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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 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” theta”/ dt, to determine how quickly an object is approaching. Barton & Cohn 1 1 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 TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 2 2 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. TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 3 3 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 workzone 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 TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 4 4 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 offreedom 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 TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 5 5 ( 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 TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 6 6 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. TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 7 7 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 TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 8 8 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. TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 9 9 ACKNOWLEDGEMENTS The authors wish to acknowledge the California Department of Transportation ( Driver Behavior at Rail Crossings). TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 10 10 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.
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Title  A 3D computer simulation test of the Leibowitz Hypothesis 
Subject  Motion perception (Vision)Computer simulation. 
Description  Title from PDF title page (viewed on August 7, 2007).; At head of title: Institute of Transportation Studies.; "April 1, 2007"Abstract.; "UCBTSCTR200710."; Includes bibliographical references (p. 10).; Harvested from the web on 8/8/07 
Creator  Barton, Joseph E. 
Publisher  Traffic Safety Center, University of California, Berkeley 
Contributors  Cohn, Theodore E.; University of California, Berkeley. Traffic Safety Center.; University of California, Berkeley. Institute of Transportation Studies. 
Type  Text 
Identifier  http://repositories.cdlib.org/cgi/viewcontent.cgi?article=1032&context=its/tsc 
Language  eng 
Relation  http://repositories.cdlib.org/its/tsc/UCBTSCTR200710/ 
TitleAlternative  Threedimensional computer simulation test of the Leibowitz Hypothesis 
DateIssued  c2007 
FormatExtent  10 p. : digital, PDF file with col. ill. 
RelationRequires  Mode of access: World Wide Web. 
Transcript  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 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” theta”/ dt, to determine how quickly an object is approaching. Barton & Cohn 1 1 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 TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 2 2 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. TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 3 3 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 workzone 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 TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 4 4 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 offreedom 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 TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 5 5 ( 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 TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 6 6 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. TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 7 7 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 TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 8 8 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. TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 9 9 ACKNOWLEDGEMENTS The authors wish to acknowledge the California Department of Transportation ( Driver Behavior at Rail Crossings). TRB 2007 Annual Meeting CD ROM Paper revised from original submittal. Barton & Cohn 10 10 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. 
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