2002;

Schwald, 2005; Ribo et al., 2001) and are also com-

mercially available (A.R.T., 2006; Fakespace Sys-

tems, 2006). The main issue in designing tracking

system is to advance both hardware as well as algo-

rithms in order to increase the accuracy and robust-

ness. These factors are the most important properties

to make such a system usable for applications, e.g., in

medicine or MR environments in general.

When using such systems objects or devices to

be tracked are associated with so-called rigid bodies,

sometimes denoted as targets (Schwald, 2005; Kato

and Billinghurst, 1999; Dorfm

uller-Ulhaas, 2002). A

rigid body is a ﬁxed geometrical arrangement of at

least three passive or active IR markers. A calibrated

camera system allows to reconstruct the 3D coordi-

nates of IR point-based markers in the tracking coor-

387

Steinicke F., Jansen C., Hinrichs K., Vahrenhold J. and Schwald B. (2007).

GENERATING OPTIMIZED MARKER-BASED RIGID BODIES FOR OPTICAL TRACKING SYSTEMS.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IU/MTSV, pages 387-395

 SciTePress

dinate system if at least two cameras can detect the

same marker (Zhang, 2000). A rigid body registered

by the tracking system, can be tracked by means of

the ﬁxed known distances between its markers.

In many MR applications, devices like head

mounted displays, stereo glasses or interaction de-

vices, e.g., gloves or wands, have to be tracked (Bow-

man et al., 2004). Often these devices do not require

more than three, four or ﬁve markers, arranged in a

target attached to such a device. For instance, a typi-

cal setup in a co-located MR environment consists of

two users, each equipped with an interaction device

and stereo glasses each including approximately three

to ﬁve markers. However, some applications need

to track more complicated objects requiring a much

larger number of markers. This occurs, for example,

when tracking real objects having several markers at-

tached on different sides in order to avoid occlusions.

Usually, rigid bodies are deﬁned by the developer

or an interaction device designer by arranging mark-

ers around the object respectively device to be tracked

(Davis et al., 2004). When arranging markers on such

an object it is essential to arrange them in such a way

that the distances between all markers are pairwise

different if possible. Otherwise the tracking system

may mistake distances and the corresponding device

will not be recognized. For each new marker to be

integrated into a conﬁguration consisting of n mark-

ers n new distances have to be considered. For ex-

ample, when building a simple target including three

markers, adding a fourth marker requires the designer

to consider three new distances. Moving one marker

in a conﬁguration of four markers may change three

distances that have to be pairwise different and differ-

ent from each of the distances of the remaining three

markers. Thus, although ﬁnding a well-deﬁned con-

ﬁguration seems to be simple it involves a non-trivial

task of arranging the markers especially if several de-

vices with numerous markers are included.

However, when constructing a target usually the

markers are arranged by trial-and-error. After a pro-

totype rigid body has been built, the application de-

veloper or user has to test the corresponding device in

a laboratory setup. If the test shows bad rigid body

performance, the designer has to rebuild the device.

A bad rigid body performance means that the device

is often not tracked or it is tracked with position or

orientation errors which do not result from accuracy

errors of the used tracking system; these mistakes re-

sult from confusing distances within the same conﬁg-

uration or between different conﬁgurations.

After several iterations of building, testing and re-

deﬁning, the designer may have constructed a con-

ﬁguration that provides sufﬁcient tracking properties.

However, even when ready-made targets are tracked

well, it is not ensured that the corresponding arrange-

ments of markers are optimal in terms of reliability

and robustness and if it is possible to add further tar-

gets to the setup that are distinguishable from the al-

ready designed ones. Typically, targets built via such

a procedure consist of distances which have the po-

tential to disturb the tracking process. For instance,

distances within the same target or distances of dif-

ferent targets used for the interaction may be equal.

In order to support the arrangement of markers in

a target we present a procedure to semi-automatically

generate marker-based rigid bodies in an iterative

way. When using our approach the proposed con-

ﬁgurations are adapted to the properties of the corre-

sponding tracking system, e.g., granularity, accuracy,

jitter etc., and thus the described concepts enhance the

tracking process. Our procedure allows to generate

rigid bodies for several devices associated with an ar-

bitrary set of markers. This paper describes the tech-

nical background of our approach and the results of an

evaluation comparing commercially available devices

with their associated targets to rigid bodies proposed

by our approach.

The remainder of this paper is structured as fol-

lows. Section 2 outlines the concepts of optical track-

ing and explains how 3D points are generated from

2D images of point-based markers grabbed with at

least two cameras. In Section 3 we describe how rigid

bodies are deﬁned and how the detection of a conﬁg-

uration is performed by the tracking system. Section

4 explains our algorithm to generate conﬁgurations

semi-automatically in an iterative way. In Section 5

we present an evaluation of our concept and show how

we could increase the performance of the devices by

redeﬁning two example targets. Section 6 concludes

the paper and gives an overview about future research

directions.

2 INFRARED-BASED OPTICAL

TRACKING

Since the brightness of most MR systems, e.g.,

CAVEs, PowerWalls, etc., is relatively limited, many

projection-based environments require a signiﬁcant

reduction of the ambient light. To overcome the re-

sulting lighting problem for the cameras, an infrared

(IR) optical tracking systems illuminates the scene us-

ing infrared light, and IR pass ﬁlters are attached to

the lenses of the tracking system’s cameras. Infrared

optical tracking systems aim at measuring the (real-

world) positions of numerous markers in the envi-

ronment. Since active markers such as light-emitting

VISAPP 2007 - International Conference on Computer Vision Theory and Applications

388

During the tracking process, the relation between

the current position and orientation of a given target

and the position of its reference position/orientation

speciﬁes the rigid body transformation of the target.

This rigid body transformation is applied to a virtual

object which is associated with a corresponding tar-

get, i.e., the transformation matrix describing the rigid

body transformation is applied to the associated scene

object visualized in the virtual environment.

To further increase the effectiveness of the track-

ing algorithm, learning algorithms may be used that

aim at enhancing the tracking process, e.g., by tuning

the distances predeﬁned manually in the reference tar-

get with respect to measured distances (Figueriredo,

2002; Kanbara et al., 2001).

3.1 Description

A target used to track an input device is a conﬁgu-

ration of a set of markers where one of the markers

serves as a local reference point. As an example, con-

sider the six-degrees-of-freedom interaction device

depicted in Figure 3. The target attached to this de-

vice consists of three markers where the point m

(0, 0, 0) is associated with the marker at the top of

the input device, point m

:= (−73.0, 0.0, −188.72)

is associated with the marker in front of the handle,

and m

:= (0.0, 0.0, −181.25) is associated with the

marker at the top of the stick branched to the left. The

positions of the markers are given as relative coordi-

nates (in millimeters) in the Cartesian coordinate sys-

tem with the origin at m

Figure 3: Example device with an associated with a rigid

body consisting of three markers.

In general, such targets can be deﬁned by two dif-

ferent procedures: (1) storing the (absolute) positions

of the markers or (2) storing the relative distances

between the markers as well as its reference posi-

tion/orientation. We use the second alternative and

apply a matrix representation for the distances

= {d(m

, m

) | m

, m

∈ M ; i, j = 1, ..., n} (1)

where M is the set of all markers in a conﬁguration

consisting of n markers, and d(m

, m

) denotes the

Euclidean distance between two markers m

and m

in 3D-space.

The quadratic n × n-dimensional matrix D =

i j

:= d(m

, m

))

i, j=1,...,n

stores the distances induced

by the conﬁguration and has the following form:







0 d

. . . d

0 d

. . . d

0 . . . d

. . . 0







(2)

From (1) and (2), it follows:

1. d(m

, m

) = 0, and

2. d is symmetric, i.e., d(m

, m

) = d(m

, m

To ensure that no errors occur in determining the cor-

rect rigid body transformation, we also desire that

d(m

, m

) are pairwise different for all 1 ≤ i < j ≤ n.

3.2 Detection

The above matrix-based description of rigid bodies

can be used to identify the rigid body by means of

distance detection. When tracking a target taken from

a set of several predeﬁned rigid bodies, the tracking

algorithm scans the 3D point cloud P resulting from

the reconstruction of the detected markers. While do-

ing so, the algorithm tries to match point-to-point dis-

tances in P to distances d

i j

= d(m

, m

) stored in the

i-th row and j-th column of each matrix D that de-

scribes one of the predeﬁned rigid bodies (see (2)).

If any d

i j

varies by at most a distance threshold of

ε from some d ∈ D

, i.e., if

|d − d

i j

| < ε, (3)

it is assumed that d = d

i j

, and — assuming that

all distances are unique — the ﬁrst two markers of

a target T , m

and m

, are detected within the point

cloud P . Otherwise the algorithm scans P for the next

distance stored in D

The algorithm continues searching until a third

marker m

has been found or the search space is ex-

hausted. If a third marker m

is identiﬁed by detect-

ing the distances d

or d

in the point cloud, the

rigid body transformation of this target with respect to

its reference position can be determined (Dorfm

uller-

Ulhaas, 2002). From Equation (3), we see that if

we choose ε to small, distances may not be found

due to accuracy errors. If the threshold is chosen too

high, there may be ambiguities with other distances.

This may result in tracking of targets with wrong po-

sition/orientation or to confusions between targets.

Hence, the threshold must be carefully adapted in or-

der to increase the reliability of the tracked data.

VISAPP 2007 - International Conference on Computer Vision Theory and Applications

390

4 GENERATION OF TARGETS

Assuming that the shape and size of the markers are

identical for each rigid body we focus on improv-

ing the tracking performance by means of redeﬁning

positions of markers within a target in order to be

able to support the largest possible distance thresh-

old. Hence, the main idea is to arrange the markers in

such a way that the resulting distances between each

pair of markers are as diverse as possible with respect

to the granularity of the tracking system and already

existing conﬁgurations.

4.1 Iterative Approach

Let D

denote the set of all distances of the r-th

rigid body, which consists of the marker set M

. Fur-

thermore, let D denote the union of all D

for all

rigid bodies registered at the tracking system. With

increasing difference between the distances contained

in D, the threshold ε from Equation (3) can be chosen

larger. We calculate the threshold ε by

ε = min

{

| d

i j

− d

, d

i j

, d

∈ D

i j

6= d

}. (4)

Hence, ε is given by the minimal difference between

two distances within a conﬁguration of a rigid body

involved in the tracking process. As it can easily be

seen from Equation (3) a large threshold improves

the tracking robustness, it is beneﬁcial to arrange the

markers in such a way that this minimal difference is

as large as possible. However, the distances within

rigid bodies are constrained by the maximum size of

the target with respect to the device to which it is at-

tached; a large target may be inconvenient, heavy and

it may restrict the user’s degrees of freedom. In gen-

eral for hand-held devices the maximum size of a tar-

get is about 20 − 30cm.

Our approach provides the largest possible thresh-

old with respect to the granularity of the used tracking

system. The granularity deﬁnes the minimal distance

between two points that can be measured. If a point P

is tracked with an accuracy determined by the granu-

larity g, it can be ensured that P is located in a sphere

around P with radius g. For this reason, two distances

between markers are well-deﬁned only if they distin-

guish at least by 2 · g. While in optical tracking sys-

tems the granularity is in the area of submillimeters,

the sizes of the markers itself measures at least about

4mm, and therefore we approximate the granularity to

8mm upwards.

Assume that the maximal distance in a conﬁgu-

ration is predeﬁned by y , the interval of distances

[0, y] is decomposed into subintervals with length of

2 · g. Hence, the set of all distinguishable distances

between markers C is given by

C = {d

| d

:= 2 ·i ·g; i = 1, ...,



2 ·g



}. (5)

The number of markers that can be integrated into a

tracking system providing that all resulting distances

must vary is constrained to max{i |

∑

j=1

( j − 1) <

2·g

4.1.1 Designing a New Target

We start with a tracking system without any regis-

tered rigid bodies, i.e., D is an empty set. In order

to deﬁne a new target, the developer has to specify

the properties of the tracking system, i.e., granular-

ity g and maximum distance y of the target. Now, our

approach supports the designer in deciding which dis-

tances from C should be taken into account for a new

marker conﬁguration.

Only one distance, namely d

, results from the

ﬁrst two markers m

and m

. Using our approach

the designer can either specify the largest distance

∈ C that should be used in the conﬁguration, or

our algorithm starts with 2 · g and iterates through C

until an optimal conﬁguration is found as described

following.

For simplicity, we place m

to the origin and m

onto the z-axis at a distance of d

:= (0.0, 0.0, 0.0)

:= (0.0, 0.0, −d

)

Since, D is the set of all distances which are already

included in rigid bodies, using our approach implies

D ⊂ C . After the ﬁrst distance of the target is deter-

mined, we add d

to D.

As mentioned above, to allow six-degrees-of-

freedom tracking, at least three markers are required

in a ﬁxed conﬁguration. Hence, our approach deter-

mines the best position for the marker m

, such that

the resulting distances d

, d

∈ C are as diverse as

possible from d

and each other. This is done by

choosing d

and d

such that they are uniformly dis-

tributed within the subintervalls of C.

...

Figure 4: Example conﬁguration of used distances from C

for a rigid body consisting of three markers.

Figure 4 illustrates this procedure. Let d

, d

etc.

denote the distances from C (see Equation (5)). After

the user has speciﬁed, for example, d

:= d

as ﬁrst

GENERATING OPTIMIZED MARKER-BASED RIGID BODIES FOR OPTICAL TRACKING SYSTEMS

391

distance between m

and m

, the next marker is in-

serted in such a way that both new distances d

:= d

and d

:= d

result in the best distribution possi-

ble. When determining these markers by means of

calculating the corresponding distances, it has to be

ensured that the spheres around m

with radius d

and around m

with radius d

intersect at least in one

point which deﬁnes the position of marker m

. Oth-

erwise new radii, i.e., distances from C , have to be

tested for intersections. This procedure is illustrated

in Figure 5. When more than one intersection point

exists, the marker m

is chosen to be located in the

local xy-plane of the conﬁguration having a positive

y-value.

Figure 5: 2D Illustration of determination of third point us-

ing spheres.

Again, we add d

and d

to D. Now, that

three markers are conﬁgured, the rigid body can be

tracked. In order to enable robust tracking even if

several markers are occluded, e.g., by the device it-

self, further markers can be added.

4.1.2 Adding a Further Marker

We continue with a set of targets each consisting

of several markers, where the resulting distances are

stored in D. Let the current target contain i −1 mark-

ers. If the user wants to add the i-th marker to the

target, i − 1 distances from C have to be selected uni-

formly analogously to the procedure described in Sec-

tion 4.1.1. Furthermore, the new distances shall be el-

ements of C \ D, i.e., they are used neither within the

current target nor within another conﬁguration. Again

the spheres surrounding the markers of the current tar-

get with radii determined by the corresponding dis-

tances have to intersect in one point at least. If one or

more intersection points exist, the desired distances

can be satisﬁed and the new marker can be inserted.

Otherwise, all distances have to be redeﬁned, i.e., the

next distance from C is used as ﬁrst distance in the

conﬁguration and it is continued as described above.

An error degree is calculated by means of the sum

of the reciprocal values of the squares of differences

between the distances in D. This error degree in-

dicates how well the distances are distributed. The

aforementioned procedure is done in an optimization

step until the error degree is minimized, and the best

conﬁguration for the target results.

If no intersection point exists for any distance of

C \ D, a compromise has to be accepted and also dis-

tances from D are allowed as long as no symmetrical

triangle constellations result. Thus, another marker

can be added although the resulting distances have

the potential to introduce ambiguities when track-

ing. However, since most marker positions have been

added using the described iterative way, our approach

provides a sufﬁcient set of well-deﬁned markers that

enhances tracking.

4.2 Example Conﬁgurations

4.2.1 Haptic Input Device

We have evaluated our approach for a hand-held inter-

action device, called haptic input device that is used

for multimodal VR-based interactions in several ap-

plication domains (Steinicke et al., 2005). The asso-

ciated target is illustrated in Figure 3 and Figure 6 (a).

The distances d

= 181.25mm, d

= 73.38, and

= 202.35 result from the marker positions as de-

scribed in Section 3.1. Since the minimal difference

between these distances is about 20mm, we have cho-

sen a distance threshold of ε = 9mm as explained in

Equation (4).

(a) original target

(b) proposed target

Figure 6: Conﬁguration of markers for (a) original target

and (b) target generated by our approach. The markers have

been scaled for illustration purposes.

We have used our approach to redeﬁne the po-

sition of marker m

, because we wanted to provide

the same starting distance d

in the resulting conﬁg-

uration. The algorithm produces the target depicted

in Figure 6 (b). The new position of marker m

(−17.2, 0.0, −124.1). Now, we can increase the dis-

tance threshold ε to 22mm, and the error degree could

be decreased to 0.001904 from 0.009276 for the orig-

inal distances.

4.2.2 Medarpa Display

Since the device in Section 4.2.1 is constrained to

only three markers, we have tested the approach also

VISAPP 2007 - International Conference on Computer Vision Theory and Applications

392

ments as combination of translations and rotations. In

both series we have used a distance threshold of 9mm.

Table 2: Results for original (HID-old) and proposed (HID-

new) target of HID for rotational movements in (A) inner,

(B) middle, and (C) outer third of interaction region.

device: HID-old HID-new

A abs/rel 326 / 82% 326 / 82%

B abs/rel 282 / 71% 307 / 77%

C abs/rel 231 / 58% 263 / 66%

Table 3: Results for original (HID-old) and proposed (HID-

new) target of HID for complex movements in (A) inner,

(B) middle, and (C) outer third of interaction region.

device: HID-old HID-new

A abs/rel 318 / 80% 326 / 82%

B abs/rel 249 / 62% 287 / 72%

C abs/rel 239 / 60% 274 / 69%

We have performed a similar analysis for the orig-

inal and proposed target of the MEDARPA display.

Therefore, we have taken 5 series with 2000 measure-

ments using a 2.5mm as well as 6mm distance thresh-

old.

While the original target has been correctly

tracked 1525 in average (76.5%), our proposed con-

ﬁguration has reached in average 1662 correct track-

ing events (83.1%) when using the 2.5mm distance

threshold. For 6mm distance threshold, the results are

even better: 1529 in contrast to 1752 correct tracking

events for the original target respectively the proposed

conﬁguration, which corresponds to an improvement

of 14%.

The results clearly show that our proposed con-

ﬁguration de facto improves the tracking robustness.

In all regions of the interaction space, and with all

distance thresholds, our proposed conﬁguration per-

formed better than the original rigid bodies.

6 DISCUSSION

In this paper we have proposed an approach to au-

tomatically generate rigid bodies for arbitrary MR

applications running with optical-based tracking sys-

tems. The approach determines the optimal conﬁgu-

ration for a target consisting of an arbitrary number

of markers with respect to the properties of the used

tracking system. We have tested the approach by re-

deﬁning marker positions for existing devices. An

evaluation of the proposed conﬁgurations shows the

beneﬁts of the approach; improvements of up to 20%

could be achieved without any modiﬁcations to the

tracking system. The considered devices have proven

their beneﬁts for many applications in research as well

as industrial usage, and they have been revealed hav-

ing well-deﬁned tracking properties. Nevertheless,

our simple approach enhances the tracking for them.

For the future we want to expand our system by

exporting a construction plan for the designer in or-

der to improve also the build process. This is due to

the fact that during our evaluation we ﬁgured out that

it is essential that the arrangement of markers allows

the cameras to see as many markers as possible si-

multaneously. A bad construction results in markers

occluding themselves, which yields reconstruction er-

rors. Furthermore, the size have the markers have to

be considered therefore. Moreover, the used camera

setup has to be taken into account since it has a major

impact on the tracking performance.

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