Institute of Transportation Studies
UC Berkeley Traffic Safety Center
( University of California, Berkeley)
Year 2007 Paper UCB - TSC - TR - 2007 - 10
A 3D Computer Simulation Test of the
Leibowitz Hypothesis
Joseph E. Barton  Theodore E. Cohn†
 UC Berkeley Department of Mechanical Engineering
† School of Vision Science and Optometry, UC Berkeley
This paper is posted at the eScholarship Repository, University of California.
http:// repositories. cdlib. org/ its/ tsc/ UCB- TSC- TR- 2007- 10
Copyright  c 2007 by the authors.
A 3D Computer Simulation Test of the
Leibowitz Hypothesis
Abstract
Do large objects appear to approach more slowly than smaller objects trav-eling
at the same speed? If so then this might help explain the inordinately
high accident rates involving large vehicles such as buses and trains. To test
this, this study constructed an experiment using a 3D visual simulator in which
different sized textured spheres approached at different speeds. We found that
observers consistently judged the smaller sphere to be the faster, even in cases
where the larger sphere was traveling at up to twice the speed of the smaller.
Analysis of these results suggests that the brain relies upon the perceived rate
of change of an object’s visual angle, d” theta”/ dt, to determine how quickly an
object is approaching.
Barton & Cohn 1
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A 3D Computer Simulation Test of the Leibowitz Hypothesis
Submission Date: August 4, 2006
Word Count: 3261 ( including figures and tables)
Tables and Figures: 5
Joseph E. Barton, PhD
Department of Mechanical Engineering
University of California, Berkeley
687 Minor Hall # 2020
Berkeley, CA 94720
Phone: ( 510) 642- 0655 / Fax: ( 510) 643- 9922
jebarton@ northwestern. edu
Theodore E. Cohn, PhD
School of Vision Science and Optometry
University of California, Berkeley
689 Minor Hall
Berkeley, CA 94720
Phone: ( 510) 642- 5076 / Fax: ( 510) 643- 5109
tecohn@ berkeley. edu
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ABSTRACT
Do large objects appear to approach more slowly than smaller objects traveling at the same
speed? If so then this might help explain the inordinately high accident rates involving large
vehicles such as buses and trains. To test this, this study constructed an experiment using a 3D
visual simulator in which different sized textured spheres approached at different speeds. We
found that observers consistently judged the smaller sphere to be the faster, even in cases where
the larger sphere was traveling at up to twice the speed of the smaller. Analysis of these results
suggests that the brain relies upon the perceived rate of change of an object’s visual angle, d  / dt,
to determine how quickly an object is approaching.
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INTRODUCTION
Rear- end collisions with buses and collisions with trains at railroad crossings occur at
significantly higher rates than the corresponding cases involving only automobiles. This has long
puzzled accident investigators, since one would expect the movements of larger objects to be
more easily noticed and interpreted by motorists. In 1985, Leibowitz observed that large aircraft
at airports appeared to move more slowly than smaller aircraft, even though the former were
traveling much faster ( 1). He went on to hypothesize that this misperception must in turn be
caused by the way in which the brain processed and interpreted the visual information provided
in this scenario. To our knowledge, Leibowitz’ hypothesis has received only limited testing. A
study by Cohn and Nguyen in 2002 studied a narrow aspect of this concept by looking at human
perception of size increases ( 2). Their conclusion that the initial approach of larger objects is
seen slower or later than for smaller ones suggests that larger objects are perceived to move at
slower speeds, thus offering credibility to Leibowitz’s theory.
This study takes research in this area one step further by testing human perception of the
relative speed of two comparable objects, distinguished only by size and speed at various times.
Our goal was to lend further support to Leibowitz’s theory that people perceive larger objects to
travel slower than smaller ones. The use of 3D visual simulators to assess perceptions of
roadway safety has been tested in studies looking at construction work zones safety ( 3). The
results indicate that the vision model- based tool can assess the relative conspicuity of individual
elements of a roadway or roadside scene. It holds potential value in virtual prototyping of work-zone
sight lines, colors, and placement of hazard warning cues, which can have significant
implications along railroad crossings.
METHODOLOGY
We constructed a two alternative forced choice ( 2AFC) experiment consisting of two sequential
time epochs. In one of the epochs, chosen at random, a five foot diameter sphere approached the
observer at eye level, traveling at 35 mph ( 56.35 km/ h). In the other epoch, a ten foot diameter
sphere approached at one of the speeds given in Table 1. The observer’s task was to indicate by
pressing a button which epoch contained the faster approaching sphere. An experiment consisted
of 270 such trials. The number of times that the ten foot diameter sphere assumed each approach
speed ( also selected randomly) is also indicated in Table 1. ( As we explain later, we were able to
run fewer trials for the slower approach speeds while maintaining approximately equal
measurement errors.)
Speed in mph ( km/ h) # Trials ( Out of 270)
25 ( 40.25) 40
35 ( 56.35) 40
45 ( 72.45) 40
55 ( 88.55) 50
65 ( 104.65) 50
75 ( 120.75) 50
TABLE 1: 10 ft Diameter Sphere Approach Speeds
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A faceted white sphere was constructed using OpenGL ( 10 longitudinal slices, 10 lateral
slices, with a black wire- frame coinciding with the edges formed by the slices). It was presented
against a black ground plane and horizon. The ground plane was delineated with yellow
longitudinal lines 5 feet ( 1.53 meters) apart. Twenty white, 6 feet ( 1.83 meters) high, .25 feet
(. 076 meters) diameter cylinders were randomly placed throughout the ground plane ( but not in
the path of the approaching sphere) to give the observer a sense of perspective and proportion.
The scene was presented on a projection- based virtual reality ( VR) system ( see Figure 1). The
viewer, seated in front of the projection screen, wears stereo shutter glasses and a six- degrees- of-freedom
head- tracking device.
As the viewer moves his or her head, the correct stereoscopic perspective projections are
calculated for each eye and presented. The scene was presented with a frame rate in excess of 60
Hz ( resulting in a greater than 30 Hz frame rate for each eye.) A frame from one such
presentation is shown in Figure 1. Each time epoch started with the sphere 6.5 seconds away
from the observer, and ended with the sphere 0.25 seconds away, so that it remained in view for
6.25 sec. Since tests of this type are fatiguing, the experiment was divided into four segments of
approximately 67 trials each to give test subjects a chance to rest in between. Subjects can also
stop and rest within a segment if necessary.
FIGURE 1: Visual Scene
Four visual cues are available to the observer in judging the faster of the two approaching
spheres: monocular image expansion, binocular cues deriving from stereopsis, texture dilation,
and reference to the static cylindrical posts and ground plane lines. Even though we have
included binocular effects in these experiments, we do not expect them to play much, if any, role
in this task. Since such effects are not noticeable at distances greater than approximately 30 feet
( 9.15 meters), this information will not be available to observers until the final 0.33 sec of a
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( 6.25 seconds) 35 mph approach, and only the final 0.02 seconds of a 75 mph ( 121 km/ h)
approach. It seems highly unlikely that the brain would be able to utilize such a small quantity
of information, occurring at the very end of the presentation. Here we allow the sphere to come
within .25 seconds of the observer before occluding it. In practical applications where decisions
would have to be made 2 to 4 seconds before collision, binocular cues would be entirely
unavailable.
FINDINGS
We tested the ability of five males ( labeled S1 to S5, ranging in age from the early 20’ s to the
mid 50’ s, with corrected normal eyesight) to identify the faster of two, different sized
approaching spheres. The results of these tests are summarized in Table 2, which shows the total
number of trials n ( across all subjects) that were conducted for each large- sphere speed ( V10), the
proportion of times P5 that subjects judged the smaller diameter sphere to be the faster, and the
standard error ( SE) of the measurement, given by.
s s P ( 1 P )
SE .
n

=
Since SE increases from zero at Ps = 0 to a maximum at Ps = .5 and then decreases to zero again at
Ps= 1.0, correspondingly more trials are required when Ps  5 than when Ps is at either extreme to
achieve the same standard error. From a series of pre- trials we determined that when V10 was
low subjects would almost always judge the smaller sphere to be the faster ( Ps  1), hence we ran
correspondingly fewer trials for the lower values for V10 than for the larger values. The final two
columns show the difference  P5 and its associated standard error, given by
5 C 5 C 5 C 1
5 5 5 5
P
C C 1
( P ) ( P ) ( P ) ,
P ( 1 P ) P ( 1 P )
SE ,
n n



 = 
=     	 +     	

  
 
where C is the Case. The 95% confidence intervals associated with the P are thus
95% 5 C P CI ( P ) 1.96 SE .  =  ±
Since these confidence limits lie to the right of zero for every case except Case 1, we can state
with 95% confidence that the differences  P5 are statistically significant. Figure 2 plots P5 as a
function of V10. Also shown are ± 1 SE bars.
Case V10 ( mph) V5 n n5 P5 SE  P5 SE  P
1 25 35 198 188 0.949 0.016
2 35 35 200 191 0.955 0.015 - 0.006 0.021
3 45 35 197 164 0.832 0.027 0.123 0.030
4 55 35 236 164 0.695 0.030 0.138 0.040
5 65 35 237 119 0.502 0.032 0.330 0.044
6 75 35 199 95 0.477 0.035 0.218 0.048
Overall 1267 921 0.727 0.013
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TABLE 2: Experimental Results
0.0
0.2
0.4
0.6
0.8
1.0
25 35 45 55 65 75
10 ft Sphere Speed ( mph)
PS
S1
S2
S3
S4
S5
Overall
FIGURE 2: Experimental Results.
The results show that the average person displays a strong tendency to judge the smaller sphere
as the faster, even when the actual approach speed of the larger sphere is 20 mph greater
( V10 = 55 mph). Only when V10 reaches speeds of 65- 75 mph ( twice that of the smaller sphere)
does the observer become unsure as to which is approaching faster ( P5  0.5). If we let z( t) be
the distance from the observer to the sphere at any time t ( 0  t  6.5 sec). Then z( t) = V( 6.5- t),
where V = dz/ dt is the approach velocity of the sphere. If r is the sphere’s radius, then the visual
angle  r that it subtends is
 r
( t) = 2tan  1 r
z( t)


 

 	
= 2tan  1 r
V( 6.5  t)


 

 	
while
d  r
dt
= r
z2 + r2 V
These are plotted as functions of time in Figures 3.
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0
5
10
15
20
25
0 1 2 3 4 5 6
Time ( sec)
 ( deg)
25 mph
35 mph
45 mph
55 mph
65 mph
75 mph
5' Dia Sphere
95 mph
135 mph
FIGURE 3a:  vs. time
0
5
10
15
0 1 2 3 4 5 6
Time ( sec)
d  / dt ( deg/ sec)
25 mph
35 mph
45 mph
55 mph
65 mph
75 mph
5' Dia Sphere
95 mph
135 mph
FIGURE 3b: d  / dot vs. time
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From these we see that for V10 < 70 mph ( = 2 V5),  10 >  5 and d  10/ dt > d  5/ dt for all t.
For V10 > 2 V5 the opposite holds, and for V10 = 2 V5 the two sets of profiles coincide with one
another. This final observation demonstrates the obvious fact that the monocular view of the
smaller sphere’s approach is exactly matched by that of a sphere twice as large, approaching
twice as fast, from twice as far away. This, along with our experimental results, suggests that
observers rely heavily on the monocular cues when making judgments about the speeds of
approaching objects. In this case they could be relying exclusively on  ( i. e., comparing  for
various t), exclusively on d  / dt, or they could be using both in some combination.
If it is true that observers place heavy emphasis on monocular cues in performing this
task, then it is easy to see why judgments about approaching objects are so unreliable. It is
interesting to note that for V10 < 2 V5 the brain judges the larger sphere to be approaching more
slowly, even though its associated subtended angle and expansion rate are both greater than
those associated with the smaller sphere.
We also note that compared with the final 2- 3 seconds of the approach, the information
provided in the first 3- 4 seconds appears barely distinguishable in going from one speed to the
next.
CONCLUSIONS
This study examines the perception of the speed of moving objects given their relative
size. The results further lend credibility to Leibowitz’s theory that bigger objects are perceived to
move slower than smaller objects. In transportation safety research, the results can have
implications on the design and improvement of rail crossings or signals at major traffic
intersections.
While the design and testing of a computer model presented in this short study may yield
reliable results, the research into Leibowitz’s theory can advance much further if researchers
empirically tested the validity of the concept. One area lacking and yet to be explored is the
gauging of people’s perceptions of actual moving vehicles of various sizes, such as the
perception of fast approaching trains. Given that the safety implications involve the misjudgment
of speeds by motorists and pedestrians at rail crossings, measuring whether there is significant
discrepancy between movement by computer model simulations and actual physical objects
would shed light on the future direction of related research.
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ACKNOWLEDGEMENTS
The authors wish to acknowledge the California Department of Transportation ( Driver Behavior
at Rail Crossings).
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REFERENCES
1. Cohn, T. E., and T. Nguyen. A Sensory Cause of Railroad Grade- Crossing Collisions: Test
of the Leibowitz Hypothesis. In Transportation Research Record: Journal of the
Transportation Research Board, No. 1843, TRB, National Research Council, Washington,
D. C., 2003, 24- 30.
2. Leibowitz, H. W. Grade Crossing Accidents and Human Factors Engineering. American
Scientists, Vol. 95, 1985, pp. 558- 562.
3. Barton, J., J. A. Misener, & T. Cohn. Computational Vision Model to Assess Work Zone
Conspicuity. In Transportation Research Record: Journal of the Transportation Research
Board, No. 1801, TRB, National Research Council, Washington, D. C., 2002, 73- 79.
TRB 2007 Annual Meeting CD- ROM Paper revised from original submittal.